[{"data":1,"prerenderedAt":125311},["ShallowReactive",2],{"post-\u002Fblog\u002F2026\u002F0015-the-bellman-equation-and-its-mathematical-properties":3,"i-ci:tag":-1,"i-mingcute:time-line":-1},{"post":4,"nextPost":72770,"prevPost":72781},{"id":5,"title":6,"body":7,"description":13,"draft":72767,"enableCodeHighlight":72767,"enableCodeHightlight":72767,"enableComment":72768,"enableD2":72767,"enableKatex":72768,"extension":72769,"image":72720,"important":72767,"location":72770,"meta":72771,"navigation":72768,"ogImage":72770,"path":72772,"seo":72773,"stem":72774,"summary":72775,"tags":72776,"time":72779,"__hash__":72780},"blog\u002Fblog\u002F2026\u002F0015-the-bellman-equation-and-its-mathematical-properties.md","贝尔曼方程及其数学性质",{"type":8,"value":9,"toc":72719},"minimark",[10,14,17,20,25,30,1608,2077,2584,2588,3454,4159,5012,5591,5597,6359,6483,6486,6490,7414,7637,8391,8756,9913,9916,10036,10040,10044,10052,10823,10830,11681,11684,11746,11750,11817,12495,12498,12652,12926,12930,13350,14171,14354,14547,15194,15818,16122,16176,16421,16636,17185,17187,17328,17332,17411,17585,17774,17988,17991,18108,18111,18385,18388,18738,18741,19117,19120,19654,19741,19745,19748,20049,20140,20569,20830,21086,21229,21347,21351,21355,21432,21642,21804,22090,22615,22677,23476,23675,23679,24635,24788,25145,25673,25676,26441,26567,26629,26633,27446,27658,27914,27921,28202,28233,28596,28627,28916,29394,29398,30463,30587,30947,31251,31551,32562,32908,33592,33595,34078,34080,34084,34322,34520,34760,34763,35606,35772,36087,36416,36533,36537,36541,37246,37347,37351,37415,37934,38055,39566,39858,40122,40370,41149,41427,41994,42667,43159,43637,44048,44110,44232,45024,45344,45348,45518,45884,46550,46553,47101,47166,47170,48066,48412,48415,48419,48535,48838,49052,49141,49357,49417,49627,49820,50057,50289,50291,50418,50654,50956,51026,51030,51034,51145,51381,51529,52468,52473,53141,53172,53364,53368,53445,54744,54749,55892,55895,56284,56287,56863,56866,57437,57440,57866,57869,57924,58372,58581,58585,58704,59089,59092,59799,59854,60383,60438,60836,60953,61164,61433,61599,61603,61707,61877,62036,62143,62175,62381,62428,62631,62685,62689,62693,63056,63765,64013,64620,64962,65553,65848,67125,67707,68007,68011,68076,68295,68299,68302,68722,68795,68900,68903,68907,68911,68958,69020,69644,69735,69896,69900,70014,70602,71096,71201,71205,71381,71551,72082,72139,72174,72345,72348,72351,72627,72716],[11,12,13],"p",{},"贝尔曼方程（Bellman Equation）是动态规划（Dynamic Programming）和强化学习（Reinforcement Learning）中的核心数学方程，用于描述一个系统在当前状态下的最优价值，如何由当前收益和未来状态的价值共同决定。",[11,15,16],{},"简单来说：当前状态的价值 = 当前获得的收益 + 未来可能获得的折扣收益。它的核心思想是：一个复杂的长期决策问题，可以分解为一系列相互关联的短期决策问题。",[11,18,19],{},"它不是某一个孤立的等式，而是一族把值函数与自身联系起来的泛函方程。它的数学内容可以概括为三个层次。第一，折扣回报在时间上的递归分解。第二，一个定义在有界函数空间上的算子。第三，该算子的不动点。动态规划与强化学习中的许多结论，本质上都是这三个层次的不同推论。",[21,22,24],"h2",{"id":23},"一形式化","一、形式化",[26,27,29],"h3",{"id":28},"_11-折扣马尔可夫决策过程","1.1 折扣马尔可夫决策过程",[31,32,35,743,1485],"theorem",{"id":33,"title":34},"definition-mdp","定义 1.1（折扣马尔可夫决策过程）",[11,36,37,38,86,87,116,117,264,265,294,295,323,324,386,387,516,517,599,600,683,684,713,714,742],{},"设 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为动作集，",[39,118,120,165],{"className":119,"translate":43},[42],[39,121,123],{"className":122},[47],[49,124,125],{"xmlns":51},[53,126,127,162],{},[56,128,129,132,137,148,151,153,156,159],{},[59,130,131],{},"P",[133,134,136],"mo",{"stretchy":135},"false","(",[138,139,140,143],"msup",{},[59,141,142],{},"s",[133,144,147],{"mathvariant":145,"lspace":146,"rspace":146},"normal","0em","′",[133,149,150],{},"∣",[59,152,142],{},[133,154,155],{"separator":70},",",[59,157,158],{},"a",[133,160,161],{"stretchy":135},")",[63,163,164],{"encoding":65},"P(s'\\mid s,a)",[39,166,168,239],{"className":167,"ariaHidden":70},[69],[39,169,171,175,179,183,227,232,236],{"className":170},[74],[39,172],{"className":173,"style":174},[78],"height:1.0019em;vertical-align:-0.25em;",[39,176,131],{"className":177,"style":178},[83,84],"margin-right:0.1389em;",[39,180,136],{"className":181},[182],"mopen",[39,184,186,189],{"className":185},[83],[39,187,142],{"className":188},[83,84],[39,190,193],{"className":191},[192],"msupsub",[39,194,197],{"className":195},[196],"vlist-t",[39,198,201],{"className":199},[200],"vlist-r",[39,202,206],{"className":203,"style":205},[204],"vlist","height:0.7519em;",[39,207,209,214],{"style":208},"top:-3.063em;margin-right:0.05em;",[39,210],{"className":211,"style":213},[212],"pstrut","height:2.7em;",[39,215,221],{"className":216},[217,218,219,220],"sizing","reset-size6","size3","mtight",[39,222,224],{"className":223},[83,220],[39,225,147],{"className":226},[83,220],[39,228],{"className":229,"style":231},[230],"mspace","margin-right:0.2778em;",[39,233,150],{"className":234},[235],"mrel",[39,237],{"className":238,"style":231},[230],[39,240,242,246,249,253,257,260],{"className":241},[74],[39,243],{"className":244,"style":245},[78],"height:1em;vertical-align:-0.25em;",[39,247,142],{"className":248},[83,84],[39,250,155],{"className":251},[252],"mpunct",[39,254],{"className":255,"style":256},[230],"margin-right:0.1667em;",[39,258,158],{"className":259},[83,84],[39,261,161],{"className":262},[263],"mclose"," 为在状态 ",[39,266,268,281],{"className":267,"translate":43},[42],[39,269,271],{"className":270},[47],[49,272,273],{"xmlns":51},[53,274,275,279],{},[56,276,277],{},[59,278,142],{},[63,280,142],{"encoding":65},[39,282,284],{"className":283,"ariaHidden":70},[69],[39,285,287,291],{"className":286},[74],[39,288],{"className":289,"style":290},[78],"height:0.4306em;",[39,292,142],{"className":293},[83,84]," 执行动作 ",[39,296,298,311],{"className":297,"translate":43},[42],[39,299,301],{"className":300},[47],[49,302,303],{"xmlns":51},[53,304,305,309],{},[56,306,307],{},[59,308,158],{},[63,310,158],{"encoding":65},[39,312,314],{"className":313,"ariaHidden":70},[69],[39,315,317,320],{"className":316},[74],[39,318],{"className":319,"style":290},[78],[39,321,158],{"className":322},[83,84]," 后转移到状态 ",[39,325,327,345],{"className":326,"translate":43},[42],[39,328,330],{"className":329},[47],[49,331,332],{"xmlns":51},[53,333,334,342],{},[56,335,336],{},[138,337,338,340],{},[59,339,142],{},[133,341,147],{"mathvariant":145,"lspace":146,"rspace":146},[63,343,344],{"encoding":65},"s'",[39,346,348],{"className":347,"ariaHidden":70},[69],[39,349,351,354],{"className":350},[74],[39,352],{"className":353,"style":205},[78],[39,355,357,360],{"className":356},[83],[39,358,142],{"className":359},[83,84],[39,361,363],{"className":362},[192],[39,364,366],{"className":365},[196],[39,367,369],{"className":368},[200],[39,370,372],{"className":371,"style":205},[204],[39,373,374,377],{"style":208},[39,375],{"className":376,"style":213},[212],[39,378,380],{"className":379},[217,218,219,220],[39,381,383],{"className":382},[83,220],[39,384,147],{"className":385},[83,220]," 的概率，",[39,388,390,426],{"className":389,"translate":43},[42],[39,391,393],{"className":392},[47],[49,394,395],{"xmlns":51},[53,396,397,423],{},[56,398,399,402,405,407,410,412,414,416,419],{},[59,400,401],{},"r",[133,403,404],{},":",[59,406,61],{},[133,408,409],{},"×",[59,411,101],{},[133,413,409],{},[59,415,61],{},[133,417,418],{},"→",[59,420,422],{"mathvariant":421},"double-struck","R",[63,424,425],{"encoding":65},"r:S\\times A\\times S\\to\\mathbb{R}",[39,427,429,448,469,487,505],{"className":428,"ariaHidden":70},[69],[39,430,432,435,439,442,445],{"className":431},[74],[39,433],{"className":434,"style":290},[78],[39,436,401],{"className":437,"style":438},[83,84],"margin-right:0.0278em;",[39,440],{"className":441,"style":231},[230],[39,443,404],{"className":444},[235],[39,446],{"className":447,"style":231},[230],[39,449,451,455,458,462,466],{"className":450},[74],[39,452],{"className":453,"style":454},[78],"height:0.7667em;vertical-align:-0.0833em;",[39,456,61],{"className":457,"style":85},[83,84],[39,459],{"className":460,"style":461},[230],"margin-right:0.2222em;",[39,463,409],{"className":464},[465],"mbin",[39,467],{"className":468,"style":461},[230],[39,470,472,475,478,481,484],{"className":471},[74],[39,473],{"className":474,"style":454},[78],[39,476,101],{"className":477},[83,84],[39,479],{"className":480,"style":461},[230],[39,482,409],{"className":483},[465],[39,485],{"className":486,"style":461},[230],[39,488,490,493,496,499,502],{"className":489},[74],[39,491],{"className":492,"style":79},[78],[39,494,61],{"className":495,"style":85},[83,84],[39,497],{"className":498,"style":231},[230],[39,500,418],{"className":501},[235],[39,503],{"className":504,"style":231},[230],[39,506,508,512],{"className":507},[74],[39,509],{"className":510,"style":511},[78],"height:0.6889em;",[39,513,422],{"className":514},[83,515],"mathbb"," 为奖励函数，",[39,518,520,552],{"className":519,"translate":43},[42],[39,521,523],{"className":522},[47],[49,524,525],{"xmlns":51},[53,526,527,549],{},[56,528,529,532,535,538,542,544,547],{},[59,530,531],{},"γ",[133,533,534],{},"∈",[133,536,537],{"stretchy":135},"[",[539,540,541],"mn",{},"0",[133,543,155],{"separator":70},[539,545,546],{},"1",[133,548,161],{"stretchy":135},[63,550,551],{"encoding":65},"\\gamma\\in[0,1)",[39,553,555,575],{"className":554,"ariaHidden":70},[69],[39,556,558,562,566,569,572],{"className":557},[74],[39,559],{"className":560,"style":561},[78],"height:0.7335em;vertical-align:-0.1944em;",[39,563,531],{"className":564,"style":565},[83,84],"margin-right:0.0556em;",[39,567],{"className":568,"style":231},[230],[39,570,534],{"className":571},[235],[39,573],{"className":574,"style":231},[230],[39,576,578,581,584,587,590,593,596],{"className":577},[74],[39,579],{"className":580,"style":245},[78],[39,582,537],{"className":583},[182],[39,585,541],{"className":586},[83],[39,588,155],{"className":589},[252],[39,591],{"className":592,"style":256},[230],[39,594,546],{"className":595},[83],[39,597,161],{"className":598},[263]," 为折扣因子。记 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为截至时刻 ",[39,685,687,700],{"className":686,"translate":43},[42],[39,688,690],{"className":689},[47],[49,691,692],{"xmlns":51},[53,693,694,698],{},[56,695,696],{},[59,697,621],{},[63,699,621],{"encoding":65},[39,701,703],{"className":702,"ariaHidden":70},[69],[39,704,706,710],{"className":705},[74],[39,707],{"className":708,"style":709},[78],"height:0.6151em;",[39,711,621],{"className":712},[83,84]," 的历史信息。若对任意 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Markov decision process，折扣 MDP），记作 ",[39,1494,1496,1535],{"className":1495,"translate":43},[42],[39,1497,1499],{"className":1498},[47],[49,1500,1501],{"xmlns":51},[53,1502,1503,1532],{},[56,1504,1505,1508,1510,1512,1514,1516,1518,1520,1522,1524,1526,1528,1530],{},[59,1506,1507],{"mathvariant":617},"M",[133,1509,793],{},[133,1511,136],{"stretchy":135},[59,1513,61],{},[133,1515,155],{"separator":70},[59,1517,101],{},[133,1519,155],{"separator":70},[59,1521,131],{},[133,1523,155],{"separator":70},[59,1525,401],{},[133,1527,155],{"separator":70},[59,1529,531],{},[133,1531,161],{"stretchy":135},[63,1533,1534],{"encoding":65},"\\mathcal{M}=(S,A,P,r,\\gamma)",[39,1536,1538,1556],{"className":1537,"ariaHidden":70},[69],[39,1539,1541,1544,1547,1550,1553],{"className":1540},[74],[39,1542],{"className":1543,"style":79},[78],[39,1545,1507],{"className":1546},[83,641],[39,1548],{"className":1549,"style":231},[230],[39,1551,793],{"className":1552},[235],[39,1554],{"className":1555,"style":231},[230],[39,1557,1559,1562,1565,1568,1571,1574,1577,1580,1583,1586,1589,1592,1595,1598,1601,1604],{"className":1558},[74],[39,1560],{"className":1561,"style":245},[78],[39,1563,136],{"className":1564},[182],[39,1566,61],{"className":1567,"style":85},[83,84],[39,1569,155],{"className":1570},[252],[39,1572],{"className":1573,"style":256},[230],[39,1575,101],{"className":1576},[83,84],[39,1578,155],{"className":1579},[252],[39,1581],{"className":1582,"style":256},[230],[39,1584,131],{"className":1585,"style":178},[83,84],[39,1587,155],{"className":1588},[252],[39,1590],{"className":1591,"style":256},[230],[39,1593,401],{"className":1594,"style":438},[83,84],[39,1596,155],{"className":1597},[252],[39,1599],{"className":1600,"style":256},[230],[39,1602,531],{"className":1603,"style":565},[83,84],[39,1605,161],{"className":1606},[263],"。",[11,1609,1610,1611,1639,1640,1668,1669,1785,1786,1977,1978,2076],{},"本文默认 ",[39,1612,1614,1627],{"className":1613,"translate":43},[42],[39,1615,1617],{"className":1616},[47],[49,1618,1619],{"xmlns":51},[53,1620,1621,1625],{},[56,1622,1623],{},[59,1624,61],{},[63,1626,61],{"encoding":65},[39,1628,1630],{"className":1629,"ariaHidden":70},[69],[39,1631,1633,1636],{"className":1632},[74],[39,1634],{"className":1635,"style":79},[78],[39,1637,61],{"className":1638,"style":85},[83,84]," 与 ",[39,1641,1643,1656],{"className":1642,"translate":43},[42],[39,1644,1646],{"className":1645},[47],[49,1647,1648],{"xmlns":51},[53,1649,1650,1654],{},[56,1651,1652],{},[59,1653,101],{},[63,1655,101],{"encoding":65},[39,1657,1659],{"className":1658,"ariaHidden":70},[69],[39,1660,1662,1665],{"className":1661},[74],[39,1663],{"className":1664,"style":79},[78],[39,1666,101],{"className":1667},[83,84]," 均为有限集，且存在常数 ",[39,1670,1672,1700],{"className":1671,"translate":43},[42],[39,1673,1675],{"className":1674},[47],[49,1676,1677],{"xmlns":51},[53,1678,1679,1697],{},[56,1680,1681,1691,1694],{},[613,1682,1683,1685,1688],{},[59,1684,422],{},[59,1686,1687],{},"max",[133,1689,1690],{},"⁡",[133,1692,1693],{},"\u003C",[59,1695,1696],{"mathvariant":145},"∞",[63,1698,1699],{"encoding":65},"R_{\\max}\u003C\\infty",[39,1701,1703,1776],{"className":1702,"ariaHidden":70},[69],[39,1704,1706,1709,1767,1770,1773],{"className":1705},[74],[39,1707],{"className":1708,"style":634},[78],[39,1710,1712,1716],{"className":1711},[83],[39,1713,422],{"className":1714,"style":1715},[83,84],"margin-right:0.0077em;",[39,1717,1719],{"className":1718},[192],[39,1720,1722,1759],{"className":1721},[196,649],[39,1723,1725,1756],{"className":1724},[200],[39,1726,1729],{"className":1727,"style":1728},[204],"height:0.1514em;",[39,1730,1732,1735],{"style":1731},"top:-2.55em;margin-left:-0.0077em;margin-right:0.05em;",[39,1733],{"className":1734,"style":213},[212],[39,1736,1738],{"className":1737},[217,218,219,220],[39,1739,1741],{"className":1740},[83,220],[39,1742,1745,1749,1752],{"className":1743},[1744,220],"mop",[39,1746,1748],{"className":1747},[220],"m",[39,1750,158],{"className":1751},[220],[39,1753,1755],{"className":1754},[220],"x",[39,1757,673],{"className":1758},[672],[39,1760,1762],{"className":1761},[200],[39,1763,1765],{"className":1764,"style":680},[204],[39,1766],{},[39,1768],{"className":1769,"style":231},[230],[39,1771,1693],{"className":1772},[235],[39,1774],{"className":1775,"style":231},[230],[39,1777,1779,1782],{"className":1778},[74],[39,1780],{"className":1781,"style":290},[78],[39,1783,1696],{"className":1784},[83]," 使得 ",[39,1787,1789,1836],{"className":1788,"translate":43},[42],[39,1790,1792],{"className":1791},[47],[49,1793,1794],{"xmlns":51},[53,1795,1796,1833],{},[56,1797,1798,1800,1802,1804,1806,1808,1810,1812,1818,1820,1822,1825],{},[59,1799,150],{"mathvariant":145},[59,1801,401],{},[133,1803,136],{"stretchy":135},[59,1805,142],{},[133,1807,155],{"separator":70},[59,1809,158],{},[133,1811,155],{"separator":70},[138,1813,1814,1816],{},[59,1815,142],{},[133,1817,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,1819,161],{"stretchy":135},[59,1821,150],{"mathvariant":145},[133,1823,1824],{},"≤",[613,1826,1827,1829,1831],{},[59,1828,422],{},[59,1830,1687],{},[133,1832,1690],{},[63,1834,1835],{"encoding":65},"|r(s,a,s')|\\le R_{\\max}",[39,1837,1839,1919],{"className":1838,"ariaHidden":70},[69],[39,1840,1842,1845,1848,1851,1854,1857,1860,1863,1866,1869,1872,1904,1907,1910,1913,1916],{"className":1841},[74],[39,1843],{"className":1844,"style":174},[78],[39,1846,150],{"className":1847},[83],[39,1849,401],{"className":1850,"style":438},[83,84],[39,1852,136],{"className":1853},[182],[39,1855,142],{"className":1856},[83,84],[39,1858,155],{"className":1859},[252],[39,1861],{"className":1862,"style":256},[230],[39,1864,158],{"className":1865},[83,84],[39,1867,155],{"className":1868},[252],[39,1870],{"className":1871,"style":256},[230],[39,1873,1875,1878],{"className":1874},[83],[39,1876,142],{"className":1877},[83,84],[39,1879,1881],{"className":1880},[192],[39,1882,1884],{"className":1883},[196],[39,1885,1887],{"className":1886},[200],[39,1888,1890],{"className":1889,"style":205},[204],[39,1891,1892,1895],{"style":208},[39,1893],{"className":1894,"style":213},[212],[39,1896,1898],{"className":1897},[217,218,219,220],[39,1899,1901],{"className":1900},[83,220],[39,1902,147],{"className":1903},[83,220],[39,1905,161],{"className":1906},[263],[39,1908,150],{"className":1909},[83],[39,1911],{"className":1912,"style":231},[230],[39,1914,1824],{"className":1915},[235],[39,1917],{"className":1918,"style":231},[230],[39,1920,1922,1925],{"className":1921},[74],[39,1923],{"className":1924,"style":634},[78],[39,1926,1928,1931],{"className":1927},[83],[39,1929,422],{"className":1930,"style":1715},[83,84],[39,1932,1934],{"className":1933},[192],[39,1935,1937,1969],{"className":1936},[196,649],[39,1938,1940,1966],{"className":1939},[200],[39,1941,1943],{"className":1942,"style":1728},[204],[39,1944,1945,1948],{"style":1731},[39,1946],{"className":1947,"style":213},[212],[39,1949,1951],{"className":1950},[217,218,219,220],[39,1952,1954],{"className":1953},[83,220],[39,1955,1957,1960,1963],{"className":1956},[1744,220],[39,1958,1748],{"className":1959},[220],[39,1961,158],{"className":1962},[220],[39,1964,1755],{"className":1965},[220],[39,1967,673],{"className":1968},[672],[39,1970,1972],{"className":1971},[200],[39,1973,1975],{"className":1974,"style":680},[204],[39,1976],{}," 对一切 ",[39,1979,1981,2011],{"className":1980,"translate":43},[42],[39,1982,1984],{"className":1983},[47],[49,1985,1986],{"xmlns":51},[53,1987,1988,2008],{},[56,1989,1990,1992,1994,1996,1998,2000,2006],{},[133,1991,136],{"stretchy":135},[59,1993,142],{},[133,1995,155],{"separator":70},[59,1997,158],{},[133,1999,155],{"separator":70},[138,2001,2002,2004],{},[59,2003,142],{},[133,2005,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,2007,161],{"stretchy":135},[63,2009,2010],{"encoding":65},"(s,a,s')",[39,2012,2014],{"className":2013,"ariaHidden":70},[69],[39,2015,2017,2020,2023,2026,2029,2032,2035,2038,2041,2073],{"className":2016},[74],[39,2018],{"className":2019,"style":174},[78],[39,2021,136],{"className":2022},[182],[39,2024,142],{"className":2025},[83,84],[39,2027,155],{"className":2028},[252],[39,2030],{"className":2031,"style":256},[230],[39,2033,158],{"className":2034},[83,84],[39,2036,155],{"className":2037},[252],[39,2039],{"className":2040,"style":256},[230],[39,2042,2044,2047],{"className":2043},[83],[39,2045,142],{"className":2046},[83,84],[39,2048,2050],{"className":2049},[192],[39,2051,2053],{"className":2052},[196],[39,2054,2056],{"className":2055},[200],[39,2057,2059],{"className":2058,"style":205},[204],[39,2060,2061,2064],{"style":208},[39,2062],{"className":2063,"style":213},[212],[39,2065,2067],{"className":2066},[217,218,219,220],[39,2068,2070],{"className":2069},[83,220],[39,2071,147],{"className":2072},[83,220],[39,2074,161],{"className":2075},[263]," 成立。有限性假设保证最大值可以取到，后文会在第 7 节说明连续状态空间中的相应修改。",[11,2078,2079,2080,2083,2084,2115,2116,2183,2184,2228,2229,2295,2296,2324,2325,2353,2354,742,2382,2447,2448,2477,2478,2506,2507,2510,2511,1607],{},"一个",[1489,2081,2082],{},"策略"," ",[39,2085,2087,2102],{"className":2086,"translate":43},[42],[39,2088,2090],{"className":2089},[47],[49,2091,2092],{"xmlns":51},[53,2093,2094,2099],{},[56,2095,2096],{},[59,2097,2098],{},"π",[63,2100,2101],{"encoding":65},"\\pi",[39,2103,2105],{"className":2104,"ariaHidden":70},[69],[39,2106,2108,2111],{"className":2107},[74],[39,2109],{"className":2110,"style":290},[78],[39,2112,2098],{"className":2113,"style":2114},[83,84],"margin-right:0.0359em;"," 是映射 ",[39,2117,2119,2144],{"className":2118,"translate":43},[42],[39,2120,2122],{"className":2121},[47],[49,2123,2124],{"xmlns":51},[53,2125,2126,2141],{},[56,2127,2128,2130,2132,2135,2137,2139],{},[59,2129,61],{},[133,2131,418],{},[59,2133,2134],{"mathvariant":145},"Δ",[133,2136,136],{"stretchy":135},[59,2138,101],{},[133,2140,161],{"stretchy":135},[63,2142,2143],{"encoding":65},"S\\to\\Delta(A)",[39,2145,2147,2165],{"className":2146,"ariaHidden":70},[69],[39,2148,2150,2153,2156,2159,2162],{"className":2149},[74],[39,2151],{"className":2152,"style":79},[78],[39,2154,61],{"className":2155,"style":85},[83,84],[39,2157],{"className":2158,"style":231},[230],[39,2160,418],{"className":2161},[235],[39,2163],{"className":2164,"style":231},[230],[39,2166,2168,2171,2174,2177,2180],{"className":2167},[74],[39,2169],{"className":2170,"style":245},[78],[39,2172,2134],{"className":2173},[83],[39,2175,136],{"className":2176},[182],[39,2178,101],{"className":2179},[83,84],[39,2181,161],{"className":2182},[263],"，其中 ",[39,2185,2187,2207],{"className":2186,"translate":43},[42],[39,2188,2190],{"className":2189},[47],[49,2191,2192],{"xmlns":51},[53,2193,2194,2204],{},[56,2195,2196,2198,2200,2202],{},[59,2197,2134],{"mathvariant":145},[133,2199,136],{"stretchy":135},[59,2201,101],{},[133,2203,161],{"stretchy":135},[63,2205,2206],{"encoding":65},"\\Delta(A)",[39,2208,2210],{"className":2209,"ariaHidden":70},[69],[39,2211,2213,2216,2219,2222,2225],{"className":2212},[74],[39,2214],{"className":2215,"style":245},[78],[39,2217,2134],{"className":2218},[83],[39,2220,136],{"className":2221},[182],[39,2223,101],{"className":2224},[83,84],[39,2226,161],{"className":2227},[263]," 是动作集上的概率单纯形。",[39,2230,2232,2256],{"className":2231,"translate":43},[42],[39,2233,2235],{"className":2234},[47],[49,2236,2237],{"xmlns":51},[53,2238,2239,2253],{},[56,2240,2241,2243,2245,2247,2249,2251],{},[59,2242,2098],{},[133,2244,136],{"stretchy":135},[59,2246,158],{},[133,2248,150],{},[59,2250,142],{},[133,2252,161],{"stretchy":135},[63,2254,2255],{"encoding":65},"\\pi(a\\mid s)",[39,2257,2259,2283],{"className":2258,"ariaHidden":70},[69],[39,2260,2262,2265,2268,2271,2274,2277,2280],{"className":2261},[74],[39,2263],{"className":2264,"style":245},[78],[39,2266,2098],{"className":2267,"style":2114},[83,84],[39,2269,136],{"className":2270},[182],[39,2272,158],{"className":2273},[83,84],[39,2275],{"className":2276,"style":231},[230],[39,2278,150],{"className":2279},[235],[39,2281],{"className":2282,"style":231},[230],[39,2284,2286,2289,2292],{"className":2285},[74],[39,2287],{"className":2288,"style":245},[78],[39,2290,142],{"className":2291},[83,84],[39,2293,161],{"className":2294},[263]," 表示在状态 ",[39,2297,2299,2312],{"className":2298,"translate":43},[42],[39,2300,2302],{"className":2301},[47],[49,2303,2304],{"xmlns":51},[53,2305,2306,2310],{},[56,2307,2308],{},[59,2309,142],{},[63,2311,142],{"encoding":65},[39,2313,2315],{"className":2314,"ariaHidden":70},[69],[39,2316,2318,2321],{"className":2317},[74],[39,2319],{"className":2320,"style":290},[78],[39,2322,142],{"className":2323},[83,84]," 选择动作 ",[39,2326,2328,2341],{"className":2327,"translate":43},[42],[39,2329,2331],{"className":2330},[47],[49,2332,2333],{"xmlns":51},[53,2334,2335,2339],{},[56,2336,2337],{},[59,2338,158],{},[63,2340,158],{"encoding":65},[39,2342,2344],{"className":2343,"ariaHidden":70},[69],[39,2345,2347,2350],{"className":2346},[74],[39,2348],{"className":2349,"style":290},[78],[39,2351,158],{"className":2352},[83,84]," 的概率。若对每个 ",[39,2355,2357,2370],{"className":2356,"translate":43},[42],[39,2358,2360],{"className":2359},[47],[49,2361,2362],{"xmlns":51},[53,2363,2364,2368],{},[56,2365,2366],{},[59,2367,142],{},[63,2369,142],{"encoding":65},[39,2371,2373],{"className":2372,"ariaHidden":70},[69],[39,2374,2376,2379],{"className":2375},[74],[39,2377],{"className":2378,"style":290},[78],[39,2380,142],{"className":2381},[83,84],[39,2383,2385,2408],{"className":2384,"translate":43},[42],[39,2386,2388],{"className":2387},[47],[49,2389,2390],{"xmlns":51},[53,2391,2392,2406],{},[56,2393,2394,2396,2398,2400,2402,2404],{},[59,2395,2098],{},[133,2397,136],{"stretchy":135},[59,2399,158],{},[133,2401,150],{},[59,2403,142],{},[133,2405,161],{"stretchy":135},[63,2407,2255],{"encoding":65},[39,2409,2411,2435],{"className":2410,"ariaHidden":70},[69],[39,2412,2414,2417,2420,2423,2426,2429,2432],{"className":2413},[74],[39,2415],{"className":2416,"style":245},[78],[39,2418,2098],{"className":2419,"style":2114},[83,84],[39,2421,136],{"className":2422},[182],[39,2424,158],{"className":2425},[83,84],[39,2427],{"className":2428,"style":231},[230],[39,2430,150],{"className":2431},[235],[39,2433],{"className":2434,"style":231},[230],[39,2436,2438,2441,2444],{"className":2437},[74],[39,2439],{"className":2440,"style":245},[78],[39,2442,142],{"className":2443},[83,84],[39,2445,161],{"className":2446},[263]," 只在某一个动作处取 ",[39,2449,2451,2464],{"className":2450,"translate":43},[42],[39,2452,2454],{"className":2453},[47],[49,2455,2456],{"xmlns":51},[53,2457,2458,2462],{},[56,2459,2460],{},[539,2461,546],{},[63,2463,546],{"encoding":65},[39,2465,2467],{"className":2466,"ariaHidden":70},[69],[39,2468,2470,2474],{"className":2469},[74],[39,2471],{"className":2472,"style":2473},[78],"height:0.6444em;",[39,2475,546],{"className":2476},[83],"，则称 ",[39,2479,2481,2494],{"className":2480,"translate":43},[42],[39,2482,2484],{"className":2483},[47],[49,2485,2486],{"xmlns":51},[53,2487,2488,2492],{},[56,2489,2490],{},[59,2491,2098],{},[63,2493,2101],{"encoding":65},[39,2495,2497],{"className":2496,"ariaHidden":70},[69],[39,2498,2500,2503],{"className":2499},[74],[39,2501],{"className":2502,"style":290},[78],[39,2504,2098],{"className":2505,"style":2114},[83,84]," 为",[1489,2508,2509],{},"确定性策略","，并可将其视为映射 ",[39,2512,2514,2536],{"className":2513,"translate":43},[42],[39,2515,2517],{"className":2516},[47],[49,2518,2519],{"xmlns":51},[53,2520,2521,2533],{},[56,2522,2523,2525,2527,2529,2531],{},[59,2524,2098],{},[133,2526,404],{},[59,2528,61],{},[133,2530,418],{},[59,2532,101],{},[63,2534,2535],{"encoding":65},"\\pi:S\\to A",[39,2537,2539,2557,2575],{"className":2538,"ariaHidden":70},[69],[39,2540,2542,2545,2548,2551,2554],{"className":2541},[74],[39,2543],{"className":2544,"style":290},[78],[39,2546,2098],{"className":2547,"style":2114},[83,84],[39,2549],{"className":2550,"style":231},[230],[39,2552,404],{"className":2553},[235],[39,2555],{"className":2556,"style":231},[230],[39,2558,2560,2563,2566,2569,2572],{"className":2559},[74],[39,2561],{"className":2562,"style":79},[78],[39,2564,61],{"className":2565,"style":85},[83,84],[39,2567],{"className":2568,"style":231},[230],[39,2570,418],{"className":2571},[235],[39,2573],{"className":2574,"style":231},[230],[39,2576,2578,2581],{"className":2577},[74],[39,2579],{"className":2580,"style":79},[78],[39,2582,101],{"className":2583},[83,84],[26,2585,2587],{"id":2586},"_12-回报与值函数","1.2 回报与值函数",[31,2589,2592,2844,3126,3162],{"id":2590,"title":2591},"definition-return","定义 1.2（折扣回报）",[11,2593,2594,2595,2843],{},"设轨迹为 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开始的",[1489,3159,3160],{},"折扣回报","定义为",[39,3163,3165],{"className":3164,"translate":43},[746],[39,3166,3168,3226],{"className":3167,"translate":43},[42],[39,3169,3171],{"className":3170},[47],[49,3172,3173],{"xmlns":51,"display":755},[53,3174,3175,3223],{},[56,3176,3177,3184,3186,3203,3209,3221],{},[613,3178,3179,3182],{},[59,3180,3181],{},"G",[59,3183,621],{},[133,3185,793],{},[3187,3188,3189,3192,3201],"munderover",{},[133,3190,3191],{},"∑",[56,3193,3194,3197,3199],{},[59,3195,3196],{},"k",[133,3198,793],{},[539,3200,541],{},[59,3202,1696],{"mathvariant":145},[138,3204,3205,3207],{},[59,3206,531],{},[59,3208,3196],{},[613,3210,3211,3213],{},[59,3212,401],{},[56,3214,3215,3217,3219],{},[59,3216,621],{},[133,3218,788],{},[59,3220,3196],{},[59,3222,2903],{"mathvariant":145},[63,3224,3225],{"encoding":65},"G_t=\\sum_{k=0}^{\\infty}\\gamma^k 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是在从 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转移到 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时获得的奖励。若奖励只依赖于 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成立，则对任意轨迹与任意 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由三角不等式与几何级数求和，",[39,5598,5600],{"className":5599,"translate":43},[746],[39,5601,5603,5747],{"className":5602,"translate":43},[42],[39,5604,5606],{"className":5605},[47],[49,5607,5608],{"xmlns":51,"display":755},[53,5609,5610,5744],{},[759,5611,5612,5676],{"rowspacing":761,"columnalign":762,"columnspacing":146},[764,5613,5614,5630],{},[767,5615,5616],{},[770,5617,5618],{"scriptlevel":541,"displaystyle":70},[56,5619,5620,5622,5628],{},[59,5621,150],{"mathvariant":145},[613,5623,5624,5626],{},[59,5625,3181],{},[59,5627,621],{},[59,5629,150],{"mathvariant":145},[767,5631,5632],{},[770,5633,5634],{"scriptlevel":541,"displaystyle":70},[56,5635,5636,5638,5640,5654,5660,5662,5674],{},[56,5637],{},[133,5639,1824],{},[3187,5641,5642,5644,5652],{},[133,5643,3191],{},[56,5645,5646,5648,5650],{},[59,5647,3196],{},[133,5649,793],{},[539,5651,541],{},[59,5653,1696],{"mathvariant":145},[138,5655,5656,5658],{},[59,5657,531],{},[59,5659,3196],{},[59,5661,150],{"mathvariant":145},[613,5663,5664,5666],{},[59,5665,401],{},[56,5667,5668,5670,5672],{},[59,5669,621],{},[133,5671,788],{},[59,5673,3196],{},[59,5675,150],{"mathvariant":145},[764,5677,5678,5684],{},[767,5679,5680],{},[770,5681,5682],{"scriptlevel":541,"displaystyle":70},[56,5683],{},[767,5685,5686],{},[770,5687,5688],{"scriptlevel":541,"displaystyle":70},[56,5689,5690,5692,5694,5702,5716,5722,5724,5742],{},[56,5691],{},[133,5693,1824],{},[613,5695,5696,5698,5700],{},[59,5697,422],{},[59,5699,1687],{},[133,5701,1690],{},[3187,5703,5704,5706,5714],{},[133,5705,3191],{},[56,5707,5708,5710,5712],{},[59,5709,3196],{},[133,5711,793],{},[539,5713,541],{},[59,5715,1696],{"mathvariant":145},[138,5717,5718,5720],{},[59,5719,531],{},[59,5721,3196],{},[133,5723,793],{},[5362,5725,5726,5734],{},[613,5727,5728,5730,5732],{},[59,5729,422],{},[59,5731,1687],{},[133,5733,1690],{},[56,5735,5736,5738,5740],{},[539,5737,546],{},[133,5739,5378],{},[59,5741,531],{},[59,5743,2903],{"mathvariant":145},[63,5745,5746],{"encoding":65},"\\begin{aligned}\n|G_t|\n&\\le\\sum_{k=0}^{\\infty}\\gamma^k|r_{t+k}|\\\\\n&\\le R_{\\max}\\sum_{k=0}^{\\infty}\\gamma^k\n=\\frac{R_{\\max}}{1-\\gamma}.\n\\end{aligned}",[39,5748,5750],{"className":5749,"ariaHidden":70},[69],[39,5751,5753,5757],{"className":5752},[74],[39,5754],{"className":5755,"style":5756},[78],"height:6.207em;vertical-align:-2.8535em;",[39,5758,5760],{"className":5759},[83],[39,5761,5763,5853],{"className":5762},[759],[39,5764,5766],{"className":5765},[918],[39,5767,5769,5844],{"className":5768},[196,649],[39,5770,5772,5841],{"className":5771},[200],[39,5773,5776,5832],{"className":5774,"style":5775},[204],"height:3.3535em;",[39,5777,5779,5783],{"style":5778},"top:-5.3535em;",[39,5780],{"className":5781,"style":5782},[212],"height:3.6514em;",[39,5784,5786,5789,5829],{"className":5785},[83],[39,5787,150],{"className":5788},[83],[39,5790,5792,5795],{"className":5791},[83],[39,5793,3181],{"className":5794},[83,84],[39,5796,5798],{"className":5797},[192],[39,5799,5801,5821],{"className":5800},[196,649],[39,5802,5804,5818],{"className":5803},[200],[39,5805,5807],{"className":5806,"style":656},[204],[39,5808,5809,5812],{"style":966},[39,5810],{"className":5811,"style":213},[212],[39,5813,5815],{"className":5814},[217,218,219,220],[39,5816,621],{"className":5817},[83,84,220],[39,5819,673],{"className":5820},[672],[39,5822,5824],{"className":5823},[200],[39,5825,5827],{"className":5826,"style":680},[204],[39,5828],{},[39,5830,150],{"className":5831},[83],[39,5833,5835,5838],{"style":5834},"top:-2.1em;",[39,5836],{"className":5837,"style":5782},[212],[39,5839],{"className":5840},[83],[39,5842,673],{"className":5843},[672],[39,5845,5847],{"className":5846},[200],[39,5848,5851],{"className":5849,"style":5850},[204],"height:2.8535em;",[39,5852],{},[39,5854,5856],{"className":5855},[1223],[39,5857,5859,6351],{"className":5858},[196,649],[39,5860,5862,6348],{"className":5861},[200],[39,5863,5865,6039],{"className":5864,"style":5775},[204],[39,5866,5867,5870],{"style":5778},[39,5868],{"className":5869,"style":5782},[212],[39,5871,5873,5876,5879,5882,5885,5952,5955,5984,5987,6036],{"className":5872},[83],[39,5874],{"className":5875},[83],[39,5877],{"className":5878,"style":231},[230],[39,5880,1824],{"className":5881},[235],[39,5883],{"className":5884,"style":231},[230],[39,5886,5888],{"className":5887},[1744,3294],[39,5889,5891,5944],{"className":5890},[196,649],[39,5892,5894,5941],{"className":5893},[200],[39,5895,5897,5917,5927],{"className":5896,"style":3304},[204],[39,5898,5899,5902],{"style":3307},[39,5900],{"className":5901,"style":3311},[212],[39,5903,5905],{"className":5904},[217,218,219,220],[39,5906,5908,5911,5914],{"className":5907},[83,220],[39,5909,3196],{"className":5910,"style":3321},[83,84,220],[39,5912,793],{"className":5913},[235,220],[39,5915,541],{"className":5916},[83,220],[39,5918,5919,5922],{"style":3330},[39,5920],{"className":5921,"style":3311},[212],[39,5923,5924],{},[39,5925,3191],{"className":5926},[1744,3339,3340],[39,5928,5929,5932],{"style":3343},[39,5930],{"className":5931,"style":3311},[212],[39,5933,5935],{"className":5934},[217,218,219,220],[39,5936,5938],{"className":5937},[83,220],[39,5939,1696],{"className":5940},[83,220],[39,5942,673],{"className":5943},[672],[39,5945,5947],{"className":5946},[200],[39,5948,5950],{"className":5949,"style":3365},[204],[39,5951],{},[39,5953],{"className":5954,"style":256},[230],[39,5956,5958,5961],{"className":5957},[83],[39,5959,531],{"className":5960,"style":565},[83,84],[39,5962,5964],{"className":5963},[192],[39,5965,5967],{"className":5966},[196],[39,5968,5970],{"className":5969},[200],[39,5971,5973],{"className":5972,"style":3389},[204],[39,5974,5975,5978],{"style":1027},[39,5976],{"className":5977,"style":213},[212],[39,5979,5981],{"className":5980},[217,218,219,220],[39,5982,3196],{"className":5983,"style":3321},[83,84,220],[39,5985,150],{"className":5986},[83],[39,5988,5990,5993],{"className":5989},[83],[39,5991,401],{"className":5992,"style":438},[83,84],[39,5994,5996],{"className":5995},[192],[39,5997,5999,6028],{"className":5998},[196,649],[39,6000,6002,6025],{"className":6001},[200],[39,6003,6005],{"className":6004,"style":3419},[204],[39,6006,6007,6010],{"style":2937},[39,6008],{"className":6009,"style":213},[212],[39,6011,6013],{"className":6012},[217,218,219,220],[39,6014,6016,6019,6022],{"className":6015},[83,220],[39,6017,621],{"className":6018},[83,84,220],[39,6020,788],{"className":6021},[465,220],[39,6023,3196],{"className":6024,"style":3321},[83,84,220],[39,6026,673],{"className":6027},[672],[39,6029,6031],{"className":6030},[200],[39,6032,6034],{"className":6033,"style":994},[204],[39,6035],{},[39,6037,150],{"className":6038},[83],[39,6040,6041,6044],{"style":5834},[39,6042],{"className":6043,"style":5782},[212],[39,6045,6047,6050,6053,6056,6059,6111,6114,6181,6184,6213,6216,6219,6222,6345],{"className":6046},[83],[39,6048],{"className":6049},[83],[39,6051],{"className":6052,"style":231},[230],[39,6054,1824],{"className":6055},[235],[39,6057],{"className":6058,"style":231},[230],[39,6060,6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",[39,6363,6365,6383],{"className":6364,"translate":43},[42],[39,6366,6368],{"className":6367},[47],[49,6369,6370],{"xmlns":51},[53,6371,6372,6380],{},[56,6373,6374],{},[138,6375,6376,6378],{},[59,6377,4230],{},[59,6379,2098],{},[63,6381,6382],{"encoding":65},"V^\\pi",[39,6384,6386],{"className":6385,"ariaHidden":70},[69],[39,6387,6389,6392],{"className":6388},[74],[39,6390],{"className":6391,"style":79},[78],[39,6393,6395,6398],{"className":6394},[83],[39,6396,4230],{"className":6397,"style":461},[83,84],[39,6399,6401],{"className":6400},[192],[39,6402,6404],{"className":6403},[196],[39,6405,6407],{"className":6406},[200],[39,6408,6411],{"className":6409,"style":6410},[204],"height:0.6644em;",[39,6412,6413,6416],{"style":208},[39,6414],{"className":6415,"style":213},[212],[39,6417,6419],{"className":6418},[217,218,219,220],[39,6420,2098],{"className":6421,"style":2114},[83,84,220],[39,6423,6425,6443],{"className":6424,"translate":43},[42],[39,6426,6428],{"className":6427},[47],[49,6429,6430],{"xmlns":51},[53,6431,6432,6440],{},[56,6433,6434],{},[138,6435,6436,6438],{},[59,6437,4295],{},[59,6439,2098],{},[63,6441,6442],{"encoding":65},"Q^\\pi",[39,6444,6446],{"className":6445,"ariaHidden":70},[69],[39,6447,6449,6453],{"className":6448},[74],[39,6450],{"className":6451,"style":6452},[78],"height:0.8778em;vertical-align:-0.1944em;",[39,6454,6456,6459],{"className":6455},[83],[39,6457,4295],{"className":6458},[83,84],[39,6460,6462],{"className":6461},[192],[39,6463,6465],{"className":6464},[196],[39,6466,6468],{"className":6467},[200],[39,6469,6471],{"className":6470,"style":6410},[204],[39,6472,6473,6476],{"style":208},[39,6474],{"className":6475,"style":213},[212],[39,6477,6479],{"className":6478},[217,218,219,220],[39,6480,2098],{"className":6481,"style":2114},[83,84,220]," 均为有界函数。证毕。",[11,6484,6485],{},"有界性并不是一个无关紧要的技术条件。它保证值函数属于一个有界函数空间，而该空间在 sup 范数下是完备的。后文的压缩映射论证正是在这个空间上进行的。",[26,6487,6489],{"id":6488},"_13-两个基本恒等式","1.3 两个基本恒等式",[31,6491,6494,6526,6791,6794],{"id":6492,"title":6493},"proposition-basic-identities","命题 1.5（值函数的基本恒等式）",[11,6495,6496,6497,6525],{},"对任意策略 ",[39,6498,6500,6513],{"className":6499,"translate":43},[42],[39,6501,6503],{"className":6502},[47],[49,6504,6505],{"xmlns":51},[53,6506,6507,6511],{},[56,6508,6509],{},[59,6510,2098],{},[63,6512,2101],{"encoding":65},[39,6514,6516],{"className":6515,"ariaHidden":70},[69],[39,6517,6519,6522],{"className":6518},[74],[39,6520],{"className":6521,"style":290},[78],[39,6523,2098],{"className":6524,"style":2114},[83,84],"，有",[39,6527,6529],{"className":6528,"translate":43},[746],[39,6530,6532,6595],{"className":6531,"translate":43},[42],[39,6533,6535],{"className":6534},[47],[49,6536,6537],{"xmlns":51,"display":755},[53,6538,6539,6592],{},[56,6540,6541,6547,6549,6551,6553,6555,6562,6564,6566,6568,6570,6572,6574,6580,6582,6584,6586,6588,6590],{},[138,6542,6543,6545],{},[59,6544,4230],{},[59,6546,2098],{},[133,6548,136],{"stretchy":135},[59,6550,142],{},[133,6552,161],{"stretchy":135},[133,6554,793],{},[6556,6557,6558,6560],"munder",{},[133,6559,3191],{},[59,6561,158],{},[59,6563,2098],{},[133,6565,136],{"stretchy":135},[59,6567,158],{},[133,6569,150],{},[59,6571,142],{},[133,6573,161],{"stretchy":135},[138,6575,6576,6578],{},[59,6577,4295],{},[59,6579,2098],{},[133,6581,136],{"stretchy":135},[59,6583,142],{},[133,6585,155],{"separator":70},[59,6587,158],{},[133,6589,161],{"stretchy":135},[133,6591,155],{"separator":70},[63,6593,6594],{"encoding":65},"V^\\pi(s)=\\sum_{a}\\pi(a\\mid s)Q^\\pi(s,a),",[39,6596,6598,6651,6729],{"className":6597,"ariaHidden":70},[69],[39,6599,6601,6604,6633,6636,6639,6642,6645,6648],{"className":6600},[74],[39,6602],{"className":6603,"style":245},[78],[39,6605,6607,6610],{"className":6606},[83],[39,6608,4230],{"className":6609,"style":461},[83,84],[39,6611,6613],{"className":6612},[192],[39,6614,6616],{"className":6615},[196],[39,6617,6619],{"className":6618},[200],[39,6620,6622],{"className":6621,"style":4421},[204],[39,6623,6624,6627],{"style":1027},[39,6625],{"className":6626,"style":213},[212],[39,6628,6630],{"className":6629},[217,218,219,220],[39,6631,2098],{"className":6632,"style":2114},[83,84,220],[39,6634,136],{"className":6635},[182],[39,6637,142],{"className":6638},[83,84],[39,6640,161],{"className":6641},[263],[39,6643],{"className":6644,"style":231},[230],[39,6646,793],{"className":6647},[235],[39,6649],{"className":6650,"style":231},[230],[39,6652,6654,6658,6708,6711,6714,6717,6720,6723,6726],{"className":6653},[74],[39,6655],{"className":6656,"style":6657},[78],"height:2.3em;vertical-align:-1.25em;",[39,6659,6661],{"className":6660},[1744,3294],[39,6662,6664,6699],{"className":6663},[196,649],[39,6665,6667,6696],{"className":6666},[200],[39,6668,6671,6686],{"className":6669,"style":6670},[204],"height:1.05em;",[39,6672,6674,6677],{"style":6673},"top:-1.9em;margin-left:0em;",[39,6675],{"className":6676,"style":3311},[212],[39,6678,6680],{"className":6679},[217,218,219,220],[39,6681,6683],{"className":6682},[83,220],[39,6684,158],{"className":6685},[83,84,220],[39,6687,6688,6691],{"style":3330},[39,6689],{"className":6690,"style":3311},[212],[39,6692,6693],{},[39,6694,3191],{"className":6695},[1744,3339,3340],[39,6697,673],{"className":6698},[672],[39,6700,6702],{"className":6701},[200],[39,6703,6706],{"className":6704,"style":6705},[204],"height:1.25em;",[39,6707],{},[39,6709],{"className":6710,"style":256},[230],[39,6712,2098],{"className":6713,"style":2114},[83,84],[39,6715,136],{"className":6716},[182],[39,6718,158],{"className":6719},[83,84],[39,6721],{"className":6722,"style":231},[230],[39,6724,150],{"className":6725},[235],[39,6727],{"className":6728,"style":231},[230],[39,6730,6732,6735,6738,6741,6770,6773,6776,6779,6782,6785,6788],{"className":6731},[74],[39,6733],{"className":6734,"style":245},[78],[39,6736,142],{"className":6737},[83,84],[39,6739,161],{"className":6740},[263],[39,6742,6744,6747],{"className":6743},[83],[39,6745,4295],{"className":6746},[83,84],[39,6748,6750],{"className":6749},[192],[39,6751,6753],{"className":6752},[196],[39,6754,6756],{"className":6755},[200],[39,6757,6759],{"className":6758,"style":4421},[204],[39,6760,6761,6764],{"style":1027},[39,6762],{"className":6763,"style":213},[212],[39,6765,6767],{"className":6766},[217,218,219,220],[39,6768,2098],{"className":6769,"style":2114},[83,84,220],[39,6771,136],{"className":6772},[182],[39,6774,142],{"className":6775},[83,84],[39,6777,155],{"className":6778},[252],[39,6780],{"className":6781,"style":256},[230],[39,6783,158],{"className":6784},[83,84],[39,6786,161],{"className":6787},[263],[39,6789,155],{"className":6790},[252],[11,6792,6793],{},"以及",[39,6795,6797],{"className":6796,"translate":43},[746],[39,6798,6800,6945],{"className":6799,"translate":43},[42],[39,6801,6803],{"className":6802},[47],[49,6804,6805],{"xmlns":51,"display":755},[53,6806,6807,6942],{},[759,6808,6809,6873],{"rowspacing":761,"columnalign":762,"columnspacing":146},[764,6810,6811,6833],{},[767,6812,6813],{},[770,6814,6815],{"scriptlevel":541,"displaystyle":70},[56,6816,6817,6823,6825,6827,6829,6831],{},[138,6818,6819,6821],{},[59,6820,4295],{},[59,6822,2098],{},[133,6824,136],{"stretchy":135},[59,6826,142],{},[133,6828,155],{"separator":70},[59,6830,158],{},[133,6832,161],{"stretchy":135},[767,6834,6835],{},[770,6836,6837],{"scriptlevel":541,"displaystyle":70},[56,6838,6839,6841,6843,6853,6855,6857,6863,6865,6867,6869,6871],{},[56,6840],{},[133,6842,793],{},[6556,6844,6845,6847],{},[133,6846,3191],{},[138,6848,6849,6851],{},[59,6850,142],{},[133,6852,147],{"mathvariant":145,"lspace":146,"rspace":146},[59,6854,131],{},[133,6856,136],{"stretchy":135},[138,6858,6859,6861],{},[59,6860,142],{},[133,6862,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,6864,150],{},[59,6866,142],{},[133,6868,155],{"separator":70},[59,6870,158],{},[133,6872,161],{"stretchy":135},[764,6874,6875,6881],{},[767,6876,6877],{},[770,6878,6879],{"scriptlevel":541,"displaystyle":70},[56,6880],{},[767,6882,6883],{},[770,6884,6885],{"scriptlevel":541,"displaystyle":70},[56,6886,6887,6889,6892,6894,6940],{},[56,6888],{},[230,6890],{"width":6891},"1em",[133,6893,409],{},[56,6895,6896,6898,6900,6902,6904,6906,6908,6910,6916,6918,6920,6922,6928,6930,6936,6938],{},[133,6897,537],{"fence":70},[59,6899,401],{},[133,6901,136],{"stretchy":135},[59,6903,142],{},[133,6905,155],{"separator":70},[59,6907,158],{},[133,6909,155],{"separator":70},[138,6911,6912,6914],{},[59,6913,142],{},[133,6915,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,6917,161],{"stretchy":135},[133,6919,788],{},[59,6921,531],{},[138,6923,6924,6926],{},[59,6925,4230],{},[59,6927,2098],{},[133,6929,136],{"stretchy":135},[138,6931,6932,6934],{},[59,6933,142],{},[133,6935,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,6937,161],{"stretchy":135},[133,6939,4280],{"fence":70},[59,6941,2903],{"mathvariant":145},[63,6943,6944],{"encoding":65},"\\begin{aligned}\nQ^\\pi(s,a)\n&=\\sum_{s'}P(s'\\mid s,a)\\\\\n&\\quad\\times\\left[r(s,a,s')+\\gamma V^\\pi(s')\\right].\n\\end{aligned}",[39,6946,6948],{"className":6947,"ariaHidden":70},[69],[39,6949,6951,6955],{"className":6950},[74],[39,6952],{"className":6953,"style":6954},[78],"height:3.844em;vertical-align:-1.672em;",[39,6956,6958],{"className":6957},[83],[39,6959,6961,7051],{"className":6960},[759],[39,6962,6964],{"className":6963},[918],[39,6965,6967,7042],{"className":6966},[196,649],[39,6968,6970,7039],{"className":6969},[200],[39,6971,6974,7030],{"className":6972,"style":6973},[204],"height:2.172em;",[39,6975,6977,6980],{"style":6976},"top:-4.172em;",[39,6978],{"className":6979,"style":3311},[212],[39,6981,6983,7012,7015,7018,7021,7024,7027],{"className":6982},[83],[39,6984,6986,6989],{"className":6985},[83],[39,6987,4295],{"className":6988},[83,84],[39,6990,6992],{"className":6991},[192],[39,6993,6995],{"className":6994},[196],[39,6996,6998],{"className":6997},[200],[39,6999,7001],{"className":7000,"style":4421},[204],[39,7002,7003,7006],{"style":1027},[39,7004],{"className":7005,"style":213},[212],[39,7007,7009],{"className":7008},[217,218,219,220],[39,7010,2098],{"className":7011,"style":2114},[83,84,220],[39,7013,136],{"className":7014},[182],[39,7016,142],{"className":7017},[83,84],[39,7019,155],{"className":7020},[252],[39,7022],{"className":7023,"style":256},[230],[39,7025,158],{"className":7026},[83,84],[39,7028,161],{"className":7029},[263],[39,7031,7033,7036],{"style":7032},"top:-1.738em;",[39,7034],{"className":7035,"style":3311},[212],[39,7037],{"className":7038},[83],[39,7040,673],{"className":7041},[672],[39,7043,7045],{"className":7044},[200],[39,7046,7049],{"className":7047,"style":7048},[204],"height:1.672em;",[39,7050],{},[39,7052,7054],{"className":7053},[1223],[39,7055,7057,7406],{"className":7056},[196,649],[39,7058,7060,7403],{"className":7059},[200],[39,7061,7063,7226],{"className":7062,"style":6973},[204],[39,7064,7065,7068],{"style":6976},[39,7066],{"className":7067,"style":3311},[212],[39,7069,7071,7074,7077,7080,7083,7161,7164,7167,7170,7202,7205,7208,7211,7214,7217,7220,7223],{"className":7070},[83],[39,7072],{"className":7073},[83],[39,7075],{"className":7076,"style":231},[230],[39,7078,793],{"className":7079},[235],[39,7081],{"className":7082,"style":231},[230],[39,7084,7086],{"className":7085},[1744,3294],[39,7087,7089,7152],{"className":7088},[196,649],[39,7090,7092,7149],{"className":7091},[200],[39,7093,7095,7139],{"className":7094,"style":6670},[204],[39,7096,7098,7101],{"style":7097},"top:-1.856em;margin-left:0em;",[39,7099],{"className":7100,"style":3311},[212],[39,7102,7104],{"className":7103},[217,218,219,220],[39,7105,7107],{"className":7106},[83,220],[39,7108,7110,7113],{"className":7109},[83,220],[39,7111,142],{"className":7112},[83,84,220],[39,7114,7116],{"className":7115},[192],[39,7117,7119],{"className":7118},[196],[39,7120,7122],{"className":7121},[200],[39,7123,7125],{"className":7124,"style":3997},[204],[39,7126,7127,7130],{"style":4000},[39,7128],{"className":7129,"style":4004},[212],[39,7131,7133],{"className":7132},[217,4008,4009,220],[39,7134,7136],{"className":7135},[83,220],[39,7137,147],{"className":7138},[83,220],[39,7140,7141,7144],{"style":3330},[39,7142],{"className":7143,"style":3311},[212],[39,7145,7146],{},[39,7147,3191],{"className":7148},[1744,3339,3340],[39,7150,673],{"className":7151},[672],[39,7153,7155],{"className":7154},[200],[39,7156,7159],{"className":7157,"style":7158},[204],"height:1.294em;",[39,7160],{},[39,7162],{"className":7163,"style":256},[230],[39,7165,131],{"className":7166,"style":178},[83,84],[39,7168,136],{"className":7169},[182],[39,7171,7173,7176],{"className":7172},[83],[39,7174,142],{"className":7175},[83,84],[39,7177,7179],{"className":7178},[192],[39,7180,7182],{"className":7181},[196],[39,7183,7185],{"className":7184},[200],[39,7186,7188],{"className":7187,"style":1024},[204],[39,7189,7190,7193],{"style":1027},[39,7191],{"className":7192,"style":213},[212],[39,7194,7196],{"className":7195},[217,218,219,220],[39,7197,7199],{"className":7198},[83,220],[39,7200,147],{"className":7201},[83,220],[39,7203],{"className":7204,"style":231},[230],[39,7206,150],{"className":7207},[235],[39,7209],{"className":7210,"style":231},[230],[39,7212,142],{"className":7213},[83,84],[39,7215,155],{"className":7216},[252],[39,7218],{"className":7219,"style":256},[230],[39,7221,158],{"className":7222},[83,84],[39,7224,161],{"className":7225},[263],[39,7227,7228,7231],{"style":7032},[39,7229],{"className":7230,"style":3311},[212],[39,7232,7234,7237,7241,7244,7247,7250,7397,7400],{"className":7233},[83],[39,7235],{"className":7236},[83],[39,7238],{"className":7239,"style":7240},[230],"margin-right:1em;",[39,7242],{"className":7243,"style":461},[230],[39,7245,409],{"className":7246},[465],[39,7248],{"className":7249,"style":461},[230],[39,7251,7253,7256,7259,7262,7265,7268,7271,7274,7277,7280,7312,7315,7318,7321,7324,7327,7356,7359,7391,7394],{"className":7252},[2842],[39,7254,537],{"className":7255,"style":4592},[182,4591],[39,7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第一式由全期望公式直接得到。对于第二式，利用 ",[39,7420,7422,7464],{"className":7421,"translate":43},[42],[39,7423,7425],{"className":7424},[47],[49,7426,7427],{"xmlns":51},[53,7428,7429,7461],{},[56,7430,7431,7437,7439,7445,7447,7449],{},[613,7432,7433,7435],{},[59,7434,3181],{},[59,7436,621],{},[133,7438,793],{},[613,7440,7441,7443],{},[59,7442,401],{},[59,7444,621],{},[133,7446,788],{},[59,7448,531],{},[613,7450,7451,7453],{},[59,7452,3181],{},[56,7454,7455,7457,7459],{},[59,7456,621],{},[133,7458,788],{},[539,7460,546],{},[63,7462,7463],{"encoding":65},"G_t=r_t+\\gamma 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s,a)r(s,a,s')",[39,8460,8462,8590],{"className":8461,"ariaHidden":70},[69],[39,8463,8465,8468,8540,8543,8546,8549,8581,8584,8587],{"className":8464},[74],[39,8466],{"className":8467,"style":3945},[78],[39,8469,8471,8474],{"className":8470},[1744],[39,8472,3191],{"className":8473,"style":3953},[1744,3339,3952],[39,8475,8477],{"className":8476},[192],[39,8478,8480,8532],{"className":8479},[196,649],[39,8481,8483,8529],{"className":8482},[200],[39,8484,8486],{"className":8485,"style":3966},[204],[39,8487,8488,8491],{"style":3969},[39,8489],{"className":8490,"style":213},[212],[39,8492,8494],{"className":8493},[217,218,219,220],[39,8495,8497],{"className":8496},[83,220],[39,8498,8500,8503],{"className":8499},[83,220],[39,8501,142],{"className":8502},[83,84,220],[39,8504,8506],{"className":8505},[192],[39,8507,8509],{"className":8508},[196],[39,8510,8512],{"className":8511},[200],[39,8513,8515],{"className":8514,"style":3997},[204],[39,8516,8517,8520],{"style":4000},[39,8518],{"className":8519,"style":4004},[212],[39,8521,8523],{"className":8522},[217,4008,4009,220],[39,8524,8526],{"className":8525},[83,220],[39,8527,147],{"className":8528},[83,220],[39,8530,673],{"className":8531},[672],[39,8533,8535],{"className":8534},[200],[39,8536,8538],{"className":8537,"style":4025},[204],[39,8539],{},[39,8541],{"className":8542,"style":256},[230],[39,8544,131],{"className":8545,"style":178},[83,84],[39,8547,136],{"className":8548},[182],[39,8550,8552,8555],{"className":8551},[83],[39,8553,142],{"className":8554},[83,84],[39,8556,8558],{"className":8557},[192],[39,8559,8561],{"className":8560},[196],[39,8562,8564],{"className":8563},[200],[39,8565,8567],{"className":8566,"style":205},[204],[39,8568,8569,8572],{"style":208},[39,8570],{"className":8571,"style":213},[212],[39,8573,8575],{"className":8574},[217,218,219,220],[39,8576,8578],{"className":8577},[83,220],[39,8579,147],{"className":8580},[83,220],[39,8582],{"className":8583,"style":231},[230],[39,8585,150],{"className":8586},[235],[39,8588],{"className":8589,"style":231},[230],[39,8591,8593,8596,8599,8602,8605,8608,8611,8614,8617,8620,8623,8626,8629,8632,8635,8667],{"className":8592},[74],[39,8594],{"className":8595,"style":174},[78],[39,8597,142],{"className":8598},[83,84],[39,8600,155],{"className":8601},[252],[39,8603],{"className":8604,"style":256},[230],[39,8606,158],{"className":8607},[83,84],[39,8609,161],{"className":8610},[263],[39,8612,401],{"className":8613,"style":438},[83,84],[39,8615,136],{"className":8616},[182],[39,8618,142],{"className":8619},[83,84],[39,8621,155],{"className":8622},[252],[39,8624],{"className":8625,"style":256},[230],[39,8627,158],{"className":8628},[83,84],[39,8630,155],{"className":8631},[252],[39,8633],{"className":8634,"style":256},[230],[39,8636,8638,8641],{"className":8637},[83],[39,8639,142],{"className":8640},[83,84],[39,8642,8644],{"className":8643},[192],[39,8645,8647],{"className":8646},[196],[39,8648,8650],{"className":8649},[200],[39,8651,8653],{"className":8652,"style":205},[204],[39,8654,8655,8658],{"style":208},[39,8656],{"className":8657,"style":213},[212],[39,8659,8661],{"className":8660},[217,218,219,220],[39,8662,8664],{"className":8663},[83,220],[39,8665,147],{"className":8666},[83,220],[39,8668,161],{"className":8669},[263],"。第二项先用全期望公式对 ",[39,8672,8674,8697],{"className":8673,"translate":43},[42],[39,8675,8677],{"className":8676},[47],[49,8678,8679],{"xmlns":51},[53,8680,8681,8695],{},[56,8682,8683],{},[613,8684,8685,8687],{},[59,8686,142],{},[56,8688,8689,8691,8693],{},[59,8690,621],{},[133,8692,788],{},[539,8694,546],{},[63,8696,3625],{"encoding":65},[39,8698,8700],{"className":8699,"ariaHidden":70},[69],[39,8701,8703,8706],{"className":8702},[74],[39,8704],{"className":8705,"style":3635},[78],[39,8707,8709,8712],{"className":8708},[83],[39,8710,142],{"className":8711},[83,84],[39,8713,8715],{"className":8714},[192],[39,8716,8718,8747],{"className":8717},[196,649],[39,8719,8721,8744],{"className":8720},[200],[39,8722,8724],{"className":8723,"style":963},[204],[39,8725,8726,8729],{"style":966},[39,8727],{"className":8728,"style":213},[212],[39,8730,8732],{"className":8731},[217,218,219,220],[39,8733,8735,8738,8741],{"className":8734},[83,220],[39,8736,621],{"className":8737},[83,84,220],[39,8739,788],{"className":8740},[465,220],[39,8742,546],{"className":8743},[83,220],[39,8745,673],{"className":8746},[672],[39,8748,8750],{"className":8749},[200],[39,8751,8753],{"className":8752,"style":994},[204],[39,8754],{}," 分解，再利用马尔可夫性与策略的平稳性：",[39,8757,8759],{"className":8758,"translate":43},[746],[39,8760,8762,8999],{"className":8761,"translate":43},[42],[39,8763,8765],{"className":8764},[47],[49,8766,8767],{"xmlns":51,"display":755},[53,8768,8769,8996],{},[759,8770,8771,8865,8930],{"rowspacing":761,"columnalign":762,"columnspacing":146},[764,8772,8773,8825],{},[767,8774,8775],{},[770,8776,8777],{"scriptlevel":541,"displaystyle":70},[56,8778,8779,8785,8787,8799,8801,8807,8809,8811,8813,8819,8821,8823],{},[613,8780,8781,8783],{},[59,8782,4253],{"mathvariant":421},[59,8784,2098],{},[133,8786,537],{"stretchy":135},[613,8788,8789,8791],{},[59,8790,3181],{},[56,8792,8793,8795,8797],{},[59,8794,621],{},[133,8796,788],{},[539,8798,546],{},[133,8800,150],{},[613,8802,8803,8805],{},[59,8804,142],{},[59,8806,621],{},[133,8808,793],{},[59,8810,142],{},[133,8812,155],{"separator":70},[613,8814,8815,8817],{},[59,8816,158],{},[59,8818,621],{},[133,8820,793],{},[59,8822,158],{},[133,8824,4280],{"stretchy":135},[767,8826,8827],{},[770,8828,8829],{"scriptlevel":541,"displaystyle":70},[56,8830,8831,8833,8835,8845,8847,8849,8855,8857,8859,8861,8863],{},[56,8832],{},[133,8834,793],{},[6556,8836,8837,8839],{},[133,8838,3191],{},[138,8840,8841,8843],{},[59,8842,142],{},[133,8844,147],{"mathvariant":145,"lspace":146,"rspace":146},[59,8846,131],{},[133,8848,136],{"stretchy":135},[138,8850,8851,8853],{},[59,8852,142],{},[133,8854,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,8856,150],{},[59,8858,142],{},[133,8860,155],{"separator":70},[59,8862,158],{},[133,8864,161],{"stretchy":135},[764,8866,8867,8873],{},[767,8868,8869],{},[770,8870,8871],{"scriptlevel":541,"displaystyle":70},[56,8872],{},[767,8874,8875],{},[770,8876,8877],{"scriptlevel":541,"displaystyle":70},[56,8878,8879,8881,8883,8886,8892,8894,8906,8908,8920,8922,8928],{},[56,8880],{},[230,8882],{"width":6891},[133,8884,8885],{},"⋅",[613,8887,8888,8890],{},[59,8889,4253],{"mathvariant":421},[59,8891,2098],{},[133,8893,537],{"stretchy":135},[613,8895,8896,8898],{},[59,8897,3181],{},[56,8899,8900,8902,8904],{},[59,8901,621],{},[133,8903,788],{},[539,8905,546],{},[133,8907,150],{},[613,8909,8910,8912],{},[59,8911,142],{},[56,8913,8914,8916,8918],{},[59,8915,621],{},[133,8917,788],{},[539,8919,546],{},[133,8921,793],{},[138,8923,8924,8926],{},[59,8925,142],{},[133,8927,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,8929,4280],{"stretchy":135},[764,8931,8932,8938],{},[767,8933,8934],{},[770,8935,8936],{"scriptlevel":541,"displaystyle":70},[56,8937],{},[767,8939,8940],{},[770,8941,8942],{"scriptlevel":541,"displaystyle":70},[56,8943,8944,8946,8948,8958,8960,8962,8968,8970,8972,8974,8976,8978,8984,8986,8992,8994],{},[56,8945],{},[133,8947,793],{},[6556,8949,8950,8952],{},[133,8951,3191],{},[138,8953,8954,8956],{},[59,8955,142],{},[133,8957,147],{"mathvariant":145,"lspace":146,"rspace":146},[59,8959,131],{},[133,8961,136],{"stretchy":135},[138,8963,8964,8966],{},[59,8965,142],{},[133,8967,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,8969,150],{},[59,8971,142],{},[133,8973,155],{"separator":70},[59,8975,158],{},[133,8977,161],{"stretchy":135},[138,8979,8980,8982],{},[59,8981,4230],{},[59,8983,2098],{},[133,8985,136],{"stretchy":135},[138,8987,8988,8990],{},[59,8989,142],{},[133,8991,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,8993,161],{"stretchy":135},[59,8995,2903],{"mathvariant":145},[63,8997,8998],{"encoding":65},"\\begin{aligned}\n\\mathbb{E}_\\pi[G_{t+1}\\mid s_t=s,a_t=a]\n&=\\sum_{s'}P(s'\\mid s,a)\\\\\n&\\quad\\cdot\\mathbb{E}_\\pi[G_{t+1}\\mid s_{t+1}=s']\\\\\n&=\\sum_{s'}P(s'\\mid s,a)V^\\pi(s').\n\\end{aligned}",[39,9000,9002],{"className":9001,"ariaHidden":70},[69],[39,9003,9005,9009],{"className":9004},[74],[39,9006],{"className":9007,"style":9008},[78],"height:6.488em;vertical-align:-2.994em;",[39,9010,9012],{"className":9011},[83],[39,9013,9015,9281],{"className":9014},[759],[39,9016,9018],{"className":9017},[918],[39,9019,9021,9272],{"className":9020},[196,649],[39,9022,9024,9269],{"className":9023},[200],[39,9025,9028,9251,9260],{"className":9026,"style":9027},[204],"height:3.494em;",[39,9029,9031,9034],{"style":9030},"top:-5.494em;",[39,9032],{"className":9033,"style":3311},[212],[39,9035,9037,9077,9080,9129,9132,9135,9138,9178,9181,9184,9187,9190,9193,9196,9236,9239,9242,9245,9248],{"className":9036},[83],[39,9038,9040,9043],{"className":9039},[83],[39,9041,4253],{"className":9042},[83,515],[39,9044,9046],{"className":9045},[192],[39,9047,9049,9069],{"className":9048},[196,649],[39,9050,9052,9066],{"className":9051},[200],[39,9053,9055],{"className":9054,"style":1728},[204],[39,9056,9057,9060],{"style":966},[39,9058],{"className":9059,"style":213},[212],[39,9061,9063],{"className":9062},[217,218,219,220],[39,9064,2098],{"className":9065,"style":2114},[83,84,220],[39,9067,673],{"className":9068},[672],[39,9070,9072],{"className":9071},[200],[39,9073,9075],{"className":9074,"style":680},[204],[39,9076],{},[39,9078,537],{"className":9079},[182],[39,9081,9083,9086],{"className":9082},[83],[39,9084,3181],{"className":9085},[83,84],[39,9087,9089],{"className":9088},[192],[39,9090,9092,9121],{"className":9091},[196,649],[39,9093,9095,9118],{"className":9094},[200],[39,9096,9098],{"className":9097,"style":963},[204],[39,9099,9100,9103],{"style":966},[39,9101],{"className":9102,"style":213},[212],[39,9104,9106],{"className":9105},[217,218,219,220],[39,9107,9109,9112,9115],{"className":9108},[83,220],[39,9110,621],{"className":9111},[83,84,220],[39,9113,788],{"className":9114},[465,220],[39,9116,546],{"className":9117},[83,220],[39,9119,673],{"className":9120},[672],[39,9122,9124],{"className":9123},[200],[39,9125,9127],{"className":9126,"style":994},[204],[39,9128],{},[39,9130],{"className":9131,"style":231},[230],[39,9133,150],{"className":9134},[235],[39,9136],{"className":9137,"style":231},[230],[39,9139,9141,9144],{"className":9140},[83],[39,9142,142],{"className":9143},[83,84],[39,9145,9147],{"className":9146},[192],[39,9148,9150,9170],{"className":9149},[196,649],[39,9151,9153,9167],{"className":9152},[200],[39,9154,9156],{"className":9155,"style":656},[204],[39,9157,9158,9161],{"style":966},[39,9159],{"className":9160,"style":213},[212],[39,9162,9164],{"className":9163},[217,218,219,220],[39,9165,621],{"className":9166},[83,84,220],[39,9168,673],{"className":9169},[672],[39,9171,9173],{"className":9172},[200],[39,9174,9176],{"className":9175,"style":680},[204],[39,9177],{},[39,9179],{"className":9180,"style":231},[230],[39,9182,793],{"className":9183},[235],[39,9185],{"className":9186,"style":231},[230],[39,9188,142],{"className":9189},[83,84],[39,9191,155],{"className":9192},[252],[39,9194],{"className":9195,"style":256},[230],[39,9197,9199,9202],{"className":9198},[83],[39,9200,158],{"className":9201},[83,84],[39,9203,9205],{"className":9204},[192],[39,9206,9208,9228],{"className":9207},[196,649],[39,9209,9211,9225],{"className":9210},[200],[39,9212,9214],{"className":9213,"style":656},[204],[39,9215,9216,9219],{"style":966},[39,9217],{"className":9218,"style":213},[212],[39,9220,9222],{"className":9221},[217,218,219,220],[39,9223,621],{"className":9224},[83,84,220],[39,9226,673],{"className":9227},[672],[39,9229,9231],{"className":9230},[200],[39,9232,9234],{"className":9233,"style":680},[204],[39,9235],{},[39,9237],{"className":9238,"style":231},[230],[39,9240,793],{"className":9241},[235],[39,9243],{"className":9244,"style":231},[230],[39,9246,158],{"className":9247},[83,84],[39,9249,4280],{"className":9250},[263],[39,9252,9254,9257],{"style":9253},"top:-3.06em;",[39,9255],{"className":9256,"style":3311},[212],[39,9258],{"className":9259},[83],[39,9261,9263,9266],{"style":9262},"top:-1.35em;",[39,9264],{"className":9265,"style":3311},[212],[39,9267],{"className":9268},[83],[39,9270,673],{"className":9271},[672],[39,9273,9275],{"className":9274},[200],[39,9276,9279],{"className":9277,"style":9278},[204],"height:2.994em;",[39,9280],{},[39,9282,9284],{"className":9283},[1223],[39,9285,9287,9905],{"className":9286},[196,649],[39,9288,9290,9902],{"className":9289},[200],[39,9291,9293,9454,9671],{"className":9292,"style":9027},[204],[39,9294,9295,9298],{"style":9030},[39,9296],{"className":9297,"style":3311},[212],[39,9299,9301,9304,9307,9310,9313,9389,9392,9395,9398,9430,9433,9436,9439,9442,9445,9448,9451],{"className":9300},[83],[39,9302],{"className":9303},[83],[39,9305],{"className":9306,"style":231},[230],[39,9308,793],{"className":9309},[235],[39,9311],{"className":9312,"style":231},[230],[39,9314,9316],{"className":9315},[1744,3294],[39,9317,9319,9381],{"className":9318}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",[39,9920,9922,9943],{"className":9921,"translate":43},[42],[39,9923,9925],{"className":9924},[47],[49,9926,9927],{"xmlns":51},[53,9928,9929,9940],{},[56,9930,9931],{},[9932,9933,9934,9936,9938],"msubsup",{},[59,9935,4230],{},[59,9937,621],{},[59,9939,2098],{},[63,9941,9942],{"encoding":65},"V_t^\\pi",[39,9944,9946],{"className":9945,"ariaHidden":70},[69],[39,9947,9949,9953],{"className":9948},[74],[39,9950],{"className":9951,"style":9952},[78],"height:0.9303em;vertical-align:-0.247em;",[39,9954,9956,9959],{"className":9955},[83],[39,9957,4230],{"className":9958,"style":461},[83,84],[39,9960,9962],{"className":9961},[192],[39,9963,9965,9997],{"className":9964},[196,649],[39,9966,9968,9994],{"className":9967},[200],[39,9969,9971,9983],{"className":9970,"style":6410},[204],[39,9972,9974,9977],{"style":9973},"top:-2.453em;margin-left:-0.2222em;margin-right:0.05em;",[39,9975],{"className":9976,"style":213},[212],[39,9978,9980],{"className":9979},[217,218,219,220],[39,9981,621],{"className":9982},[83,84,220],[39,9984,9985,9988],{"style":208},[39,9986],{"className":9987,"style":213},[212],[39,9989,9991],{"className":9990},[217,218,219,220],[39,9992,2098],{"className":9993,"style":2114},[83,84,220],[39,9995,673],{"className":9996},[672],[39,9998,10000],{"className":9999},[200],[39,10001,10004],{"className":10002,"style":10003},[204],"height:0.247em;",[39,10005],{},"，递归形式仍然成立，但函数会随 ",[39,10008,10010,10023],{"className":10009,"translate":43},[42],[39,10011,10013],{"className":10012},[47],[49,10014,10015],{"xmlns":51},[53,10016,10017,10021],{},[56,10018,10019],{},[59,10020,621],{},[63,10022,621],{"encoding":65},[39,10024,10026],{"className":10025,"ariaHidden":70},[69],[39,10027,10029,10032],{"className":10028},[74],[39,10030],{"className":10031,"style":709},[78],[39,10033,621],{"className":10034},[83,84]," 变化。本文只讨论平稳策略，因此可以省略时间下标。",[21,10037,10039],{"id":10038},"二贝尔曼期望方程","二、贝尔曼期望方程",[26,10041,10043],{"id":10042},"_21-方程的推导","2.1 方程的推导",[11,10045,10046,10047,10051],{},"将",[158,10048,10050],{"href":10049},"#proposition-basic-identities","命题 1.5"," 的第二式代入第一式，立即得到策略评估的核心方程。",[31,10053,10056,10088],{"id":10054,"title":10055},"theorem-bellman-expectation","定理 2.1（贝尔曼期望方程）",[11,10057,10058,10059,10087],{},"对任意平稳策略 ",[39,10060,10062,10075],{"className":10061,"translate":43},[42],[39,10063,10065],{"className":10064},[47],[49,10066,10067],{"xmlns":51},[53,10068,10069,10073],{},[56,10070,10071],{},[59,10072,2098],{},[63,10074,2101],{"encoding":65},[39,10076,10078],{"className":10077,"ariaHidden":70},[69],[39,10079,10081,10084],{"className":10080},[74],[39,10082],{"className":10083,"style":290},[78],[39,10085,2098],{"className":10086,"style":2114},[83,84],"，状态值函数满足",[39,10089,10091],{"className":10090,"translate":43},[746],[39,10092,10094,10266],{"className":10093,"translate":43},[42],[39,10095,10097],{"className":10096},[47],[49,10098,10099],{"xmlns":51,"display":755},[53,10100,10101,10263],{},[759,10102,10103,10181],{"rowspacing":761,"columnalign":762,"columnspacing":146},[764,10104,10105,10123],{},[767,10106,10107],{},[770,10108,10109],{"scriptlevel":541,"displaystyle":70},[56,10110,10111,10117,10119,10121],{},[138,10112,10113,10115],{},[59,10114,4230],{},[59,10116,2098],{},[133,10118,136],{"stretchy":135},[59,10120,142],{},[133,10122,161],{"stretchy":135},[767,10124,10125],{},[770,10126,10127],{"scriptlevel":541,"displaystyle":70},[56,10128,10129,10131,10133,10139,10141,10143,10145,10147,10149,10151,10161,10163,10165,10171,10173,10175,10177,10179],{},[56,10130],{},[133,10132,793],{},[6556,10134,10135,10137],{},[133,10136,3191],{},[59,10138,158],{},[59,10140,2098],{},[133,10142,136],{"stretchy":135},[59,10144,158],{},[133,10146,150],{},[59,10148,142],{},[133,10150,161],{"stretchy":135},[6556,10152,10153,10155],{},[133,10154,3191],{},[138,10156,10157,10159],{},[59,10158,142],{},[133,10160,147],{"mathvariant":145,"lspace":146,"rspace":146},[59,10162,131],{},[133,10164,136],{"stretchy":135},[138,10166,10167,10169],{},[59,10168,142],{},[133,10170,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,10172,150],{},[59,10174,142],{},[133,10176,155],{"separator":70},[59,10178,158],{},[133,10180,161],{"stretchy":135},[764,10182,10183,10189],{},[767,10184,10185],{},[770,10186,10187],{"scriptlevel":541,"displaystyle":70},[56,10188],{},[767,10190,10191],{},[770,10192,10193],{"scriptlevel":541,"displaystyle":70},[56,10194,10195,10197,10199,10201,10247,10249,10252,10255,10257,10259,10261],{},[56,10196],{},[230,10198],{"width":6891},[133,10200,409],{},[56,10202,10203,10205,10207,10209,10211,10213,10215,10217,10223,10225,10227,10229,10235,10237,10243,10245],{},[133,10204,537],{"fence":70},[59,10206,401],{},[133,10208,136],{"stretchy":135},[59,10210,142],{},[133,10212,155],{"separator":70},[59,10214,158],{},[133,10216,155],{"separator":70},[138,10218,10219,10221],{},[59,10220,142],{},[133,10222,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,10224,161],{"stretchy":135},[133,10226,788],{},[59,10228,531],{},[138,10230,10231,10233],{},[59,10232,4230],{},[59,10234,2098],{},[133,10236,136],{"stretchy":135},[138,10238,10239,10241],{},[59,10240,142],{},[133,10242,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,10244,161],{"stretchy":135},[133,10246,4280],{"fence":70},[133,10248,155],{"separator":70},[230,10250],{"width":10251},"2em",[59,10253,10254],{"mathvariant":145},"∀",[59,10256,142],{},[133,10258,534],{},[59,10260,61],{},[59,10262,2903],{"mathvariant":145},[63,10264,10265],{"encoding":65},"\\begin{aligned}\nV^\\pi(s)\n&=\\sum_{a}\\pi(a\\mid s)\\sum_{s'}P(s'\\mid s,a)\\\\\n&\\quad\\times\\left[r(s,a,s')+\\gamma V^\\pi(s')\\right],\n\\qquad \\forall s\\in S.\n\\end{aligned}",[39,10267,10269],{"className":10268,"ariaHidden":70},[69],[39,10270,10272,10275],{"className":10271},[74],[39,10273],{"className":10274,"style":6954},[78],[39,10276,10278],{"className":10277},[83],[39,10279,10281,10358],{"className":10280},[759],[39,10282,10284],{"className":10283},[918],[39,10285,10287,10350],{"className":10286},[196,649],[39,10288,10290,10347],{"className":10289},[200],[39,10291,10293,10339],{"className":10292,"style":6973},[204],[39,10294,10295,10298],{"style":6976},[39,10296],{"className":10297,"style":3311},[212],[39,10299,10301,10330,10333,10336],{"className":10300},[83],[39,10302,10304,10307],{"className":10303},[83],[39,10305,4230],{"className":10306,"style":461},[83,84],[39,10308,10310],{"className":10309},[192],[39,10311,10313],{"className":10312},[196],[39,10314,10316],{"className":10315},[200],[39,10317,10319],{"className":10318,"style":4421},[204],[39,10320,10321,10324],{"style":1027},[39,10322],{"className":10323,"style":213},[212],[39,10325,10327],{"className":10326},[217,218,219,220],[39,10328,2098],{"className":10329,"style":2114},[83,84,220],[39,10331,136],{"className":10332},[182],[39,10334,142],{"className":10335},[83,84],[39,10337,161],{"className":10338},[263],[39,10340,10341,10344],{"style":7032},[39,10342],{"className":10343,"style":3311},[212],[39,10345],{"className":10346},[83],[39,10348,673],{"className":10349},[672],[39,10351,10353],{"className":10352},[200],[39,10354,10356],{"className":10355,"style":7048},[204],[39,10357],{},[39,10359,10361],{"className":10360},[1223],[39,10362,10364,10815],{"className":10363},[196,649],[39,10365,10367,10812],{"className":10366},[200],[39,10368,10370,10608],{"className":10369,"style":6973},[204],[39,10371,10372,10375],{"style":6976},[39,10373],{"className":10374,"style":3311},[212],[39,10376,10378,10381,10384,10387,10390,10437,10440,10443,10446,10449,10452,10455,10458,10461,10464,10467,10543,10546,10549,10552,10584,10587,10590,10593,10596,10599,10602,10605],{"className":10377},[83],[39,10379],{"className":10380},[83],[39,10382],{"className":10383,"style":231},[230],[39,10385,793],{"className":10386},[235],[39,10388],{"className":10389,"style":231},[230],[39,10391,10393],{"className":10392},[1744,3294],[39,10394,10396,10429],{"className":10395},[196,649],[39,10397,10399,10426],{"className":10398},[200],[39,10400,10402,10416],{"className":10401,"style":6670},[204],[39,10403,10404,10407],{"style":6673},[39,10405],{"className":10406,"style":3311},[212],[39,10408,10410],{"className":10409},[217,218,219,220],[39,10411,10413],{"className":10412},[83,220],[39,10414,158],{"className":10415},[83,84,220],[39,10417,10418,10421],{"style":3330},[39,10419],{"className":10420,"style":3311},[212],[39,10422,10423],{},[39,10424,3191],{"className":10425},[1744,3339,3340],[39,10427,673],{"className":10428},[672],[39,10430,10432],{"className":10431},[200],[39,10433,10435],{"className":10434,"style":6705},[204],[39,10436],{},[39,10438],{"className":10439,"style":256},[230],[39,10441,2098],{"className":10442,"style":2114},[83,84],[39,10444,136],{"className":10445},[182],[39,10447,158],{"className":10448},[83,84],[39,10450],{"className":10451,"style":231},[230],[39,10453,150],{"className":10454},[235],[39,10456],{"className":10457,"style":231},[230],[39,10459,142],{"className":10460},[83,84],[39,10462,161],{"className":10463},[263],[39,10465],{"className":10466,"style":256},[230],[39,10468,10470],{"className":10469},[1744,3294],[39,10471,10473,10535],{"className":10472},[196,649],[39,10474,10476,10532],{"className":10475},[200],[39,10477,10479,10522],{"className":10478,"style":6670},[204],[39,10480,10481,10484],{"style":7097},[39,10482],{"className":10483,"style":3311},[212],[39,10485,10487],{"className":10486},[217,218,219,220],[39,10488,10490],{"className":10489},[83,220],[39,10491,10493,10496],{"className":10492},[83,220],[39,10494,142],{"className":10495},[83,84,220],[39,10497,10499],{"className":10498},[192],[39,10500,10502],{"className":10501},[196],[39,10503,10505],{"className":10504},[200],[39,10506,10508],{"className":10507,"style":3997},[204],[39,10509,10510,10513],{"style":4000},[39,10511],{"className":10512,"style":4004},[212],[39,10514,10516],{"className":10515},[217,4008,4009,220],[39,10517,10519],{"className":10518},[83,220],[39,10520,147],{"className":10521},[83,220],[39,10523,10524,10527],{"style":3330},[39,10525],{"className":10526,"style":3311},[212],[39,10528,10529],{},[39,10530,3191],{"className":10531},[1744,3339,3340],[39,10533,673],{"className":10534},[672],[39,10536,10538],{"className":10537},[200],[39,10539,10541],{"className":10540,"style":7158},[204],[39,10542],{},[39,10544],{"className":10545,"style":256},[230],[39,10547,131],{"className":10548,"style":178},[83,84],[39,10550,136],{"className":10551},[182],[39,10553,10555,10558],{"className":10554},[83],[39,10556,142],{"className":10557},[83,84],[39,10559,10561],{"className":10560},[192],[39,10562,10564],{"className":10563},[196],[39,10565,10567],{"className":10566},[200],[39,10568,10570],{"className":10569,"style":1024},[204],[39,10571,10572,10575],{"style":1027},[39,10573],{"className":10574,"style":213},[212],[39,10576,10578],{"className":10577},[217,218,219,220],[39,10579,10581],{"className":10580},[83,220],[39,10582,147],{"className":10583},[83,220],[39,10585],{"className":10586,"style":231},[230],[39,10588,150],{"className":10589},[235],[39,10591],{"className":10592,"style":231},[230],[39,10594,142],{"className":10595},[83,84],[39,10597,155],{"className":10598},[252],[39,10600],{"className":10601,"style":256},[230],[39,10603,158],{"className":10604},[83,84],[39,10606,161],{"className":10607},[263],[39,10609,10610,10613],{"style":7032},[39,10611],{"className":10612,"style":3311},[212],[39,10614,10616,10619,10622,10625,10628,10631,10778,10781,10784,10788,10791,10794,10797,10800,10803,10806,10809],{"className":10615},[83],[39,10617],{"className":10618},[83],[39,10620],{"className":10621,"style":7240},[230],[39,10623],{"className":10624,"style":461},[230],[39,10626,409],{"className":10627},[465],[39,10629],{"className":10630,"style":461},[230],[39,10632,10634,10637,10640,10643,10646,10649,10652,10655,10658,10661,10693,10696,10699,10702,10705,10708,10737,10740,10772,10775],{"className":10633},[2842],[39,10635,537],{"className":10636,"style":4592},[182,4591],[39,10638,401],{"className":10639,"style":438},[83,84],[39,10641,136],{"className":10642},[182],[39,10644,142],{"className":10645},[83,84],[39,10647,155],{"className":10648},[252],[39,10650],{"className":10651,"style":256},[230],[39,10653,158],{"className":10654},[83,84],[39,10656,155],{"className":10657},[252],[39,10659],{"className":10660,"style":256},[230],[39,10662,10664,10667],{"className":10663},[83],[39,10665,142],{"className":10666},[83,84],[39,10668,10670],{"className":10669},[192],[39,10671,10673],{"className":10672},[196],[39,10674,10676],{"className":10675},[200],[39,10677,10679],{"className":10678,"style":1024},[204],[39,10680,10681,10684],{"style":1027},[39,10682],{"className":10683,"style":213},[212],[39,10685,10687],{"className":10686},[217,218,219,220],[39,10688,10690],{"className":10689},[83,220],[39,10691,147],{"className":10692},[83,220],[39,10694,161],{"className":10695},[263],[39,10697],{"className":10698,"style":461},[230],[39,10700,788],{"className":10701},[465],[39,10703],{"className":10704,"style":461},[230],[39,10706,531],{"className":10707,"style":565},[83,84],[39,10709,10711,10714],{"className":10710},[83],[39,10712,4230],{"className":10713,"style":461},[83,84],[39,10715,10717],{"className":10716},[192],[39,10718,10720],{"className":10719},[196],[39,10721,10723],{"className":10722},[200],[39,10724,10726],{"className":10725,"style":4421},[204],[39,10727,10728,10731],{"style":1027},[39,10729],{"className":10730,"style":213},[212],[39,10732,10734],{"className":10733},[217,218,219,220],[39,10735,2098],{"className":10736,"style":2114},[83,84,220],[39,10738,136],{"className":10739},[182],[39,10741,10743,10746],{"className":10742},[83],[39,10744,142],{"className":10745},[83,84],[39,10747,10749],{"className":10748},[192],[39,10750,10752],{"className":10751},[196],[39,10753,10755],{"className":10754},[200],[39,10756,10758],{"className":10757,"style":1024},[204],[39,10759,10760,10763],{"style":1027},[39,10761],{"className":10762,"style":213},[212],[39,10764,10766],{"className":10765},[217,218,219,220],[39,10767,10769],{"className":10768},[83,220],[39,10770,147],{"className":10771},[83,220],[39,10773,161],{"className":10774},[263],[39,10776,4280],{"className":10777,"style":4592},[263,4591],[39,10779],{"className":10780,"style":256},[230],[39,10782,155],{"className":10783},[252],[39,10785],{"className":10786,"style":10787},[230],"margin-right:2em;",[39,10789],{"className":10790,"style":256},[230],[39,10792,10254],{"className":10793},[83],[39,10795,142],{"className":10796},[83,84],[39,10798],{"className":10799,"style":231},[230],[39,10801,534],{"className":10802},[235],[39,10804],{"className":10805,"style":231},[230],[39,10807,61],{"className":10808,"style":85},[83,84],[39,10810,2903],{"className":10811},[83],[39,10813,673],{"className":10814},[672],[39,10816,10818],{"className":10817},[200],[39,10819,10821],{"className":10820,"style":7048},[204],[39,10822],{},[11,10824,10825,10827,10828,742],{},[1489,10826,5595],{}," 由",[158,10829,10050],{"href":10049},[39,10831,10833],{"className":10832,"translate":43},[746],[39,10834,10836,11026],{"className":10835,"translate":43},[42],[39,10837,10839],{"className":10838},[47],[49,10840,10841],{"xmlns":51,"display":755},[53,10842,10843,11023],{},[759,10844,10845,10909],{"rowspacing":761,"columnalign":762,"columnspacing":146},[764,10846,10847,10865],{},[767,10848,10849],{},[770,10850,10851],{"scriptlevel":541,"displaystyle":70},[56,10852,10853,10859,10861,10863],{},[138,10854,10855,10857],{},[59,10856,4230],{},[59,10858,2098],{},[133,10860,136],{"stretchy":135},[59,10862,142],{},[133,10864,161],{"stretchy":135},[767,10866,10867],{},[770,10868,10869],{"scriptlevel":541,"displaystyle":70},[56,10870,10871,10873,10875,10881,10883,10885,10887,10889,10891,10893,10899,10901,10903,10905,10907],{},[56,10872],{},[133,10874,793],{},[6556,10876,10877,10879],{},[133,10878,3191],{},[59,10880,158],{},[59,10882,2098],{},[133,10884,136],{"stretchy":135},[59,10886,158],{},[133,10888,150],{},[59,10890,142],{},[133,10892,161],{"stretchy":135},[138,10894,10895,10897],{},[59,10896,4295],{},[59,10898,2098],{},[133,10900,136],{"stretchy":135},[59,10902,142],{},[133,10904,155],{"separator":70},[59,10906,158],{},[133,10908,161],{"stretchy":135},[764,10910,10911,10917],{},[767,10912,10913],{},[770,10914,10915],{"scriptlevel":541,"displaystyle":70},[56,10916],{},[767,10918,10919],{},[770,10920,10921],{"scriptlevel":541,"displaystyle":70},[56,10922,10923,10925,10927,10933,10935,10937,10939,10941,10943,10945,10955,10957,10959,10965,10967,10969,10971,10973,10975,11021],{},[56,10924],{},[133,10926,793],{},[6556,10928,10929,10931],{},[133,10930,3191],{},[59,10932,158],{},[59,10934,2098],{},[133,10936,136],{"stretchy":135},[59,10938,158],{},[133,10940,150],{},[59,10942,142],{},[133,10944,161],{"stretchy":135},[6556,10946,10947,10949],{},[133,10948,3191],{},[138,10950,10951,10953],{},[59,10952,142],{},[133,10954,147],{"mathvariant":145,"lspace":146,"rspace":146},[59,10956,131],{},[133,10958,136],{"stretchy":135},[138,10960,10961,10963],{},[59,10962,142],{},[133,10964,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,10966,150],{},[59,10968,142],{},[133,10970,155],{"separator":70},[59,10972,158],{},[133,10974,161],{"stretchy":135},[56,10976,10977,10979,10981,10983,10985,10987,10989,10991,10997,10999,11001,11003,11009,11011,11017,11019],{},[133,10978,537],{"fence":70},[59,10980,401],{},[133,10982,136],{"stretchy":135},[59,10984,142],{},[133,10986,155],{"separator":70},[59,10988,158],{},[133,10990,155],{"separator":70},[138,10992,10993,10995],{},[59,10994,142],{},[133,10996,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,10998,161],{"stretchy":135},[133,11000,788],{},[59,11002,531],{},[138,11004,11005,11007],{},[59,11006,4230],{},[59,11008,2098],{},[133,11010,136],{"stretchy":135},[138,11012,11013,11015],{},[59,11014,142],{},[133,11016,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,11018,161],{"stretchy":135},[133,11020,4280],{"fence":70},[59,11022,2903],{"mathvariant":145},[63,11024,11025],{"encoding":65},"\\begin{aligned}\nV^\\pi(s)\n&=\\sum_{a}\\pi(a\\mid s)Q^\\pi(s,a)\\\\\n&=\\sum_{a}\\pi(a\\mid s)\\sum_{s'}P(s'\\mid s,a)\n\\left[r(s,a,s')+\\gamma V^\\pi(s')\\right].\n\\end{aligned}",[39,11027,11029],{"className":11028,"ariaHidden":70},[69],[39,11030,11032,11036],{"className":11031},[74],[39,11033],{"className":11034,"style":11035},[78],"height:4.944em;vertical-align:-2.222em;",[39,11037,11039],{"className":11038},[83],[39,11040,11042,11123],{"className":11041},[759],[39,11043,11045],{"className":11044},[918],[39,11046,11048,11114],{"className":11047},[196,649],[39,11049,11051,11111],{"className":11050},[200],[39,11052,11055,11102],{"className":11053,"style":11054},[204],"height:2.722em;",[39,11056,11058,11061],{"style":11057},"top:-4.722em;",[39,11059],{"className":11060,"style":3311},[212],[39,11062,11064,11093,11096,11099],{"className":11063},[83],[39,11065,11067,11070],{"className":11066},[83],[39,11068,4230],{"className":11069,"style":461},[83,84],[39,11071,11073],{"className":11072},[192],[39,11074,11076],{"className":11075},[196],[39,11077,11079],{"className":11078},[200],[39,11080,11082],{"className":11081,"style":4421},[204],[39,11083,11084,11087],{"style":1027},[39,11085],{"className":11086,"style":213},[212],[39,11088,11090],{"className":11089},[217,218,219,220],[39,11091,2098],{"className":11092,"style":2114},[83,84,220],[39,11094,136],{"className":11095},[182],[39,11097,142],{"className":11098},[83,84],[39,11100,161],{"className":11101},[263],[39,11103,11105,11108],{"style":11104},"top:-2.122em;",[39,11106],{"className":11107,"style":3311},[212],[39,11109],{"className":11110},[83],[39,11112,673],{"className":11113},[672],[39,11115,11117],{"className":11116},[200],[39,11118,11121],{"className":11119,"style":11120},[204],"height:2.222em;",[39,11122],{},[39,11124,11126],{"className":11125},[1223],[39,11127,11129,11673],{"className":11128},[196,649],[39,11130,11132,11670],{"className":11131},[200],[39,11133,11135,11276],{"className":11134,"style":11054},[204],[39,11136,11137,11140],{"style":11057},[39,11138],{"className":11139,"style":3311},[212],[39,11141,11143,11146,11149,11152,11155,11202,11205,11208,11211,11214,11217,11220,11223,11226,11229,11258,11261,11264,11267,11270,11273],{"className":11142},[83],[39,11144],{"className":11145},[83],[39,11147],{"className":11148,"style":231},[230],[39,11150,793],{"className":11151},[235],[39,11153],{"className":11154,"style":231},[230],[39,11156,11158],{"className":11157},[1744,3294],[39,11159,11161,11194],{"className":11160},[196,649],[39,11162,11164,11191],{"className":11163},[200],[39,11165,11167,11181],{"className":11166,"style":6670},[204],[39,11168,11169,11172],{"style":6673},[39,11170],{"className":11171,"style":3311},[212],[39,11173,11175],{"className":11174},[217,218,219,220],[39,11176,11178],{"className":11177},[83,220],[39,11179,158],{"className":11180},[83,84,220],[39,11182,11183,11186],{"style":3330},[39,11184],{"className":11185,"style":3311},[212],[39,11187,11188],{},[39,11189,3191],{"className":11190},[1744,3339,3340],[39,11192,673],{"className":11193},[672],[39,11195,11197],{"className":11196},[200],[39,11198,11200],{"className":11199,"style":6705},[204],[39,11201],{},[39,11203],{"className":11204,"style":256},[230],[39,11206,2098],{"className":11207,"style":2114},[83,84],[39,11209,136],{"className":11210},[182],[39,11212,158],{"className":11213},[83,84],[39,11215],{"className":11216,"style":231},[230],[39,11218,150],{"className":11219},[235],[39,11221],{"className":11222,"style":231},[230],[39,11224,142],{"className":11225},[83,84],[39,11227,161],{"className":11228},[263],[39,11230,11232,11235],{"className":11231},[83],[39,11233,4295],{"className":11234},[83,84],[39,11236,11238],{"className":11237},[192],[39,11239,11241],{"className":11240},[196],[39,11242,11244],{"className":11243},[200],[39,11245,11247],{"className":11246,"style":4421},[204],[39,11248,11249,11252],{"style":1027},[39,11250],{"className":11251,"style":213},[212],[39,11253,11255],{"className":11254},[217,218,219,220],[39,11256,2098],{"className":11257,"style":2114},[83,84,220],[39,11259,136],{"className":11260},[182],[39,11262,142],{"className":11263},[83,84],[39,11265,155],{"className":11266},[252],[39,11268],{"className":11269,"style":256},[230],[39,11271,158],{"className":11272},[83,84],[39,11274,161],{"className":11275},[263],[39,11277,11278,11281],{"style":11104},[39,11279],{"className":11280,"style":3311},[212],[39,11282,11284,11287,11290,11293,11296,11343,11346,11349,11352,11355,11358,11361,11364,11367,11370,11373,11449,11452,11455,11458,11490,11493,11496,11499,11502,11505,11508,11511,11514,11517,11664,11667],{"className":11283},[83],[39,11285],{"className":11286},[83],[39,11288],{"className":11289,"style":231},[230],[39,11291,793],{"className":11292},[235],[39,11294],{"className":11295,"style":231},[230],[39,11297,11299],{"className":11298},[1744,3294],[39,11300,11302,11335],{"className":11301},[196,649],[39,11303,11305,11332],{"className":11304},[200],[39,11306,11308,11322],{"className":11307,"style":6670},[204],[39,11309,11310,11313],{"style":6673},[39,11311],{"className":11312,"style":3311},[212],[39,11314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tyle":2114},[83,84,220],[39,11624,136],{"className":11625},[182],[39,11627,11629,11632],{"className":11628},[83],[39,11630,142],{"className":11631},[83,84],[39,11633,11635],{"className":11634},[192],[39,11636,11638],{"className":11637},[196],[39,11639,11641],{"className":11640},[200],[39,11642,11644],{"className":11643,"style":1024},[204],[39,11645,11646,11649],{"style":1027},[39,11647],{"className":11648,"style":213},[212],[39,11650,11652],{"className":11651},[217,218,219,220],[39,11653,11655],{"className":11654},[83,220],[39,11656,147],{"className":11657},[83,220],[39,11659,161],{"className":11660},[263],[39,11662,4280],{"className":11663,"style":4592},[263,4591],[39,11665],{"className":11666,"style":256},[230],[39,11668,2903],{"className":11669},[83],[39,11671,673],{"className":11672},[672],[39,11674,11676],{"className":11675},[200],[39,11677,11679],{"className":11678,"style":11120},[204],[39,11680],{},[11,11682,11683],{},"证毕。",[11,11685,11686,11687,11745],{},"这个方程的意义在于，它把一个涉及无限期回报的期望，转化为一个只涉及当前奖励与下一状态值函数的方程。方程右端仍然含有未知函数 ",[39,11688,11690,11707],{"className":11689,"translate":43},[42],[39,11691,11693],{"className":11692},[47],[49,11694,11695],{"xmlns":51},[53,11696,11697,11705],{},[56,11698,11699],{},[138,11700,11701,11703],{},[59,11702,4230],{},[59,11704,2098],{},[63,11706,6382],{"encoding":65},[39,11708,11710],{"className":11709,"ariaHidden":70},[69],[39,11711,11713,11716],{"className":11712},[74],[39,11714],{"className":11715,"style":79},[78],[39,11717,11719,11722],{"className":11718},[83],[39,11720,4230],{"className":11721,"style":461},[83,84],[39,11723,11725],{"className":11724},[192],[39,11726,11728],{"className":11727},[196],[39,11729,11731],{"className":11730},[200],[39,11732,11734],{"className":11733,"style":6410},[204],[39,11735,11736,11739],{"style":208},[39,11737],{"className":11738,"style":213},[212],[39,11740,11742],{"className":11741},[217,218,219,220],[39,11743,2098],{"className":11744,"style":2114},[83,84,220],"，因此它不是普通的递推式，而是一个泛函方程。",[26,11747,11749],{"id":11748},"_22-算子形式","2.2 算子形式",[11,11751,11752,11753,2083,11756,2506],{},"为了把上述方程写成更紧凑的形式，定义",[1489,11754,11755],{},"贝尔曼期望算子",[39,11757,11759,11778],{"className":11758,"translate":43},[42],[39,11760,11762],{"className":11761},[47],[49,11763,11764],{"xmlns":51},[53,11765,11766,11775],{},[56,11767,11768],{},[138,11769,11770,11773],{},[59,11771,11772],{"mathvariant":617},"T",[59,11774,2098],{},[63,11776,11777],{"encoding":65},"\\mathcal{T}^\\pi",[39,11779,11781],{"className":11780,"ariaHidden":70},[69],[39,11782,11784,11787],{"className":11783},[74],[39,11785],{"className":11786,"style":79},[78],[39,11788,11790,11794],{"className":11789},[83],[39,11791,11772],{"className":11792,"style":11793},[83,641],"margin-right:0.2542em;",[39,11795,11797],{"className":11796},[192],[39,11798,11800],{"className":11799},[196],[39,11801,11803],{"className":11802},[200],[39,11804,11806],{"className":11805,"style":6410},[204],[39,11807,11808,11811],{"style":208},[39,11809],{"className":11810,"style":213},[212],[39,11812,11814],{"className":11813},[217,218,219,220],[39,11815,2098],{"className":11816,"style":2114},[83,84,220],[39,11818,11820],{"className":11819,"translate":43},[746],[39,11821,11823,11983],{"className":11822,"translate":43},[42],[39,11824,11826],{"className":11825},[47],[49,11827,11828],{"xmlns":51,"display":755},[53,11829,11830,11980],{},[759,11831,11832,11916],{"rowspacing":761,"columnalign":762,"columnspacing":146},[764,11833,11834,11858],{},[767,11835,11836],{},[770,11837,11838],{"scriptlevel":541,"displaystyle":70},[56,11839,11840,11842,11848,11850,11852,11854,11856],{},[133,11841,136],{"stretchy":135},[138,11843,11844,11846],{},[59,11845,11772],{"mathvariant":617},[59,11847,2098],{},[59,11849,4230],{},[133,11851,161],{"stretchy":135},[133,11853,136],{"stretchy":135},[59,11855,142],{},[133,11857,161],{"stretchy":135},[767,11859,11860],{},[770,11861,11862],{"scriptlevel":541,"displaystyle":70},[56,11863,11864,11866,11868,11874,11876,11878,11880,11882,11884,11886,11896,11898,11900,11906,11908,11910,11912,11914],{},[56,11865],{},[133,11867,793],{},[6556,11869,11870,11872],{},[133,11871,3191],{},[59,11873,158],{},[59,11875,2098],{},[133,11877,136],{"stretchy":135},[59,11879,158],{},[133,11881,150],{},[59,11883,142],{},[133,11885,161],{"stretchy":135},[6556,11887,11888,11890],{},[133,11889,3191],{},[138,11891,11892,11894],{},[59,11893,142],{},[133,11895,147],{"mathvariant":145,"lspace":146,"rspace":146},[59,11897,131],{},[133,11899,136],{"stretchy":135},[138,11901,11902,11904],{},[59,11903,142],{},[133,11905,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,11907,150],{},[59,11909,142],{},[133,11911,155],{"separator":70},[59,11913,158],{},[133,11915,161],{"stretchy":135},[764,11917,11918,11924],{},[767,11919,11920],{},[770,11921,11922],{"scriptlevel":541,"displaystyle":70},[56,11923],{},[767,11925,11926],{},[770,11927,11928],{"scriptlevel":541,"displaystyle":70},[56,11929,11930,11932,11934,11936,11978],{},[56,11931],{},[230,11933],{"width":6891},[133,11935,409],{},[56,11937,11938,11940,11942,11944,11946,11948,11950,11952,11958,11960,11962,11964,11966,11968,11974,11976],{},[133,11939,537],{"fence":70},[59,11941,401],{},[133,11943,136],{"stretchy":135},[59,11945,142],{},[133,11947,155],{"separator":70},[59,11949,158],{},[133,11951,155],{"separator":70},[138,11953,11954,11956],{},[59,11955,142],{},[133,11957,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,11959,161],{"stretchy":135},[133,11961,788],{},[59,11963,531],{},[59,11965,4230],{},[133,11967,136],{"stretchy":135},[138,11969,11970,11972],{},[59,11971,142],{},[133,11973,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,11975,161],{"stretchy":135},[133,11977,4280],{"fence":70},[59,11979,2903],{"mathvariant":145},[63,11981,11982],{"encoding":65},"\\begin{aligned}\n(\\mathcal{T}^\\pi V)(s)\n&=\\sum_{a}\\pi(a\\mid s)\\sum_{s'}P(s'\\mid s,a)\\\\\n&\\quad\\times\\left[r(s,a,s')+\\gamma V(s')\\right].\n\\end{aligned}",[39,11984,11986],{"className":11985,"ariaHidden":70},[69],[39,11987,11989,11992],{"className":11988},[74],[39,11990],{"className":11991,"style":6954},[78],[39,11993,11995],{"className":11994},[83],[39,11996,11998,12084],{"className":11997},[759],[39,11999,12001],{"className":12000},[918],[39,12002,12004,12076],{"className":12003},[196,649],[39,12005,12007,12073],{"className":12006},[200],[39,12008,12010,12065],{"className":12009,"style":6973},[204],[39,12011,12012,12015],{"style":6976},[39,12013],{"className":12014,"style":3311},[212],[39,12016,12018,12021,12050,12053,12056,12059,12062],{"className":12017},[83],[39,12019,136],{"className":12020},[182],[39,12022,12024,12027],{"className":12023},[83],[39,12025,11772],{"className":12026,"style":11793},[83,641],[39,12028,12030],{"className":12029},[192],[39,12031,12033],{"className":12032},[196],[39,12034,12036],{"className":12035},[200],[39,12037,12039],{"className":12038,"style":4421},[204],[39,12040,12041,12044],{"style":1027},[39,12042],{"className":12043,"style":213},[212],[39,12045,12047],{"className":12046},[217,218,219,220],[39,12048,2098],{"className":12049,"style":2114},[83,84,220],[39,12051,4230],{"className":12052,"style":461},[83,84],[39,12054,161],{"className":12055},[263],[39,12057,136],{"className":12058},[182],[39,12060,142],{"className":12061},[83,84],[39,12063,161],{"className":12064},[263],[39,12066,12067,12070],{"style":7032},[39,12068],{"className":12069,"style":3311},[212],[39,12071],{"className":12072},[83],[39,12074,673],{"className":12075},[672],[39,12077,12079],{"className":12078},[200],[39,12080,12082],{"className":12081,"style":7048},[204],[39,12083],{},[39,12085,12087],{"className":12086},[1223],[39,12088,12090,12487],{"className":12089},[196,649],[39,12091,12093,12484],{"className":12092},[200],[39,12094,12096,12334],{"className":12095,"style":6973},[204],[39,12097,12098,12101],{"style":6976},[39,12099],{"className":12100,"style":3311},[212],[39,12102,12104,12107,12110,12113,12116,12163,12166,12169,12172,12175,12178,12181,12184,12187,12190,12193,12269,12272,12275,12278,12310,12313,12316,12319,12322,12325,12328,12331],{"className":12103},[83],[39,12105],{"className":12106},[83],[39,12108],{"className":12109,"style":231},[230],[39,12111,793],{"className":12112},[235],[39,12114],{"className":12115,"style":231},[230],[39,12117,12119],{"className":12118},[1744,3294],[39,12120,12122,12155],{"className":12121},[196,649],[39,12123,12125,12152],{"className":12124},[200],[39,12126,12128,12142],{"className":12127,"style":6670},[204],[39,12129,12130,12133],{"style":6673},[39,12131],{"className":12132,"style":3311},[212],[39,12134,12136],{"className":12135},[217,218,219,220],[39,12137,12139],{"className":12138},[83,220],[39,12140,158],{"className":12141},[83,84,220],[39,12143,12144,12147],{"style":3330},[39,12145],{"className":12146,"style":3311},[212],[39,12148,12149],{},[39,12150,3191],{"className":12151},[1744,3339,3340],[39,12153,673],{"className":12154},[672],[39,12156,12158],{"className":12157},[200],[39,12159,12161],{"className":12160,"style":6705},[204],[39,12162],{},[39,12164],{"className":12165,"style":256},[230],[39,12167,2098],{"className":12168,"style":2114},[83,84],[39,12170,136],{"className":12171},[182],[39,12173,158],{"className":12174},[83,84],[39,12176],{"className":12177,"style":231},[230],[39,12179,150],{"className":12180},[235],[39,12182],{"className":12183,"style":231},[230],[39,12185,142],{"className":12186},[83,84],[39,12188,161],{"className":12189},[263],[39,12191],{"className":12192,"style":256},[230],[39,12194,12196],{"className":12195},[1744,3294],[39,12197,12199,12261],{"className":12198},[196,649],[39,12200,12202,12258],{"className":12201},[200],[39,12203,12205,12248],{"className":12204,"style":6670},[204],[39,12206,12207,12210],{"style":7097},[39,12208],{"className":12209,"style":3311},[212],[39,12211,12213],{"className":12212},[217,218,219,220],[39,12214,12216],{"className":12215},[83,220],[39,12217,12219,12222],{"className":12218},[83,220],[39,12220,142],{"className":12221},[83,84,220],[39,12223,12225],{"className":12224},[192],[39,12226,12228],{"className":12227},[196],[39,12229,12231],{"className":12230},[200],[39,12232,12234],{"className":12233,"style":3997},[204],[39,12235,12236,12239],{"style":4000},[39,12237],{"className":12238,"style":4004},[212],[39,12240,12242],{"className":12241},[217,4008,4009,220],[39,12243,12245],{"className":12244},[83,220],[39,12246,147],{"className":12247},[83,220],[39,12249,12250,12253],{"style":3330},[39,12251],{"className":12252,"style":3311},[212],[39,12254,12255],{},[39,12256,3191],{"className":12257},[1744,3339,3340],[39,12259,673],{"className":12260},[672],[39,12262,12264],{"className":12263},[200],[39,12265,12267],{"className":12266,"style":7158},[204],[39,12268],{},[39,12270],{"className":12271,"style":256},[230],[39,12273,131],{"className":12274,"style":178},[83,84],[39,12276,136],{"className":12277},[182],[39,12279,12281,12284],{"className":12280},[83],[39,12282,142],{"className":12283},[83,84],[39,12285,12287],{"className":12286},[192],[39,12288,12290],{"className":12289},[196],[39,12291,12293],{"className":12292},[200],[39,12294,12296],{"className":12295,"style":1024},[204],[39,12297,12298,12301],{"style":1027},[39,12299],{"className":12300,"style":213},[212],[39,12302,12304],{"className":12303},[217,218,219,220],[39,12305,12307],{"className":12306},[83,220],[39,12308,147],{"className":12309},[83,220],[39,12311],{"className":12312,"style":231},[230],[39,12314,150],{"className":12315},[235],[39,12317],{"className":12318,"style":231},[230],[39,12320,142],{"className":12321},[83,84],[39,12323,155],{"className":12324},[252],[39,12326],{"className":12327,"style":256},[230],[39,12329,158],{"className":12330},[83,84],[39,12332,161],{"className":12333},[263],[39,12335,12336,12339],{"style":7032},[39,12337],{"className":12338,"style":3311},[212],[39,12340,12342,12345,12348,12351,12354,12357,12478,12481],{"className":12341},[83],[39,12343],{"className":12344},[83],[39,12346],{"className":12347,"style":7240},[230],[39,12349],{"className":12350,"style":461},[230],[39,12352,409],{"className":12353},[465],[39,12355],{"className":12356,"style":461},[230],[39,12358,12360,12363,12366,12369,12372,12375,12378,12381,12384,12387,12419,12422,12425,12428,12431,12434,12437,12440,12472,12475],{"className":12359},[2842],[39,12361,537],{"className":12362,"style":4592},[182,4591],[39,12364,401],{"className":12365,"style":438},[83,84],[39,12367,136],{"className":12368},[182],[39,12370,142],{"className":12371},[83,84],[39,12373,155],{"className":12374},[252],[39,12376],{"className":12377,"style":256},[230],[39,12379,158],{"className":12380},[83,84],[39,12382,155],{"className":12383},[252],[39,12385],{"className":12386,"style":256},[230],[39,12388,12390,12393],{"className":12389},[83],[39,12391,142],{"className":12392},[83,84],[39,12394,12396],{"className":12395},[192],[39,12397,12399],{"className":12398},[196],[39,12400,12402],{"className":12401},[200],[39,12403,12405],{"className":12404,"style":1024},[204],[39,12406,12407,12410],{"style":1027},[39,12408],{"className":12409,"style":213},[212],[39,12411,12413],{"className":12412},[217,218,219,220],[39,12414,12416],{"className":12415},[83,220],[39,12417,147],{"className":12418},[83,220],[39,12420,161],{"className":12421},[263],[39,12423],{"className":12424,"style":461},[230],[39,12426,788],{"className":12427},[465],[39,12429],{"className":12430,"style":461},[230],[39,12432,531],{"className":12433,"style":565},[83,84],[39,12435,4230],{"className":12436,"style":461},[83,84],[39,12438,136],{"className":12439},[182],[39,12441,12443,12446],{"className":12442},[83],[39,12444,142],{"className":12445},[83,84],[39,12447,12449],{"className":12448},[192],[39,12450,12452],{"className":12451},[196],[39,12453,12455],{"className":12454},[200],[39,12456,12458],{"className":12457,"style":1024},[204],[39,12459,12460,12463],{"style":1027},[39,12461],{"className":12462,"style":213},[212],[39,12464,12466],{"className":12465},[217,218,219,220],[39,12467,12469],{"className":12468},[83,220],[39,12470,147],{"className":12471},[83,220],[39,12473,161],{"className":12474},[263],[39,12476,4280],{"className":12477,"style":4592},[263,4591],[39,12479],{"className":12480,"style":256},[230],[39,12482,2903],{"className":12483},[83],[39,12485,673],{"className":12486},[672],[39,12488,12490],{"className":12489},[200],[39,12491,12493],{"className":12492,"style":7048},[204],[39,12494],{},[11,12496,12497],{},"于是贝尔曼期望方程可以写成不动点形式",[39,12499,12501],{"className":12500,"translate":43},[746],[39,12502,12504,12538],{"className":12503,"translate":43},[42],[39,12505,12507],{"className":12506},[47],[49,12508,12509],{"xmlns":51,"display":755},[53,12510,12511,12535],{},[56,12512,12513,12519,12521,12527,12533],{},[138,12514,12515,12517],{},[59,12516,4230],{},[59,12518,2098],{},[133,12520,793],{},[138,12522,12523,12525],{},[59,12524,11772],{"mathvariant":617},[59,12526,2098],{},[138,12528,12529,12531],{},[59,12530,4230],{},[59,12532,2098],{},[59,12534,2903],{"mathvariant":145},[63,12536,12537],{"encoding":65},"V^\\pi=\\mathcal{T}^\\pi V^\\pi.",[39,12539,12541,12585],{"className":12540,"ariaHidden":70},[69],[39,12542,12544,12547,12576,12579,12582],{"className":12543},[74],[39,12545],{"className":12546,"style":4421},[78],[39,12548,12550,12553],{"className":12549},[83],[39,12551,4230],{"className":12552,"style":461},[83,84],[39,12554,12556],{"className":12555},[192],[39,12557,12559],{"className":12558},[196],[39,12560,12562],{"className":12561},[200],[39,12563,12565],{"className":12564,"style":4421},[204],[39,12566,12567,12570],{"style":1027},[39,12568],{"className":12569,"style":213},[212],[39,12571,12573],{"className":12572},[217,218,219,220],[39,12574,2098],{"className":12575,"style":2114},[83,84,220],[39,12577],{"className":12578,"style":231},[230],[39,12580,793],{"className":12581},[235],[39,12583],{"className":12584,"style":231},[230],[39,12586,12588,12591,12620,12649],{"className":12587},[74],[39,12589],{"className":12590,"style":4421},[78],[39,12592,12594,12597],{"className":12593},[83],[39,12595,11772],{"className":12596,"style":11793},[83,641],[39,12598,12600],{"className":12599},[192],[39,12601,12603],{"className":12602},[196],[39,12604,12606],{"className":12605},[200],[39,12607,12609],{"className":12608,"style":4421},[204],[39,12610,12611,12614],{"style":1027},[39,12612],{"className":12613,"style":213},[212],[39,12615,12617],{"className":12616},[217,218,219,220],[39,12618,2098],{"className":12619,"style":2114},[83,84,220],[39,12621,12623,12626],{"className":12622},[83],[39,12624,4230],{"className":12625,"style":461},[83,84],[39,12627,12629],{"className":12628},[192],[39,12630,12632],{"className":12631},[196],[39,12633,12635],{"className":12634},[200],[39,12636,12638],{"className":12637,"style":4421},[204],[39,12639,12640,12643],{"style":1027},[39,12641],{"className":12642,"style":213},[212],[39,12644,12646],{"className":12645},[217,218,219,220],[39,12647,2098],{"className":12648,"style":2114},[83,84,220],[39,12650,2903],{"className":12651},[83],[11,12653,12654,12655,12713,12714,12772,12773,12801,12802,12866,12867,12925],{},"这里的 ",[39,12656,12658,12675],{"className":12657,"translate":43},[42],[39,12659,12661],{"className":12660},[47],[49,12662,12663],{"xmlns":51},[53,12664,12665,12673],{},[56,12666,12667],{},[138,12668,12669,12671],{},[59,12670,4230],{},[59,12672,2098],{},[63,12674,6382],{"encoding":65},[39,12676,12678],{"className":12677,"ariaHidden":70},[69],[39,12679,12681,12684],{"className":12680},[74],[39,12682],{"className":12683,"style":79},[78],[39,12685,12687,12690],{"className":12686},[83],[39,12688,4230],{"className":12689,"style":461},[83,84],[39,12691,12693],{"className":12692},[192],[39,12694,12696],{"className":12695},[196],[39,12697,12699],{"className":12698},[200],[39,12700,12702],{"className":12701,"style":6410},[204],[39,12703,12704,12707],{"style":208},[39,12705],{"className":12706,"style":213},[212],[39,12708,12710],{"className":12709},[217,218,219,220],[39,12711,2098],{"className":12712,"style":2114},[83,84,220]," 不是某个数，而是函数空间中的一个点。算子 ",[39,12715,12717,12734],{"className":12716,"translate":43},[42],[39,12718,12720],{"className":12719},[47],[49,12721,12722],{"xmlns":51},[53,12723,12724,12732],{},[56,12725,12726],{},[138,12727,12728,12730],{},[59,12729,11772],{"mathvariant":617},[59,12731,2098],{},[63,12733,11777],{"encoding":65},[39,12735,12737],{"className":12736,"ariaHidden":70},[69],[39,12738,12740,12743],{"className":12739},[74],[39,12741],{"className":12742,"style":79},[78],[39,12744,12746,12749],{"className":12745},[83],[39,12747,11772],{"className":12748,"style":11793},[83,641],[39,12750,12752],{"className":12751},[192],[39,12753,12755],{"className":12754},[196],[39,12756,12758],{"className":12757},[200],[39,12759,12761],{"className":12760,"style":6410},[204],[39,12762,12763,12766],{"style":208},[39,12764],{"className":12765,"style":213},[212],[39,12767,12769],{"className":12768},[217,218,219,220],[39,12770,2098],{"className":12771,"style":2114},[83,84,220]," 把一个有界函数 ",[39,12774,12776,12789],{"className":12775,"translate":43},[42],[39,12777,12779],{"className":12778},[47],[49,12780,12781],{"xmlns":51},[53,12782,12783,12787],{},[56,12784,12785],{},[59,12786,4230],{},[63,12788,4230],{"encoding":65},[39,12790,12792],{"className":12791,"ariaHidden":70},[69],[39,12793,12795,12798],{"className":12794},[74],[39,12796],{"className":12797,"style":79},[78],[39,12799,4230],{"className":12800,"style":461},[83,84]," 映射为另一个有界函数 ",[39,12803,12805,12825],{"className":12804,"translate":43},[42],[39,12806,12808],{"className":12807},[47],[49,12809,12810],{"xmlns":51},[53,12811,12812,12822],{},[56,12813,12814,12820],{},[138,12815,12816,12818],{},[59,12817,11772],{"mathvariant":617},[59,12819,2098],{},[59,12821,4230],{},[63,12823,12824],{"encoding":65},"\\mathcal{T}^\\pi V",[39,12826,12828],{"className":12827,"ariaHidden":70},[69],[39,12829,12831,12834,12863],{"className":12830},[74],[39,12832],{"className":12833,"style":79},[78],[39,12835,12837,12840],{"className":12836},[83],[39,12838,11772],{"className":12839,"style":11793},[83,641],[39,12841,12843],{"className":12842},[192],[39,12844,12846],{"className":12845},[196],[39,12847,12849],{"className":12848},[200],[39,12850,12852],{"className":12851,"style":6410},[204],[39,12853,12854,12857],{"style":208},[39,12855],{"className":12856,"style":213},[212],[39,12858,12860],{"className":12859},[217,218,219,220],[39,12861,2098],{"className":12862,"style":2114},[83,84,220],[39,12864,4230],{"className":12865,"style":461},[83,84],"。因此，求解贝尔曼方程等价于寻找算子 ",[39,12868,12870,12887],{"className":12869,"translate":43},[42],[39,12871,12873],{"className":12872},[47],[49,12874,12875],{"xmlns":51},[53,12876,12877,12885],{},[56,12878,12879],{},[138,12880,12881,12883],{},[59,12882,11772],{"mathvariant":617},[59,12884,2098],{},[63,12886,11777],{"encoding":65},[39,12888,12890],{"className":12889,"ariaHidden":70},[69],[39,12891,12893,12896],{"className":12892},[74],[39,12894],{"className":12895,"style":79},[78],[39,12897,12899,12902],{"className":12898},[83],[39,12900,11772],{"className":12901,"style":11793},[83,641],[39,12903,12905],{"className":12904},[192],[39,12906,12908],{"className":12907},[196],[39,12909,12911],{"className":12910},[200],[39,12912,12914],{"className":12913,"style":6410},[204],[39,12915,12916,12919],{"style":208},[39,12917],{"className":12918,"style":213},[212],[39,12920,12922],{"className":12921},[217,218,219,220],[39,12923,2098],{"className":12924,"style":2114},[83,84,220]," 的不动点。",[26,12927,12929],{"id":12928},"_23-有限状态下的矩阵解","2.3 有限状态下的矩阵解",[11,12931,12932,12933,13026,13027,1639,13109,13221,13222,13349],{},"当 ",[39,12934,12936,12969],{"className":12935,"translate":43},[42],[39,12937,12939],{"className":12938},[47],[49,12940,12941],{"xmlns":51},[53,12942,12943,12966],{},[56,12944,12945,12947,12949,12952,12954,12956,12958,12960,12963],{},[59,12946,61],{},[133,12948,793],{},[133,12950,12951],{"stretchy":135},"{",[539,12953,546],{},[133,12955,155],{"separator":70},[133,12957,2641],{},[133,12959,155],{"separator":70},[59,12961,12962],{},"n",[133,12964,12965],{"stretchy":135},"}",[63,12967,12968],{"encoding":65},"S=\\{1,\\dots,n\\}",[39,12970,12972,12990],{"className":12971,"ariaHidden":70},[69],[39,12973,12975,12978,12981,12984,12987],{"className":12974},[74],[39,12976],{"className":12977,"style":79},[78],[39,12979,61],{"className":12980,"style":85},[83,84],[39,12982],{"className":12983,"style":231},[230],[39,12985,793],{"className":12986},[235],[39,12988],{"className":12989,"style":231},[230],[39,12991,12993,12996,12999,13002,13005,13008,13011,13014,13017,13020,13023],{"className":12992},[74],[39,12994],{"className":12995,"style":245},[78],[39,12997,12951],{"className":12998},[182],[39,13000,546],{"className":13001},[83],[39,13003,155],{"className":13004},[252],[39,13006],{"className":13007,"style":256},[230],[39,13009,2641],{"className":13010},[2842],[39,13012],{"className":13013,"style":256},[230],[39,13015,155],{"className":13016},[252],[39,13018],{"className":13019,"style":256},[230],[39,13021,12962],{"className":13022},[83,84],[39,13024,12965],{"className":13025},[263]," 为有限集时，贝尔曼期望方程可以写成线性方程组。定义向量 ",[39,13028,13030,13052],{"className":13029,"translate":43},[42],[39,13031,13033],{"className":13032},[47],[49,13034,13035],{"xmlns":51},[53,13036,13037,13049],{},[56,13038,13039,13041,13043],{},[59,13040,4230],{},[133,13042,534],{},[138,13044,13045,13047],{},[59,13046,422],{"mathvariant":421},[59,13048,12962],{},[63,13050,13051],{"encoding":65},"V\\in\\mathbb{R}^n",[39,13053,13055,13074],{"className":13054,"ariaHidden":70},[69],[39,13056,13058,13062,13065,13068,13071],{"className":13057},[74],[39,13059],{"className":13060,"style":13061},[78],"height:0.7224em;vertical-align:-0.0391em;",[39,13063,4230],{"className":13064,"style":461},[83,84],[39,13066],{"className":13067,"style":231},[230],[39,13069,534],{"className":13070},[235],[39,13072],{"className":13073,"style":231},[230],[39,13075,13077,13080],{"className":13076},[74],[39,13078],{"className":13079,"style":511},[78],[39,13081,13083,13086],{"className":13082},[83],[39,13084,422],{"className":13085},[83,515],[39,13087,13089],{"className":13088},[192],[39,13090,13092],{"className":13091},[196],[39,13093,13095],{"className":13094},[200],[39,13096,13098],{"className":13097,"style":6410},[204],[39,13099,13100,13103],{"style":208},[39,13101],{"className":13102,"style":213},[212],[39,13104,13106],{"className":13105},[217,218,219,220],[39,13107,12962],{"className":13108},[83,84,220],[39,13110,13112,13138],{"className":13111,"translate":43},[42],[39,13113,13115],{"className":13114},[47],[49,13116,13117],{"xmlns":51},[53,13118,13119,13135],{},[56,13120,13121,13127,13129],{},[138,13122,13123,13125],{},[59,13124,401],{},[59,13126,2098],{},[133,13128,534],{},[138,13130,13131,13133],{},[59,13132,422],{"mathvariant":421},[59,13134,12962],{},[63,13136,13137],{"encoding":65},"r^\\pi\\in\\mathbb{R}^n",[39,13139,13141,13186],{"className":13140,"ariaHidden":70},[69],[39,13142,13144,13148,13177,13180,13183],{"className":13143},[74],[39,13145],{"className":13146,"style":13147},[78],"height:0.7035em;vertical-align:-0.0391em;",[39,13149,13151,13154],{"className":13150},[83],[39,13152,401],{"className":13153,"style":438},[83,84],[39,13155,13157],{"className":13156},[192],[39,13158,13160],{"className":13159},[196],[39,13161,13163],{"className":13162},[200],[39,13164,13166],{"className":13165,"style":6410},[204],[39,13167,13168,13171],{"style":208},[39,13169],{"className":13170,"style":213},[212],[39,13172,13174],{"className":13173},[217,218,219,220],[39,13175,2098],{"className":13176,"style":2114},[83,84,220],[39,13178],{"className":13179,"style":231},[230],[39,13181,534],{"className":13182},[235],[39,13184],{"className":13185,"style":231},[230],[39,13187,13189,13192],{"className":13188},[74],[39,13190],{"className":13191,"style":511},[78],[39,13193,13195,13198],{"className":13194},[83],[39,13196,422],{"className":13197},[83,515],[39,13199,13201],{"className":13200},[192],[39,13202,13204],{"className":13203},[196],[39,13205,13207],{"className":13206},[200],[39,13208,13210],{"className":13209,"style":6410},[204],[39,13211,13212,13215],{"style":208},[39,13213],{"className":13214,"style":213},[212],[39,13216,13218],{"className":13217},[217,218,219,220],[39,13219,12962],{"className":13220},[83,84,220],"，以及矩阵 ",[39,13223,13225,13257],{"className":13224,"translate":43},[42],[39,13226,13228],{"className":13227},[47],[49,13229,13230],{"xmlns":51},[53,13231,13232,13254],{},[56,13233,13234,13240,13242],{},[138,13235,13236,13238],{},[59,13237,131],{},[59,13239,2098],{},[133,13241,534],{},[138,13243,13244,13246],{},[59,13245,422],{"mathvariant":421},[56,13247,13248,13250,13252],{},[59,13249,12962],{},[133,13251,409],{},[59,13253,12962],{},[63,13255,13256],{"encoding":65},"P^\\pi\\in\\mathbb{R}^{n\\times 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i)\\sum_{j}P(j\\mid i,a)r(i,a,j),\\\\\nP^\\pi_{ij}&=\\sum_{a}\\pi(a\\mid i)P(j\\mid 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P^\\pi)V^\\pi=r^\\pi.",[39,14406,14408,14430,14509],{"className":14407,"ariaHidden":70},[69],[39,14409,14411,14414,14417,14421,14424,14427],{"className":14410},[74],[39,14412],{"className":14413,"style":245},[78],[39,14415,136],{"className":14416},[182],[39,14418,14373],{"className":14419,"style":14420},[83,84],"margin-right:0.0785em;",[39,14422],{"className":14423,"style":461},[230],[39,14425,5378],{"className":14426},[465],[39,14428],{"className":14429,"style":461},[230],[39,14431,14433,14436,14439,14468,14471,14500,14503,14506],{"className":14432},[74],[39,14434],{"className":14435,"style":245},[78],[39,14437,531],{"className":14438,"style":565},[83,84],[39,14440,14442,14445],{"className":14441},[83],[39,14443,131],{"className":14444,"style":178},[83,84],[39,14446,14448],{"className":14447},[192],[39,14449,14451],{"className":14450},[196],[39,14452,14454],{"className":14453},[200],[39,14455,14457],{"className":14456,"style":4421},[204],[39,14458,14459,14462],{"style":1027},[39,14460],{"className":14461,"style":213},[212],[39,14463,14465],{"className":14464},[217,218,219,220],[39,14466,2098],{"className":14467,"style":2114},[83,84,220],[39,14469,161],{"className":14470},[263],[39,14472,14474,14477],{"className":14473},[83],[39,14475,4230],{"className":14476,"style":461},[83,84],[39,14478,14480],{"className":14479},[192],[39,14481,14483],{"className":14482},[196],[39,14484,14486],{"className":14485},[200],[39,14487,14489],{"className":14488,"style":4421},[204],[39,14490,14491,14494],{"style":1027},[39,14492],{"className":14493,"style":213},[212],[39,14495,14497],{"className":14496},[217,218,219,220],[39,14498,2098],{"className":14499,"style":2114},[83,84,220],[39,14501],{"className":14502,"style":231},[230],[39,14504,793],{"className":14505},[235],[39,14507],{"className":14508,"style":231},[230],[39,14510,14512,14515,14544],{"className":14511},[74],[39,14513],{"className":14514,"style":4421},[78],[39,14516,14518,14521],{"className":14517},[83],[39,14519,401],{"className":14520,"style":438},[83,84],[39,14522,14524],{"className":14523},[192],[39,14525,14527],{"className":14526},[196],[39,14528,14530],{"className":14529},[200],[39,14531,14533],{"className":14532,"style":4421},[204],[39,14534,14535,14538],{"style":1027},[39,14536],{"className":14537,"style":213},[212],[39,14539,14541],{"className":14540},[217,218,219,220],[39,14542,2098],{"className":14543,"style":2114},[83,84,220],[39,14545,2903],{"className":14546},[83],[31,14548,14551,14715],{"id":14549,"title":14550},"proposition-matrix-solution","命题 2.2（矩阵解与 Neumann 级数）",[11,14552,37,14553,14627,14628,14714],{},[39,14554,14556,14578],{"className":14555,"translate":43},[42],[39,14557,14559],{"className":14558},[47],[49,14560,14561],{"xmlns":51},[53,14562,14563,14575],{},[56,14564,14565,14567,14569,14571,14573],{},[539,14566,541],{},[133,14568,1824],{},[59,14570,531],{},[133,14572,1693],{},[539,14574,546],{},[63,14576,14577],{"encoding":65},"0\\le\\gamma\u003C1",[39,14579,14581,14600,14618],{"className":14580,"ariaHidden":70},[69],[39,14582,14584,14588,14591,14594,14597],{"className":14583},[74],[39,14585],{"className":14586,"style":14587},[78],"height:0.7804em;vertical-align:-0.136em;",[39,14589,541],{"className":14590},[83],[39,14592],{"className":14593,"style":231},[230],[39,14595,1824],{"className":14596},[235],[39,14598],{"className":14599,"style":231},[230],[39,14601,14603,14606,14609,14612,14615],{"className":14602},[74],[39,14604],{"className":14605,"style":561},[78],[39,14607,531],{"className":14608,"style":565},[83,84],[39,14610],{"className":14611,"style":231},[230],[39,14613,1693],{"className":14614},[235],[39,14616],{"className":14617,"style":231},[230],[39,14619,14621,14624],{"className":14620},[74],[39,14622],{"className":14623,"style":2473},[78],[39,14625,546],{"className":14626},[83],"。则 ",[39,14629,14631,14655],{"className":14630,"translate":43},[42],[39,14632,14634],{"className":14633},[47],[49,14635,14636],{"xmlns":51},[53,14637,14638,14652],{},[56,14639,14640,14642,14644,14646],{},[59,14641,14373],{},[133,14643,5378],{},[59,14645,531],{},[138,14647,14648,14650],{},[59,14649,131],{},[59,14651,2098],{},[63,14653,14654],{"encoding":65},"I-\\gamma P^\\pi",[39,14656,14658,14676],{"className":14657,"ariaHidden":70},[69],[39,14659,14661,14664,14667,14670,14673],{"className":14660},[74],[39,14662],{"className":14663,"style":454},[78],[39,14665,14373],{"className":14666,"style":14420},[83,84],[39,14668],{"className":14669,"style":461},[230],[39,14671,5378],{"className":14672},[465],[39,14674],{"className":14675,"style":461},[230],[39,14677,14679,14682,14685],{"className":14678},[74],[39,14680],{"className":14681,"style":6452},[78],[39,14683,531],{"className":14684,"style":565},[83,84],[39,14686,14688,14691],{"className":14687},[83],[39,14689,131],{"className":14690,"style":178},[83,84],[39,14692,14694],{"className":14693},[192],[39,14695,14697],{"className":14696},[196],[39,14698,14700],{"className":14699},[200],[39,14701,14703],{"className":14702,"style":6410},[204],[39,14704,14705,14708],{"style":208},[39,14706],{"className":14707,"style":213},[212],[39,14709,14711],{"className":14710},[217,218,219,220],[39,14712,2098],{"className":14713,"style":2114},[83,84,220]," 可逆，且",[39,14716,14718],{"className":14717,"translate":43},[746],[39,14719,14721,14815],{"className":14720,"translate":43},[42],[39,14722,14724],{"className":14723},[47],[49,14725,14726],{"xmlns":51,"display":755},[53,14727,14728,14812],{},[56,14729,14730,14736,14738,14740,14742,14744,14746,14752,14762,14768,14770,14784,14790,14792,14798,14804,14810],{},[138,14731,14732,14734],{},[59,14733,4230],{},[59,14735,2098],{},[133,14737,793],{},[133,14739,136],{"stretchy":135},[59,14741,14373],{},[133,14743,5378],{},[59,14745,531],{},[138,14747,14748,14750],{},[59,14749,131],{},[59,14751,2098],{},[138,14753,14754,14756],{},[133,14755,161],{"stretchy":135},[56,14757,14758,14760],{},[133,14759,5378],{},[539,14761,546],{},[138,14763,14764,14766],{},[59,14765,401],{},[59,14767,2098],{},[133,14769,793],{},[3187,14771,14772,14774,14782],{},[133,14773,3191],{},[56,14775,14776,14778,14780],{},[59,14777,3196],{},[133,14779,793],{},[539,14781,541],{},[59,14783,1696],{"mathvariant":145},[138,14785,14786,14788],{},[59,14787,531],{},[59,14789,3196],{},[133,14791,136],{"stretchy":135},[138,14793,14794,14796],{},[59,14795,131],{},[59,14797,2098],{},[138,14799,14800,14802],{},[133,14801,161],{"stretchy":135},[59,14803,3196],{},[138,14805,14806,14808],{},[59,14807,401],{},[59,14809,2098],{},[59,14811,2903],{"mathvariant":145},[63,14813,14814],{"encoding":65},"V^\\pi=(I-\\gamma P^\\pi)^{-1}r^\\pi\n=\\sum_{k=0}^{\\infty}\\gamma^k(P^\\pi)^k r^\\pi.",[39,14816,14818,14862,14883,14996],{"className":14817,"ariaHidden":70},[69],[39,14819,14821,14824,14853,14856,14859],{"className":14820},[74],[39,14822],{"className":14823,"style":4421},[78],[39,14825,14827,14830],{"className":14826},[83],[39,14828,4230],{"className":14829,"style":461},[83,84],[39,14831,14833],{"className":14832},[192],[39,14834,14836],{"className":14835},[196],[39,14837,14839],{"className":14838},[200],[39,14840,14842],{"className":14841,"style":4421},[204],[39,14843,14844,14847],{"style":1027},[39,14845],{"className":14846,"style":213},[212],[39,14848,14850],{"className":14849},[217,218,219,220],[39,14851,2098],{"className":14852,"style":2114},[83,84,220],[39,14854],{"className":14855,"style":231},[230],[39,14857,793],{"className":14858},[235],[39,14860],{"className":14861,"style":231},[230],[39,14863,14865,14868,14871,14874,14877,14880],{"className":14864},[74],[39,14866],{"className":14867,"style":245},[78],[39,14869,136],{"className":14870},[182],[39,14872,14373],{"className":14873,"style":14420},[83,84],[39,14875],{"className":14876,"style":461},[230],[39,14878,5378],{"className":14879},[465],[39,14881],{"className":14882,"style":461},[230],[39,14884,14886,14890,14893,14922,14958,14987,14990,14993],{"className":14885},[74],[39,14887],{"className":14888,"style":14889},[78],"height:1.1141em;vertical-align:-0.25em;",[39,14891,531],{"className":14892,"style":565},[83,84],[39,14894,14896,14899],{"className":14895},[83],[39,14897,131],{"className":14898,"style":178},[83,84],[39,14900,14902],{"className":14901},[192],[39,14903,14905],{"className":14904},[196],[39,14906,14908],{"className":14907},[200],[39,14909,14911],{"className":14910,"style":4421},[204],[39,14912,14913,14916],{"style":1027},[39,14914],{"className":14915,"style":213},[212],[39,14917,14919],{"className":14918},[217,218,219,220],[39,14920,2098],{"className":14921,"style":2114},[83,84,220],[39,14923,14925,14928],{"className":14924},[263],[39,14926,161],{"className":14927},[263],[39,14929,14931],{"className":14930},[192],[39,14932,14934],{"className":14933},[196],[39,14935,14937],{"className":14936},[200],[39,14938,14941],{"className":14939,"style":14940},[204],"height:0.8641em;",[39,14942,14943,14946],{"style":1027},[39,14944],{"className":14945,"style":213},[212],[39,14947,14949],{"className":14948},[217,218,219,220],[39,14950,14952,14955],{"className":14951},[83,220],[39,14953,5378],{"className":14954},[83,220],[39,14956,546],{"className":14957},[83,220],[39,14959,14961,14964],{"className":14960},[83],[39,14962,401],{"className":14963,"style":438},[83,84],[39,14965,14967],{"className":14966},[192],[39,14968,14970],{"className":14969},[196],[39,14971,14973],{"className":14972},[200],[39,14974,14976],{"className":14975,"style":4421},[204],[39,14977,14978,14981],{"style":1027},[39,14979],{"className":14980,"style":213},[212],[39,14982,14984],{"className":14983},[217,218,219,220],[39,14985,2098],{"className":14986,"style":2114},[83,84,220],[39,14988],{"className":14989,"style":231},[230],[39,14991,793],{"className":14992},[235],[39,14994],{"className":14995,"style":231},[230],[39,14997,14999,15002,15069,15072,15101,15104,15133,15162,15191],{"className":14998},[74],[39,15000],{"className":15001,"style":3290},[78],[39,15003,15005],{"className":15004},[1744,3294],[39,15006,15008,15061],{"className":15007},[196,649],[39,15009,15011,15058],{"className":15010},[200],[39,15012,15014,15034,15044],{"className":15013,"style":3304},[204],[39,15015,15016,15019],{"style":3307},[39,15017],{"className":15018,"style":3311},[212],[39,15020,15022],{"className":15021},[217,218,219,220],[39,15023,15025,15028,15031],{"className":15024},[83,220],[39,15026,3196],{"className":15027,"style":3321},[83,84,220],[39,15029,793],{"className":15030},[235,220],[39,15032,541],{"className":15033},[83,220],[39,15035,15036,15039],{"style":3330},[39,15037],{"className":15038,"style":3311},[212],[39,15040,15041],{},[39,15042,3191],{"className":15043},[1744,3339,3340],[39,15045,15046,15049],{"style":3343},[39,15047],{"className":15048,"style":3311},[212],[39,15050,15052],{"className":15051},[217,218,219,220],[39,15053,15055],{"className":15054},[83,220],[39,15056,1696],{"className":15057},[83,220],[39,15059,673],{"className":15060},[672],[39,15062,15064],{"className":15063},[200],[39,15065,15067],{"className":15066,"style":3365},[204],[39,15068],{},[39,15070],{"className":15071,"style":256},[230],[39,15073,15075,15078],{"className":15074},[83],[39,15076,531],{"className":15077,"style":565},[83,84],[39,15079,15081],{"className":15080},[192],[39,15082,15084],{"className":15083},[196],[39,15085,15087],{"className":15086},[200],[39,15088,15090],{"className":15089,"style":3389},[204],[39,15091,15092,15095],{"style":1027},[39,15093],{"className":15094,"style":213},[212],[39,15096,15098],{"className":15097},[217,218,219,220],[39,15099,3196],{"className":15100,"style":3321},[83,84,220],[39,15102,136],{"className":15103},[182],[39,15105,15107,15110],{"className":15106},[83],[39,15108,131],{"className":15109,"style":178},[83,84],[39,15111,15113],{"className":15112},[192],[39,15114,15116],{"className":15115},[196],[39,15117,15119],{"className":15118},[200],[39,15120,15122],{"className":15121,"style":4421},[204],[39,15123,15124,15127],{"style":1027},[39,15125],{"className":15126,"style":213},[212],[39,15128,15130],{"className":15129},[217,218,219,220],[39,15131,2098],{"className":15132,"style":2114},[83,84,220],[39,15134,15136,15139],{"className":15135},[263],[39,15137,161],{"className":15138},[263],[39,15140,15142],{"className":15141},[192],[39,15143,15145],{"className":15144},[196],[39,15146,15148],{"className":15147},[200],[39,15149,15151],{"className":15150,"style":3389},[204],[39,15152,15153,15156],{"style":1027},[39,15154],{"className":15155,"style":213},[212],[39,15157,15159],{"className":15158},[217,218,219,220],[39,15160,3196],{"className":15161,"style":3321},[83,84,220],[39,15163,15165,15168],{"className":15164},[83],[39,15166,401],{"className":15167,"style":438},[83,84],[39,15169,15171],{"className":15170},[192],[39,15172,15174],{"className":15173},[196],[39,15175,15177],{"className":15176},[200],[39,15178,15180],{"className":15179,"style":4421},[204],[39,15181,15182,15185],{"style":1027},[39,15183],{"className":15184,"style":213},[212],[39,15186,15188],{"className":15187},[217,218,219,220],[39,15189,2098],{"className":15190,"style":2114},[83,84,220],[39,15192,2903],{"className":15193},[83],[11,15195,15196,15198,15199,15285,15286,15345,15346,15374,15375,15508,15509,15674,15675,15817],{},[1489,15197,5595],{}," 记 ",[39,15200,15202,15226],{"className":15201,"translate":43},[42],[39,15203,15205],{"className":15204},[47],[49,15206,15207],{"xmlns":51},[53,15208,15209,15223],{},[56,15210,15211,15213,15215,15217],{},[59,15212,4295],{},[133,15214,793],{},[59,15216,531],{},[138,15218,15219,15221],{},[59,15220,131],{},[59,15222,2098],{},[63,15224,15225],{"encoding":65},"Q=\\gamma P^\\pi",[39,15227,15229,15247],{"className":15228,"ariaHidden":70},[69],[39,15230,15232,15235,15238,15241,15244],{"className":15231},[74],[39,15233],{"className":15234,"style":6452},[78],[39,15236,4295],{"className":15237},[83,84],[39,15239],{"className":15240,"style":231},[230],[39,15242,793],{"className":15243},[235],[39,15245],{"className":15246,"style":231},[230],[39,15248,15250,15253,15256],{"className":15249},[74],[39,15251],{"className":15252,"style":6452},[78],[39,15254,531],{"className":15255,"style":565},[83,84],[39,15257,15259,15262],{"className":15258},[83],[39,15260,131],{"className":15261,"style":178},[83,84],[39,15263,15265],{"className":15264},[192],[39,15266,15268],{"className":15267},[196],[39,15269,15271],{"className":15270},[200],[39,15272,15274],{"className":15273,"style":6410},[204],[39,15275,15276,15279],{"style":208},[39,15277],{"className":15278,"style":213},[212],[39,15280,15282],{"className":15281},[217,218,219,220],[39,15283,2098],{"className":15284,"style":2114},[83,84,220],"。由于 ",[39,15287,15289,15307],{"className":15288,"translate":43},[42],[39,15290,15292],{"className":15291},[47],[49,15293,15294],{"xmlns":51},[53,15295,15296,15304],{},[56,15297,15298],{},[138,15299,15300,15302],{},[59,15301,131],{},[59,15303,2098],{},[63,15305,15306],{"encoding":65},"P^\\pi",[39,15308,15310],{"className":15309,"ariaHidden":70},[69],[39,15311,15313,15316],{"className":15312},[74],[39,15314],{"className":15315,"style":79},[78],[39,15317,15319,15322],{"className":15318},[83],[39,15320,131],{"className":15321,"style":178},[83,84],[39,15323,15325],{"className":15324},[192],[39,15326,15328],{"className":15327},[196],[39,15329,15331],{"className":15330},[200],[39,15332,15334],{"className":15333,"style":6410},[204],[39,15335,15336,15339],{"style":208},[39,15337],{"className":15338,"style":213},[212],[39,15340,15342],{"className":15341},[217,218,219,220],[39,15343,2098],{"className":15344,"style":2114},[83,84,220]," 的每一行之和为 ",[39,15347,15349,15362],{"className":15348,"translate":43},[42],[39,15350,15352],{"className":15351},[47],[49,15353,15354],{"xmlns":51},[53,15355,15356,15360],{},[56,15357,15358],{},[539,15359,546],{},[63,15361,546],{"encoding":65},[39,15363,15365],{"className":15364,"ariaHidden":70},[69],[39,15366,15368,15371],{"className":15367},[74],[39,15369],{"className":15370,"style":2473},[78],[39,15372,546],{"className":15373},[83],"，有 ",[39,15376,15378,15409],{"className":15377,"translate":43},[42],[39,15379,15381],{"className":15380},[47],[49,15382,15383],{"xmlns":51},[53,15384,15385,15406],{},[56,15386,15387,15390,15396,15402,15404],{},[59,15388,15389],{"mathvariant":145},"∥",[138,15391,15392,15394],{},[59,15393,131],{},[59,15395,2098],{},[613,15397,15398,15400],{},[59,15399,15389],{"mathvariant":145},[59,15401,1696],{"mathvariant":145},[133,15403,793],{},[539,15405,546],{},[63,15407,15408],{"encoding":65},"\\|P^\\pi\\|_\\infty=1",[39,15410,15412,15499],{"className":15411,"ariaHidden":70},[69],[39,15413,15415,15418,15421,15450,15490,15493,15496],{"className":15414},[74],[39,15416],{"className":15417,"style":245},[78],[39,15419,15389],{"className":15420},[83],[39,15422,15424,15427],{"className":15423},[83],[39,15425,131],{"className":15426,"style":178},[83,84],[39,15428,15430],{"className":15429},[192],[39,15431,15433],{"className":15432},[196],[39,15434,15436],{"className":15435},[200],[39,15437,15439],{"className":15438,"style":6410},[204],[39,15440,15441,15444],{"style":208},[39,15442],{"className":15443,"style":213},[212],[39,15445,15447],{"className":15446},[217,218,219,220],[39,15448,2098],{"className":15449,"style":2114},[83,84,220],[39,15451,15453,15456],{"className":15452},[83],[39,15454,15389],{"className":15455},[83],[39,15457,15459],{"className":15458},[192],[39,15460,15462,15482],{"className":15461},[196,649],[39,15463,15465,15479],{"className":15464},[200],[39,15466,15468],{"className":15467,"style":1728},[204],[39,15469,15470,15473],{"style":966},[39,15471],{"className":15472,"style":213},[212],[39,15474,15476],{"className":15475},[217,218,219,220],[39,15477,1696],{"className":15478},[83,220],[39,15480,673],{"className":15481},[672],[39,15483,15485],{"className":15484},[200],[39,15486,15488],{"className":15487,"style":680},[204],[39,15489],{},[39,15491],{"className":15492,"style":231},[230],[39,15494,793],{"className":15495},[235],[39,15497],{"className":15498,"style":231},[230],[39,15500,15502,15505],{"className":15501},[74],[39,15503],{"className":15504,"style":2473},[78],[39,15506,546],{"className":15507},[83],"，从而 ",[39,15510,15512,15546],{"className":15511,"translate":43},[42],[39,15513,15515],{"className":15514},[47],[49,15516,15517],{"xmlns":51},[53,15518,15519,15543],{},[56,15520,15521,15523,15529,15535,15537],{},[59,15522,15389],{"mathvariant":145},[138,15524,15525,15527],{},[59,15526,4295],{},[59,15528,3196],{},[613,15530,15531,15533],{},[59,15532,15389],{"mathvariant":145},[59,15534,1696],{"mathvariant":145},[133,15536,1824],{},[138,15538,15539,15541],{},[59,15540,531],{},[59,15542,3196],{},[63,15544,15545],{"encoding":65},"\\|Q^k\\|_\\infty\\le\\gamma^k",[39,15547,15549,15638],{"className":15548,"ariaHidden":70},[69],[39,15550,15552,15556,15559,15589,15629,15632,15635],{"className":15551},[74],[39,15553],{"className":15554,"style":15555},[78],"height:1.0991em;vertical-align:-0.25em;",[39,15557,15389],{"className":15558},[83],[39,15560,15562,15565],{"className":15561},[83],[39,15563,4295],{"className":15564},[83,84],[39,15566,15568],{"className":15567},[192],[39,15569,15571],{"className":15570},[196],[39,15572,15574],{"className":15573},[200],[39,15575,15578],{"className":15576,"style":15577},[204],"height:0.8491em;",[39,15579,15580,15583],{"style":208},[39,15581],{"className":15582,"style":213},[212],[39,15584,15586],{"className":15585},[217,218,219,220],[39,15587,3196],{"className":15588,"style":3321},[83,84,220],[39,15590,15592,15595],{"className":15591},[83],[39,15593,15389],{"className":15594},[83],[39,15596,15598],{"className":15597},[192],[39,15599,15601,15621],{"className":15600},[196,649],[39,15602,15604,15618],{"className":15603},[200],[39,15605,15607],{"className":15606,"style":1728},[204],[39,15608,15609,15612],{"style":966},[39,15610],{"className":15611,"style":213},[212],[39,15613,15615],{"className":15614},[217,218,219,220],[39,15616,1696],{"className":15617},[83,220],[39,15619,673],{"className":15620},[672],[39,15622,15624],{"className":15623},[200],[39,15625,15627],{"className":15626,"style":680},[204],[39,15628],{},[39,15630],{"className":15631,"style":231},[230],[39,15633,1824],{"className":15634},[235],[39,15636],{"className":15637,"style":231},[230],[39,15639,15641,15645],{"className":15640},[74],[39,15642],{"className":15643,"style":15644},[78],"height:1.0435em;vertical-align:-0.1944em;",[39,15646,15648,15651],{"className":15647},[83],[39,15649,531],{"className":15650,"style":565},[83,84],[39,15652,15654],{"className":15653},[192],[39,15655,15657],{"className":15656},[196],[39,15658,15660],{"className":15659},[200],[39,15661,15663],{"className":15662,"style":15577},[204],[39,15664,15665,15668],{"style":208},[39,15666],{"className":15667,"style":213},[212],[39,15669,15671],{"className":15670},[217,218,219,220],[39,15672,3196],{"className":15673,"style":3321},[83,84,220],"。因此 Neumann 级数 ",[39,15676,15678,15710],{"className":15677,"translate":43},[42],[39,15679,15681],{"className":15680},[47],[49,15682,15683],{"xmlns":51},[53,15684,15685,15707],{},[56,15686,15687,15701],{},[9932,15688,15689,15691,15699],{},[133,15690,3191],{},[56,15692,15693,15695,15697],{},[59,15694,3196],{},[133,15696,793],{},[539,15698,541],{},[59,15700,1696],{"mathvariant":145},[138,15702,15703,15705],{},[59,15704,4295],{},[59,15706,3196],{},[63,15708,15709],{"encoding":65},"\\sum_{k=0}^{\\infty}Q^k",[39,15711,15713],{"className":15712,"ariaHidden":70},[69],[39,15714,15716,15720,15785,15788],{"className":15715},[74],[39,15717],{"className":15718,"style":15719},[78],"height:1.1488em;vertical-align:-0.2997em;",[39,15721,15723,15726],{"className":15722},[1744],[39,15724,3191],{"className":15725,"style":3953},[1744,3339,3952],[39,15727,15729],{"className":15728},[192],[39,15730,15732,15777],{"className":15731},[196,649],[39,15733,15735,15774],{"className":15734},[200],[39,15736,15739,15759],{"className":15737,"style":15738},[204],"height:0.8043em;",[39,15740,15741,15744],{"style":3969},[39,15742],{"className":15743,"style":213},[212],[39,15745,15747],{"className":15746},[217,218,219,220],[39,15748,15750,15753,15756],{"className":15749},[83,220],[39,15751,3196],{"className":15752,"style":3321},[83,84,220],[39,15754,793],{"className":15755},[235,220],[39,15757,541],{"className":15758},[83,220],[39,15760,15762,15765],{"style":15761},"top:-3.2029em;margin-right:0.05em;",[39,15763],{"className":15764,"style":213},[212],[39,15766,15768],{"className":15767},[217,218,219,220],[39,15769,15771],{"className":15770},[83,220],[39,15772,1696],{"className":15773},[83,220],[39,15775,673],{"className":15776},[672],[39,15778,15780],{"className":15779},[200],[39,15781,15783],{"className":15782,"style":4025},[204],[39,15784],{},[39,15786],{"className":15787,"style":256},[230],[39,15789,15791,15794],{"className":15790},[83],[39,15792,4295],{"className":15793},[83,84],[39,15795,15797],{"className":15796},[192],[39,15798,15800],{"className":15799},[196],[39,15801,15803],{"className":15802},[200],[39,15804,15806],{"className":15805,"style":15577},[204],[39,15807,15808,15811],{"style":208},[39,15809],{"className":15810,"style":213},[212],[39,15812,15814],{"className":15813},[217,218,219,220],[39,15815,3196],{"className":15816,"style":3321},[83,84,220]," 在算子范数下绝对收敛。并且",[39,15819,15821],{"className":15820,"translate":43},[746],[39,15822,15824,15891],{"className":15823,"translate":43},[42],[39,15825,15827],{"className":15826},[47],[49,15828,15829],{"xmlns":51,"display":755},[53,15830,15831,15888],{},[56,15832,15833,15835,15837,15839,15841,15843,15857,15863,15865,15867,15869,15881,15884,15886],{},[133,15834,136],{"stretchy":135},[59,15836,14373],{},[133,15838,5378],{},[59,15840,4295],{},[133,15842,161],{"stretchy":135},[3187,15844,15845,15847,15855],{},[133,15846,3191],{},[56,15848,15849,15851,15853],{},[59,15850,3196],{},[133,15852,793],{},[539,15854,541],{},[59,15856,12962],{},[138,15858,15859,15861],{},[59,15860,4295],{},[59,15862,3196],{},[133,15864,793],{},[59,15866,14373],{},[133,15868,5378],{},[138,15870,15871,15873],{},[59,15872,4295],{},[56,15874,15875,15877,15879],{},[59,15876,12962],{},[133,15878,788],{},[539,15880,546],{},[133,15882,15883],{},"⟶",[59,15885,14373],{},[133,15887,155],{"separator":70},[63,15889,15890],{"encoding":65},"(I-Q)\\sum_{k=0}^{n}Q^k=I-Q^{n+1}\\longrightarrow I,",[39,15892,15894,15915,16038,16056,16110],{"className":15893,"ariaHidden":70},[69],[39,15895,15897,15900,15903,15906,15909,15912],{"className":15896},[74],[39,15898],{"className":15899,"style":245},[78],[39,15901,136],{"className":15902},[182],[39,15904,14373],{"className":15905,"style":14420},[83,84],[39,15907],{"className":15908,"style":461},[230],[39,15910,5378],{"className":15911},[465],[39,15913],{"className":15914,"style":461},[230],[39,15916,15918,15921,15924,15927,15930,15997,16000,16029,16032,16035],{"className":15917},[74],[39,15919],{"className":15920,"style":3290},[78],[39,15922,4295],{"className":15923},[83,84],[39,15925,161],{"className":15926},[263],[39,15928],{"className":15929,"style":256},[230],[39,15931,15933],{"className":15932},[1744,3294],[39,15934,15936,15989],{"className":15935},[196,649],[39,15937,15939,15986],{"className":15938},[200],[39,15940,15942,15962,15972],{"className":15941,"style":3304},[204],[39,15943,15944,15947],{"style":3307},[39,15945],{"className":15946,"style":3311},[212],[39,15948,15950],{"className":15949},[217,218,219,220],[39,15951,15953,15956,15959],{"className":15952},[83,220],[39,15954,3196],{"className":15955,"style":3321},[83,84,220],[39,15957,793],{"className":15958},[235,220],[39,15960,541],{"className":15961},[83,220],[39,15963,15964,15967],{"style":3330},[39,15965],{"className":15966,"style":3311},[212],[39,15968,15969],{},[39,15970,3191],{"className":15971},[1744,3339,3340],[39,15973,15974,15977],{"style":3343},[39,15975],{"className":15976,"style":3311},[212],[39,15978,15980],{"className":15979},[217,218,219,220],[39,15981,15983],{"className":15982},[83,220],[39,15984,12962],{"className":15985},[83,84,220],[39,15987,673],{"className":15988},[672],[39,15990,15992],{"className":15991},[200],[39,15993,15995],{"className":15994,"style":3365},[204],[39,15996],{},[39,15998],{"className":15999,"style":256},[230],[39,16001,16003,16006],{"className":16002},[83],[39,16004,4295],{"className":16005},[83,84],[39,16007,16009],{"className":16008},[192],[39,16010,16012],{"className":16011},[196],[39,16013,16015],{"className":16014},[200],[39,16016,16018],{"className":16017,"style":3389},[204],[39,16019,16020,16023],{"style":1027},[39,16021],{"className":16022,"style":213},[212],[39,16024,16026],{"className":16025},[217,218,219,220],[39,16027,3196],{"className":16028,"style":3321},[83,84,220],[39,16030],{"className":16031,"style":231},[230],[39,16033,793],{"className":16034},[235],[39,16036],{"className":16037,"style":231},[230],[39,16039,16041,16044,16047,16050,16053],{"className":16040},[74],[39,16042],{"className":16043,"style":454},[78],[39,16045,14373],{"className":16046,"style":14420},[83,84],[39,16048],{"className":16049,"style":461},[230],[39,16051,5378],{"className":16052},[465],[39,16054],{"className":16055,"style":461},[230],[39,16057,16059,16063,16101,16104,16107],{"className":16058},[74],[39,16060],{"className":16061,"style":16062},[78],"height:1.0585em;vertical-align:-0.1944em;",[39,16064,16066,16069],{"className":16065},[83],[39,16067,4295],{"className":16068},[83,84],[39,16070,16072],{"className":16071},[192],[39,16073,16075],{"className":16074},[196],[39,16076,16078],{"className":16077},[200],[39,16079,16081],{"className":16080,"style":14940},[204],[39,16082,16083,16086],{"style":1027},[39,16084],{"className":16085,"style":213},[212],[39,16087,16089],{"className":16088},[217,218,219,220],[39,16090,16092,16095,16098],{"className":16091},[83,220],[39,16093,12962],{"className":16094},[83,84,220],[39,16096,788],{"className":16097},[465,220],[39,16099,546],{"className":16100},[83,220],[39,16102],{"className":16103,"style":231},[230],[39,16105,15883],{"className":16106},[235],[39,16108],{"className":16109,"style":231},[230],[39,16111,16113,16116,16119],{"className":16112},[74],[39,16114],{"className":16115,"style":6452},[78],[39,16117,14373],{"className":16118,"style":14420},[83,84],[39,16120,155],{"className":16121},[252],[11,16123,16124,16125,14714],{},"故 ",[39,16126,16128,16146],{"className":16127,"translate":43},[42],[39,16129,16131],{"className":16130},[47],[49,16132,16133],{"xmlns":51},[53,16134,16135,16143],{},[56,16136,16137,16139,16141],{},[59,16138,14373],{},[133,16140,5378],{},[59,16142,4295],{},[63,16144,16145],{"encoding":65},"I-Q",[39,16147,16149,16167],{"className":16148,"ariaHidden":70},[69],[39,16150,16152,16155,16158,16161,16164],{"className":16151},[74],[39,16153],{"className":16154,"style":454},[78],[39,16156,14373],{"className":16157,"style":14420},[83,84],[39,16159],{"className":16160,"style":461},[230],[39,16162,5378],{"className":16163},[465],[39,16165],{"className":16166,"style":461},[230],[39,16168,16170,16173],{"className":16169},[74],[39,16171],{"className":16172,"style":6452},[78],[39,16174,4295],{"className":16175},[83,84],[39,16177,16179],{"className":16178,"translate":43},[746],[39,16180,16182,16236],{"className":16181,"translate":43},[42],[39,16183,16185],{"className":16184},[47],[49,16186,16187],{"xmlns":51,"display":755},[53,16188,16189,16233],{},[56,16190,16191,16193,16195,16197,16199,16209,16211,16225,16231],{},[133,16192,136],{"stretchy":135},[59,16194,14373],{},[133,16196,5378],{},[59,16198,4295],{},[138,16200,16201,16203],{},[133,16202,161],{"stretchy":135},[56,16204,16205,16207],{},[133,16206,5378],{},[539,16208,546],{},[133,16210,793],{},[3187,16212,16213,16215,16223],{},[133,16214,3191],{},[56,16216,16217,16219,16221],{},[59,16218,3196],{},[133,16220,793],{},[539,16222,541],{},[59,16224,1696],{"mathvariant":145},[138,16226,16227,16229],{},[59,16228,4295],{},[59,16230,3196],{},[59,16232,2903],{"mathvariant":145},[63,16234,16235],{"encoding":65},"(I-Q)^{-1}=\\sum_{k=0}^{\\infty}Q^k.",[39,16237,16239,16260,16313],{"className":16238,"ariaHidden":70},[69],[39,16240,16242,16245,16248,16251,16254,16257],{"className":16241},[74],[39,16243],{"className":16244,"style":245},[78],[39,16246,136],{"className":16247},[182],[39,16249,14373],{"className":16250,"style":14420},[83,84],[39,16252],{"className":16253,"style":461},[230],[39,16255,5378],{"className":16256},[465],[39,16258],{"className":16259,"style":461},[230],[39,16261,16263,16266,16269,16304,16307,16310],{"className":16262},[74],[39,16264],{"className":16265,"style":14889},[78],[39,16267,4295],{"className":16268},[83,84],[39,16270,16272,16275],{"className":16271},[263],[39,16273,161],{"className":16274},[263],[39,16276,16278],{"className":16277},[192],[39,16279,16281],{"className":16280},[196],[39,16282,16284],{"className":16283},[200],[39,16285,16287],{"className":16286,"style":14940},[204],[39,16288,16289,16292],{"style":1027},[39,16290],{"className":16291,"style":213},[212],[39,16293,16295],{"className":16294},[217,218,219,220],[39,16296,16298,16301],{"className":16297},[83,220],[39,16299,5378],{"className":16300},[83,220],[39,16302,546],{"className":16303},[83,220],[39,16305],{"className":16306,"style":231},[230],[39,16308,793],{"className":16309},[235],[39,16311],{"className":16312,"style":231},[230],[39,16314,16316,16319,16386,16389,16418],{"className":16315},[74],[39,16317],{"className":16318,"style":3290},[78],[39,16320,16322],{"className":16321},[1744,3294],[39,16323,16325,16378],{"className":16324},[196,649],[39,16326,16328,16375],{"className":16327},[200],[39,16329,16331,16351,16361],{"className":16330,"style":3304},[204],[39,16332,16333,16336],{"style":3307},[39,16334],{"className":16335,"style":3311},[212],[39,16337,16339],{"className":16338},[217,218,219,220],[39,16340,16342,16345,16348],{"className":16341},[83,220],[39,16343,3196],{"className":16344,"style":3321},[83,84,220],[39,16346,793],{"className":16347},[235,220],[39,16349,541],{"className":16350},[83,220],[39,16352,16353,16356],{"style":3330},[39,16354],{"className":16355,"style":3311},[212],[39,16357,16358],{},[39,16359,3191],{"className":16360},[1744,3339,3340],[39,16362,16363,16366],{"style":3343},[39,16364],{"className":16365,"style":3311},[212],[39,16367,16369],{"className":16368},[217,218,219,220],[39,16370,16372],{"className":16371},[83,220],[39,16373,1696],{"className":16374},[83,220],[39,16376,673],{"className":16377},[672],[39,16379,16381],{"className":16380},[200],[39,16382,16384],{"className":16383,"style":3365},[204],[39,16385],{},[39,16387],{"className":16388,"style":256},[230],[39,16390,16392,16395],{"className":16391},[83],[39,16393,4295],{"className":16394},[83,84],[39,16396,16398],{"className":16397},[192],[39,16399,16401],{"className":16400},[196],[39,16402,16404],{"className":16403},[200],[39,16405,16407],{"className":16406,"style":3389},[204],[39,16408,16409,16412],{"style":1027},[39,16410],{"className":16411,"style":213},[212],[39,16413,16415],{"className":16414},[217,218,219,220],[39,16416,3196],{"className":16417,"style":3321},[83,84,220],[39,16419,2903],{"className":16420},[83],[11,16422,16423,16424,742,16428,16486,16487,16635],{},"由",[158,16425,16427],{"href":16426},"#theorem-bellman-expectation","定理 2.1",[39,16429,16431,16448],{"className":16430,"translate":43},[42],[39,16432,16434],{"className":16433},[47],[49,16435,16436],{"xmlns":51},[53,16437,16438,16446],{},[56,16439,16440],{},[138,16441,16442,16444],{},[59,16443,4230],{},[59,16445,2098],{},[63,16447,6382],{"encoding":65},[39,16449,16451],{"className":16450,"ariaHidden":70},[69],[39,16452,16454,16457],{"className":16453},[74],[39,16455],{"className":16456,"style":79},[78],[39,16458,16460,16463],{"className":16459},[83],[39,16461,4230],{"className":16462,"style":461},[83,84],[39,16464,16466],{"className":16465},[192],[39,16467,16469],{"className":16468},[196],[39,16470,16472],{"className":16471},[200],[39,16473,16475],{"className":16474,"style":6410},[204],[39,16476,16477,16480],{"style":208},[39,16478],{"className":16479,"style":213},[212],[39,16481,16483],{"className":16482},[217,218,219,220],[39,16484,2098],{"className":16485,"style":2114},[83,84,220]," 满足 ",[39,16488,16490,16526],{"className":16489,"translate":43},[42],[39,16491,16493],{"className":16492},[47],[49,16494,16495],{"xmlns":51},[53,16496,16497,16523],{},[56,16498,16499,16501,16503,16505,16507,16509,16515,16517],{},[133,16500,136],{"stretchy":135},[59,16502,14373],{},[133,16504,5378],{},[59,16506,4295],{},[133,16508,161],{"stretchy":135},[138,16510,16511,16513],{},[59,16512,4230],{},[59,16514,2098],{},[133,16516,793],{},[138,16518,16519,16521],{},[59,16520,401],{},[59,16522,2098],{},[63,16524,16525],{"encoding":65},"(I-Q)V^\\pi=r^\\pi",[39,16527,16529,16550,16600],{"className":16528,"ariaHidden":70},[69],[39,16530,16532,16535,16538,16541,16544,16547],{"className":16531},[74],[39,16533],{"className":16534,"style":245},[78],[39,16536,136],{"className":16537},[182],[39,16539,14373],{"className":16540,"style":14420},[83,84],[39,16542],{"className":16543,"style":461},[230],[39,16545,5378],{"className":16546},[465],[39,16548],{"className":16549,"style":461},[230],[39,16551,16553,16556,16559,16562,16591,16594,16597],{"className":16552},[74],[39,16554],{"className":16555,"style":245},[78],[39,16557,4295],{"className":16558},[83,84],[39,16560,161],{"className":16561},[263],[39,16563,16565,16568],{"className":16564},[83],[39,16566,4230],{"className":16567,"style":461},[83,84],[39,16569,16571],{"className":16570},[192],[39,16572,16574],{"className":16573},[196],[39,16575,16577],{"className":16576},[200],[39,16578,16580],{"className":16579,"style":6410},[204],[39,16581,16582,16585],{"style":208},[39,16583],{"className":16584,"style":213},[212],[39,16586,16588],{"className":16587},[217,218,219,220],[39,16589,2098],{"className":16590,"style":2114},[83,84,220],[39,16592],{"className":16593,"style":231},[230],[39,16595,793],{"className":16596},[235],[39,16598],{"className":16599,"style":231},[230],[39,16601,16603,16606],{"className":16602},[74],[39,16604],{"className":16605,"style":6410},[78],[39,16607,16609,16612],{"className":16608},[83],[39,16610,401],{"className":16611,"style":438},[83,84],[39,16613,16615],{"className":16614},[192],[39,16616,16618],{"className":16617},[196],[39,16619,16621],{"className":16620},[200],[39,16622,16624],{"className":16623,"style":6410},[204],[39,16625,16626,16629],{"style":208},[39,16627],{"className":16628,"style":213},[212],[39,16630,16632],{"className":16631},[217,218,219,220],[39,16633,2098],{"className":16634,"style":2114},[83,84,220],"，于是",[39,16637,16639],{"className":16638,"translate":43},[746],[39,16640,16642,16760],{"className":16641,"translate":43},[42],[39,16643,16645],{"className":16644},[47],[49,16646,16647],{"xmlns":51,"display":755},[53,16648,16649,16757],{},[759,16650,16651,16697],{"rowspacing":761,"columnalign":762,"columnspacing":146},[764,16652,16653,16663],{},[767,16654,16655],{},[770,16656,16657],{"scriptlevel":541,"displaystyle":70},[138,16658,16659,16661],{},[59,16660,4230],{},[59,16662,2098],{},[767,16664,16665],{},[770,16666,16667],{"scriptlevel":541,"displaystyle":70},[56,16668,16669,16671,16673,16675,16677,16679,16681,16691],{},[56,16670],{},[133,16672,793],{},[133,16674,136],{"stretchy":135},[59,16676,14373],{},[133,16678,5378],{},[59,16680,4295],{},[138,16682,16683,16685],{},[133,16684,161],{"stretchy":135},[56,16686,16687,16689],{},[133,16688,5378],{},[539,16690,546],{},[138,16692,16693,16695],{},[59,16694,401],{},[59,16696,2098],{},[764,16698,16699,16705],{},[767,16700,16701],{},[770,16702,16703],{"scriptlevel":541,"displaystyle":70},[56,16704],{},[767,16706,16707],{},[770,16708,16709],{"scriptlevel":541,"displaystyle":70},[56,16710,16711,16713,16715,16729,16735,16737,16743,16749,16755],{},[56,16712],{},[133,16714,793],{},[3187,16716,16717,16719,16727],{},[133,16718,3191],{},[56,16720,16721,16723,16725],{},[59,16722,3196],{},[133,16724,793],{},[539,16726,541],{},[59,16728,1696],{"mathvariant":145},[138,16730,16731,16733],{},[59,16732,531],{},[59,16734,3196],{},[133,16736,136],{"stretchy":135},[138,16738,16739,16741],{},[59,16740,131],{},[59,16742,2098],{},[138,16744,16745,16747],{},[133,16746,161],{"stretchy":135},[59,16748,3196],{},[138,16750,16751,16753],{},[59,16752,401],{},[59,16754,2098],{},[59,16756,2903],{"mathvariant":145},[63,16758,16759],{"encoding":65},"\\begin{aligned}\nV^\\pi&=(I-Q)^{-1}r^\\pi\\\\\n&=\\sum_{k=0}^{\\infty}\\gamma^k(P^\\pi)^k r^\\pi.\n\\end{aligned}",[39,16761,16763],{"className":16762,"ariaHidden":70},[69],[39,16764,16766,16770],{"className":16765},[74],[39,16767],{"className":16768,"style":16769},[78],"height:4.4776em;vertical-align:-1.9888em;",[39,16771,16773],{"className":16772},[83],[39,16774,16776,16848],{"className":16775},[759],[39,16777,16779],{"className":16778},[918],[39,16780,16782,16839],{"className":16781},[196,649],[39,16783,16785,16836],{"className":16784},[200],[39,16786,16789,16827],{"className":16787,"style":16788},[204],"height:2.4888em;",[39,16790,16792,16795],{"style":16791},"top:-5.2761em;",[39,16793],{"className":16794,"style":5782},[212],[39,16796,16798],{"className":16797},[83],[39,16799,16801,16804],{"className":16800},[83],[39,16802,4230],{"className":16803,"style":461},[83,84],[39,16805,16807],{"className":16806},[192],[39,16808,16810],{"className":16809},[196],[39,16811,16813],{"className":16812},[200],[39,16814,16816],{"className":16815,"style":4421},[204],[39,16817,16818,16821],{"style":1027},[39,16819],{"className":16820,"style":213},[212],[39,16822,16824],{"className":16823},[217,218,219,220],[39,16825,2098],{"className":16826,"style":2114},[83,84,220],[39,16828,16830,16833],{"style":16829},"top:-2.9647em;",[39,16831],{"className":16832,"style":5782},[212],[39,16834],{"className":16835},[83],[39,16837,673],{"className":16838},[672],[39,16840,16842],{"className":16841},[200],[39,16843,16846],{"className":16844,"style":16845},[204],"height:1.9888em;",[39,16847],{},[39,16849,16851],{"className":16850},[1223],[39,16852,16854,17177],{"className":16853},[196,649],[39,16855,16857,17174],{"className":16856},[200],[39,16858,16860,16962],{"className":16859,"style":16788},[204],[39,16861,16862,16865],{"style":16791},[39,16863],{"className":16864,"style":5782},[212],[39,16866,16868,16871,16874,16877,16880,16883,16886,16889,16892,16895,16898,16933],{"className":16867},[83],[39,16869],{"className":16870},[83],[39,16872],{"className":16873,"style":231},[230],[39,16875,793],{"className":16876},[235],[39,16878],{"className":16879,"style":231},[230],[39,16881,136],{"className":16882},[182],[39,16884,14373],{"className":16885,"style":14420},[83,84],[39,16887],{"className":16888,"style":461},[230],[39,16890,5378],{"className":16891},[465],[39,16893],{"className":16894,"style":461},[230],[39,16896,4295],{"className":16897},[83,84],[39,16899,16901,16904],{"className":16900},[263],[39,16902,161],{"className":16903},[263],[39,16905,16907],{"className":16906},[192],[39,16908,16910],{"className":16909},[196],[39,16911,16913],{"className":16912},[200],[39,16914,16916],{"className":16915,"style":14940},[204],[39,16917,16918,16921],{"style":1027},[39,16919],{"className":16920,"style":213},[212],[39,16922,16924],{"className":16923},[217,218,219,220],[39,16925,16927,16930],{"className":16926},[83,220],[39,16928,5378],{"className":16929},[83,220],[39,16931,546],{"className":16932},[83,220],[39,16934,16936,16939],{"className":16935},[83],[39,16937,401],{"className":16938,"style":438},[83,84],[39,16940,16942],{"className":16941},[192],[39,16943,16945],{"className":16944},[196],[39,16946,16948],{"className":16947},[200],[39,16949,16951],{"className":16950,"style":4421},[204],[39,16952,16953,16956],{"style":1027},[39,16954],{"className":16955,"style":213},[212],[39,16957,16959],{"className":16958},[217,218,219,220],[39,16960,2098],{"className":16961,"style":2114},[83,84,220],[39,16963,16964,16967],{"style":16829},[39,16965],{"className":16966,"style":5782},[212],[39,16968,16970,16973,16976,16979,16982,17049,17052,17081,17084,17113,17142,17171],{"className":16969},[83],[39,16971],{"className":16972},[83],[39,16974],{"className":16975,"style":231},[230],[39,16977,793],{"className":16978},[235],[39,16980],{"className":16981,"style":231},[230],[39,16983,16985],{"className":16984},[1744,3294],[39,16986,16988,17041],{"className":16987},[196,649],[39,16989,16991,17038],{"className":16990},[200],[39,16992,16994,17014,17024],{"className":16993,"style":3304},[204],[39,16995,16996,16999],{"style":3307},[39,16997],{"className":16998,"style":3311},[212],[39,17000,17002],{"className":17001},[217,218,219,220],[39,17003,17005,17008,17011],{"className":17004},[83,220],[39,17006,3196],{"className":17007,"style":3321},[83,84,220],[39,17009,793],{"className":17010},[235,220],[39,17012,541],{"className":17013},[83,220],[39,17015,17016,17019],{"style":3330},[39,17017],{"className":17018,"style":3311},[212],[39,17020,17021],{},[39,17022,3191],{"className":17023},[1744,3339,3340],[39,17025,17026,17029],{"style":3343},[39,17027],{"className":17028,"style":3311},[212],[39,17030,17032],{"className":17031},[217,218,219,220],[39,17033,17035],{"className":17034},[83,220],[39,17036,1696],{"className":17037},[83,220],[39,17039,673],{"className":17040},[672],[39,17042,17044],{"className":17043},[200],[39,17045,17047],{"className":17046,"style":3365},[204],[39,17048],{},[39,17050],{"className":17051,"style":256},[230],[39,17053,17055,17058],{"className":17054},[83],[39,17056,531],{"className":17057,"style":565},[83,84],[39,17059,17061],{"className":17060},[192],[39,17062,17064],{"className":17063},[196],[39,17065,17067],{"className":17066},[200],[39,17068,17070],{"className":17069,"style":3389},[204],[39,17071,17072,17075],{"style":1027},[39,17073],{"className":17074,"style":213},[212],[39,17076,17078],{"className":17077},[217,218,219,220],[39,17079,3196],{"className":17080,"style":3321},[83,84,220],[39,17082,136],{"className":17083},[182],[39,17085,17087,17090],{"className":17086},[83],[39,17088,131],{"className":17089,"style":178},[83,84],[39,17091,17093],{"className":17092},[192],[39,17094,17096],{"className":17095},[196],[39,17097,17099],{"className":17098},[200],[39,17100,17102],{"className":17101,"style":4421},[204],[39,17103,17104,17107],{"style":1027},[39,17105],{"className":17106,"style":213},[212],[39,17108,17110],{"className":17109},[217,218,219,220],[39,17111,2098],{"className":17112,"style":2114},[83,84,220],[39,17114,17116,17119],{"className":17115},[263],[39,17117,161],{"className":17118},[263],[39,17120,17122],{"className":17121},[192],[39,17123,17125],{"className":17124},[196],[39,17126,17128],{"className":17127},[200],[39,17129,17131],{"className":17130,"style":3389},[204],[39,17132,17133,17136],{"style":1027},[39,17134],{"className":17135,"style":213},[212],[39,17137,17139],{"className":17138},[217,218,219,220],[39,17140,3196],{"className":17141,"style":3321},[83,84,220],[39,17143,17145,17148],{"className":17144},[83],[39,17146,401],{"className":17147,"style":438},[83,84],[39,17149,17151],{"className":17150},[192],[39,17152,17154],{"className":17153},[196],[39,17155,17157],{"className":17156},[200],[39,17158,17160],{"className":17159,"style":4421},[204],[39,17161,17162,17165],{"style":1027},[39,17163],{"className":17164,"style":213},[212],[39,17166,17168],{"className":17167},[217,218,219,220],[39,17169,2098],{"className":17170,"style":2114},[83,84,220],[39,17172,2903],{"className":17173},[83],[39,17175,673],{"className":17176},[672],[39,17178,17180],{"className":17179},[200],[39,17181,17183],{"className":17182,"style":16845},[204],[39,17184],{},[11,17186,11683],{},[11,17188,17189,17190,17275,17276,17327],{},"矩阵解给出了一个重要的结构性事实：在有限状态与固定策略下，贝尔曼方程是一个线性方程组，其系数矩阵为 ",[39,17191,17193,17216],{"className":17192,"translate":43},[42],[39,17194,17196],{"className":17195},[47],[49,17197,17198],{"xmlns":51},[53,17199,17200,17214],{},[56,17201,17202,17204,17206,17208],{},[59,17203,14373],{},[133,17205,5378],{},[59,17207,531],{},[138,17209,17210,17212],{},[59,17211,131],{},[59,17213,2098],{},[63,17215,14654],{"encoding":65},[39,17217,17219,17237],{"className":17218,"ariaHidden":70},[69],[39,17220,17222,17225,17228,17231,17234],{"className":17221},[74],[39,17223],{"className":17224,"style":454},[78],[39,17226,14373],{"className":17227,"style":14420},[83,84],[39,17229],{"className":17230,"style":461},[230],[39,17232,5378],{"className":17233},[465],[39,17235],{"className":17236,"style":461},[230],[39,17238,17240,17243,17246],{"className":17239},[74],[39,17241],{"className":17242,"style":6452},[78],[39,17244,531],{"className":17245,"style":565},[83,84],[39,17247,17249,17252],{"className":17248},[83],[39,17250,131],{"className":17251,"style":178},[83,84],[39,17253,17255],{"className":17254},[192],[39,17256,17258],{"className":17257},[196],[39,17259,17261],{"className":17260},[200],[39,17262,17264],{"className":17263,"style":6410},[204],[39,17265,17266,17269],{"style":208},[39,17267],{"className":17268,"style":213},[212],[39,17270,17272],{"className":17271},[217,218,219,220],[39,17273,2098],{"className":17274,"style":2114},[83,84,220],"。折扣因子 ",[39,17277,17279,17297],{"className":17278,"translate":43},[42],[39,17280,17282],{"className":17281},[47],[49,17283,17284],{"xmlns":51},[53,17285,17286,17294],{},[56,17287,17288,17290,17292],{},[59,17289,531],{},[133,17291,1693],{},[539,17293,546],{},[63,17295,17296],{"encoding":65},"\\gamma\u003C1",[39,17298,17300,17318],{"className":17299,"ariaHidden":70},[69],[39,17301,17303,17306,17309,17312,17315],{"className":17302},[74],[39,17304],{"className":17305,"style":561},[78],[39,17307,531],{"className":17308,"style":565},[83,84],[39,17310],{"className":17311,"style":231},[230],[39,17313,1693],{"className":17314},[235],[39,17316],{"className":17317,"style":231},[230],[39,17319,17321,17324],{"className":17320},[74],[39,17322],{"className":17323,"style":2473},[78],[39,17325,546],{"className":17326},[83]," 保证该矩阵可逆，因此解存在且唯一。对于一般状态空间，矩阵不再有限，但同样的结论可以由第 3 节的压缩映射论证得到。",[26,17329,17331],{"id":17330},"_24-例两状态马尔可夫链","2.4 例：两状态马尔可夫链",[11,17333,17334,17335,17410],{},"考虑两个状态 ",[39,17336,17338,17365],{"className":17337,"translate":43},[42],[39,17339,17341],{"className":17340},[47],[49,17342,17343],{"xmlns":51},[53,17344,17345,17362],{},[56,17346,17347,17349,17351,17353,17355,17357,17360],{},[59,17348,61],{},[133,17350,793],{},[133,17352,12951],{"stretchy":135},[539,17354,546],{},[133,17356,155],{"separator":70},[539,17358,17359],{},"2",[133,17361,12965],{"stretchy":135},[63,17363,17364],{"encoding":65},"S=\\{1,2\\}",[39,17366,17368,17386],{"className":17367,"ariaHidden":70},[69],[39,17369,17371,17374,17377,17380,17383],{"className":17370},[74],[39,17372],{"className":17373,"style":79},[78],[39,17375,61],{"className":17376,"style":85},[83,84],[39,17378],{"className":17379,"style":231},[230],[39,17381,793],{"className":17382},[235],[39,17384],{"className":17385,"style":231},[230],[39,17387,17389,17392,17395,17398,17401,17404,17407],{"className":17388},[74],[39,17390],{"className":17391,"style":245},[78],[39,17393,12951],{"className":17394},[182],[39,17396,546],{"className":17397},[83],[39,17399,155],{"className":17400},[252],[39,17402],{"className":17403,"style":256},[230],[39,17405,17359],{"className":17406},[83],[39,17408,12965],{"className":17409},[263],"，只有一个动作。设转移关系为",[39,17412,17414],{"className":17413,"translate":43},[746],[39,17415,17417,17467],{"className":17416,"translate":43},[42],[39,17418,17420],{"className":17419},[47],[49,17421,17422],{"xmlns":51,"display":755},[53,17423,17424,17464],{},[56,17425,17426,17428,17430,17432,17434,17436,17438,17440,17442,17444,17446,17448,17450,17452,17454,17456,17458,17460,17462],{},[59,17427,131],{},[133,17429,136],{"stretchy":135},[539,17431,17359],{},[133,17433,150],{},[539,17435,546],{},[133,17437,161],{"stretchy":135},[133,17439,793],{},[539,17441,546],{},[133,17443,155],{"separator":70},[230,17445],{"width":10251},[59,17447,131],{},[133,17449,136],{"stretchy":135},[539,17451,546],{},[133,17453,150],{},[539,17455,17359],{},[133,17457,161],{"stretchy":135},[133,17459,793],{},[539,17461,546],{},[133,17463,155],{"separator":70},[63,17465,17466],{"encoding":65},"P(2\\mid 1)=1,\\qquad P(1\\mid 2)=1,",[39,17468,17470,17494,17515,17551,17572],{"className":17469,"ariaHidden":70},[69],[39,17471,17473,17476,17479,17482,17485,17488,17491],{"className":17472},[74],[39,17474],{"className":17475,"style":245},[78],[39,17477,131],{"className":17478,"style":178},[83,84],[39,17480,136],{"className":17481},[182],[39,17483,17359],{"className":17484},[83],[39,17486],{"className":17487,"style":231},[230],[39,17489,150],{"className":17490},[235],[39,17492],{"className":17493,"style":231},[230],[39,17495,17497,17500,17503,17506,17509,17512],{"className":17496},[74],[39,17498],{"className":17499,"style":245},[78],[39,17501,546],{"className":17502},[83],[39,17504,161],{"className":17505},[263],[39,17507],{"className":17508,"style":231},[230],[39,17510,793],{"className":17511},[235],[39,17513],{"className":17514,"style":231},[230],[39,17516,17518,17521,17524,17527,17530,17533,17536,17539,17542,17545,17548],{"className":17517},[74],[39,17519],{"className":17520,"style":245},[78],[39,17522,546],{"className":17523},[83],[39,17525,155],{"className":17526},[252],[39,17528],{"className":17529,"style":10787},[230],[39,17531],{"className":17532,"style":256},[230],[39,17534,131],{"className":17535,"style":178},[83,84],[39,17537,136],{"className":17538},[182],[39,17540,546],{"className":17541},[83],[39,17543],{"className":17544,"style":231},[230],[39,17546,150],{"className":17547},[235],[39,17549],{"className":17550,"style":231},[230],[39,17552,17554,17557,17560,17563,17566,17569],{"className":17553},[74],[39,17555],{"className":17556,"style":245},[78],[39,17558,17359],{"className":17559},[83],[39,17561,161],{"className":17562},[263],[39,17564],{"className":17565,"style":231},[230],[39,17567,793],{"className":17568},[235],[39,17570],{"className":17571,"style":231},[230],[39,17573,17575,17579,17582],{"className":17574},[74],[39,17576],{"className":17577,"style":17578},[78],"height:0.8389em;vertical-align:-0.1944em;",[39,17580,546],{"className":17581},[83],[39,17583,155],{"className":17584},[252],[11,17586,17587,17588,742,17654,17720,17721,17773],{},"奖励为 ",[39,17589,17591,17615],{"className":17590,"translate":43},[42],[39,17592,17594],{"className":17593},[47],[49,17595,17596],{"xmlns":51},[53,17597,17598,17612],{},[56,17599,17600,17602,17604,17606,17608,17610],{},[59,17601,401],{},[133,17603,136],{"stretchy":135},[539,17605,546],{},[133,17607,161],{"stretchy":135},[133,17609,793],{},[539,17611,546],{},[63,17613,17614],{"encoding":65},"r(1)=1",[39,17616,17618,17645],{"className":17617,"ariaHidden":70},[69],[39,17619,17621,17624,17627,17630,17633,17636,17639,17642],{"className":17620},[74],[39,17622],{"className":17623,"style":245},[78],[39,17625,401],{"className":17626,"style":438},[83,84],[39,17628,136],{"className":17629},[182],[39,17631,546],{"className":17632},[83],[39,17634,161],{"className":17635},[263],[39,17637],{"className":17638,"style":231},[230],[39,17640,793],{"className":17641},[235],[39,17643],{"className":17644,"style":231},[230],[39,17646,17648,17651],{"className":17647},[74],[39,17649],{"className":17650,"style":2473},[78],[39,17652,546],{"className":17653},[83],[39,17655,17657,17681],{"className":17656,"translate":43},[42],[39,17658,17660],{"className":17659},[47],[49,17661,17662],{"xmlns":51},[53,17663,17664,17678],{},[56,17665,17666,17668,17670,17672,17674,17676],{},[59,17667,401],{},[133,17669,136],{"stretchy":135},[539,17671,17359],{},[133,17673,161],{"stretchy":135},[133,17675,793],{},[539,17677,541],{},[63,17679,17680],{"encoding":65},"r(2)=0",[39,17682,17684,17711],{"className":17683,"ariaHidden":70},[69],[39,17685,17687,17690,17693,17696,17699,17702,17705,17708],{"className":17686},[74],[39,17688],{"className":17689,"style":245},[78],[39,17691,401],{"className":17692,"style":438},[83,84],[39,17694,136],{"className":17695},[182],[39,17697,17359],{"className":17698},[83],[39,17700,161],{"className":17701},[263],[39,17703],{"className":17704,"style":231},[230],[39,17706,793],{"className":17707},[235],[39,17709],{"className":17710,"style":231},[230],[39,17712,17714,17717],{"className":17713},[74],[39,17715],{"className":17716,"style":2473},[78],[39,17718,541],{"className":17719},[83],"，折扣因子 ",[39,17722,17724,17743],{"className":17723,"translate":43},[42],[39,17725,17727],{"className":17726},[47],[49,17728,17729],{"xmlns":51},[53,17730,17731,17740],{},[56,17732,17733,17735,17737],{},[59,17734,531],{},[133,17736,793],{},[539,17738,17739],{},"0.9",[63,17741,17742],{"encoding":65},"\\gamma=0.9",[39,17744,17746,17764],{"className":17745,"ariaHidden":70},[69],[39,17747,17749,17752,17755,17758,17761],{"className":17748},[74],[39,17750],{"className":17751,"style":2654},[78],[39,17753,531],{"className":17754,"style":565},[83,84],[39,17756],{"className":17757,"style":231},[230],[39,17759,793],{"className":17760},[235],[39,17762],{"className":17763,"style":231},[230],[39,17765,17767,17770],{"className":17766},[74],[39,17768],{"className":17769,"style":2473},[78],[39,17771,17739],{"className":17772},[83],"。贝尔曼方程为",[39,17775,17777],{"className":17776,"translate":43},[746],[39,17778,17780,17846],{"className":17779,"translate":43},[42],[39,17781,17783],{"className":17782},[47],[49,17784,17785],{"xmlns":51,"display":755},[53,17786,17787,17843],{},[56,17788,17789,17791,17793,17795,17797,17799,17801,17803,17805,17807,17809,17811,17813,17815,17817,17819,17821,17823,17825,17827,17829,17831,17833,17835,17837,17839,17841],{},[59,17790,4230],{},[133,17792,136],{"stretchy":135},[539,17794,546],{},[133,17796,161],{"stretchy":135},[133,17798,793],{},[539,17800,546],{},[133,17802,788],{},[539,17804,17739],{},[59,17806,4230],{},[133,17808,136],{"stretchy":135},[539,17810,17359],{},[133,17812,161],{"stretchy":135},[133,17814,155],{"separator":70},[230,17816],{"width":10251},[59,17818,4230],{},[133,17820,136],{"stretchy":135},[539,17822,17359],{},[133,17824,161],{"stretchy":135},[133,17826,793],{},[539,17828,541],{},[133,17830,788],{},[539,17832,17739],{},[59,17834,4230],{},[133,17836,136],{"stretchy":135},[539,17838,546],{},[133,17840,161],{"stretchy":135},[59,17842,2903],{"mathvariant":145},[63,17844,17845],{"encoding":65},"V(1)=1+0.9V(2),\\qquad V(2)=0+0.9V(1).",[39,17847,17849,17876,17895,17946,17964],{"className":17848,"ariaHidden":70},[69],[39,17850,17852,17855,17858,17861,17864,17867,17870,17873],{"className":17851},[74],[39,17853],{"className":17854,"style":245},[78],[39,17856,4230],{"className":17857,"style":461},[83,84],[39,17859,136],{"className":17860},[182],[39,17862,546],{"className":17863},[83],[39,17865,161],{"className":17866},[263],[39,17868],{"className":17869,"style":231},[230],[39,17871,793],{"className":17872},[235],[39,17874],{"className":17875,"style":231},[230],[39,17877,17879,17883,17886,17889,17892],{"className":17878},[74],[39,17880],{"className":17881,"style":17882},[78],"height:0.7278em;vertical-align:-0.0833em;",[39,17884,546],{"className":17885},[83],[39,17887],{"className":17888,"style":461},[230],[39,17890,788],{"className":17891},[465],[39,17893],{"className":17894,"style":461},[230],[39,17896,17898,17901,17904,17907,17910,17913,17916,17919,17922,17925,17928,17931,17934,17937,17940,17943],{"className":17897},[74],[39,17899],{"className":17900,"style":245},[78],[39,17902,17739],{"className":17903},[83],[39,17905,4230],{"className":17906,"style":461},[83,84],[39,17908,136],{"className":17909},[182],[39,17911,17359],{"className":17912},[83],[39,17914,161],{"className":17915},[263],[39,17917,155],{"className":17918},[252],[39,17920],{"className":17921,"style":10787},[230],[39,17923],{"className":17924,"style":256},[230],[39,17926,4230],{"className":17927,"style":461},[83,84],[39,17929,136],{"className":17930},[182],[39,17932,17359],{"className":17933},[83],[39,17935,161],{"className":17936},[263],[39,17938],{"className":17939,"style":231},[230],[39,17941,793],{"className":17942},[235],[39,17944],{"className":17945,"style":231},[230],[39,17947,17949,17952,17955,17958,17961],{"className":17948},[74],[39,17950],{"className":17951,"style":17882},[78],[39,17953,541],{"className":17954},[83],[39,17956],{"className":17957,"style":461},[230],[39,17959,788],{"className":17960},[465],[39,17962],{"className":17963,"style":461},[230],[39,17965,17967,17970,17973,17976,17979,17982,17985],{"className":17966},[74],[39,17968],{"className":17969,"style":245},[78],[39,17971,17739],{"className":17972},[83],[39,17974,4230],{"className":17975,"style":461},[83,84],[39,17977,136],{"className":17978},[182],[39,17980,546],{"className":17981},[83],[39,17983,161],{"className":17984},[263],[39,17986,2903],{"className":17987},[83],[11,17989,17990],{},"将第二式代入第一式，得",[39,17992,17994],{"className":17993,"translate":43},[746],[39,17995,17997,18036],{"className":17996,"translate":43},[42],[39,17998,18000],{"className":17999},[47],[49,18001,18002],{"xmlns":51,"display":755},[53,18003,18004,18033],{},[56,18005,18006,18008,18010,18012,18014,18016,18018,18020,18023,18025,18027,18029,18031],{},[59,18007,4230],{},[133,18009,136],{"stretchy":135},[539,18011,546],{},[133,18013,161],{"stretchy":135},[133,18015,793],{},[539,18017,546],{},[133,18019,788],{},[539,18021,18022],{},"0.81",[59,18024,4230],{},[133,18026,136],{"stretchy":135},[539,18028,546],{},[133,18030,161],{"stretchy":135},[133,18032,155],{"separator":70},[63,18034,18035],{"encoding":65},"V(1)=1+0.81V(1),",[39,18037,18039,18066,18084],{"className":18038,"ariaHidden":70},[69],[39,18040,18042,18045,18048,18051,18054,18057,18060,18063],{"className":18041},[74],[39,18043],{"className":18044,"style":245},[78],[39,18046,4230],{"className":18047,"style":461},[83,84],[39,18049,136],{"className":18050},[182],[39,18052,546],{"className":18053},[83],[39,18055,161],{"className":18056},[263],[39,18058],{"className":18059,"style":231},[230],[39,18061,793],{"className":18062},[235],[39,18064],{"className":18065,"style":231},[230],[39,18067,18069,18072,18075,18078,18081],{"className":18068},[74],[39,18070],{"className":18071,"style":17882},[78],[39,18073,546],{"className":18074},[83],[39,18076],{"className":18077,"style":461},[230],[39,18079,788],{"className":18080},[465],[39,18082],{"className":18083,"style":461},[230],[39,18085,18087,18090,18093,18096,18099,18102,18105],{"className":18086},[74],[39,18088],{"className":18089,"style":245},[78],[39,18091,18022],{"className":18092},[83],[39,18094,4230],{"className":18095,"style":461},[83,84],[39,18097,136],{"className":18098},[182],[39,18100,546],{"className":18101},[83],[39,18103,161],{"className":18104},[263],[39,18106,155],{"className":18107},[252],[11,18109,18110],{},"因此",[39,18112,18114],{"className":18113,"translate":43},[746],[39,18115,18117,18185],{"className":18116,"translate":43},[42],[39,18118,18120],{"className":18119},[47],[49,18121,18122],{"xmlns":51,"display":755},[53,18123,18124,18182],{},[56,18125,18126,18128,18130,18132,18134,18136,18148,18150,18153,18155,18157,18159,18161,18163,18165,18167,18169,18171,18173,18175,18177,18179],{},[59,18127,4230],{},[133,18129,136],{"stretchy":135},[539,18131,546],{},[133,18133,161],{"stretchy":135},[133,18135,793],{},[5362,18137,18138,18140],{},[539,18139,546],{},[56,18141,18142,18144,18146],{},[539,18143,546],{},[133,18145,5378],{},[539,18147,18022],{},[133,18149,793],{},[539,18151,18152],{},"5.2632",[133,18154,155],{"separator":70},[230,18156],{"width":10251},[59,18158,4230],{},[133,18160,136],{"stretchy":135},[539,18162,17359],{},[133,18164,161],{"stretchy":135},[133,18166,793],{},[539,18168,17739],{},[59,18170,4230],{},[133,18172,136],{"stretchy":135},[539,18174,546],{},[133,18176,161],{"stretchy":135},[133,18178,793],{},[539,18180,18181],{},"4.7368.",[63,18183,18184],{"encoding":65},"V(1)=\\frac{1}{1-0.81}=5.2632,\\qquad\nV(2)=0.9V(1)=4.7368.",[39,18186,18188,18215,18307,18346,18376],{"className":18187,"ariaHidden":70},[69],[39,18189,18191,18194,18197,18200,18203,18206,18209,18212],{"className":18190},[74],[39,18192],{"className":18193,"style":245},[78],[39,18195,4230],{"className":18196,"style":461},[83,84],[39,18198,136],{"className":18199},[182],[39,18201,546],{"className":18202},[83],[39,18204,161],{"className":18205},[263],[39,18207],{"className":18208,"style":231},[230],[39,18210,793],{"className":18211},[235],[39,18213],{"className":18214,"style":231},[230],[39,18216,18218,18222,18298,18301,18304],{"className":18217},[74],[39,18219],{"className":18220,"style":18221},[78],"height:2.0908em;vertical-align:-0.7693em;",[39,18223,18225,18228,18295],{"className":18224},[83],[39,18226],{"className":18227},[182,5463],[39,18229,18231],{"className":18230},[5362],[39,18232,18234,18286],{"className":18233},[196,649],[39,18235,18237,18283],{"className":18236},[200],[39,18238,18241,18264,18272],{"className":18239,"style":18240},[204],"height:1.3214em;",[39,18242,18243,18246],{"style":5479},[39,18244],{"className":18245,"style":935},[212],[39,18247,18249,18252,18255,18258,18261],{"className":18248},[83],[39,18250,546],{"className":18251},[83],[39,18253],{"className":18254,"style":461},[230],[39,18256,5378],{"className":18257},[465],[39,18259],{"className":18260,"style":461},[230],[39,18262,18022],{"className":18263},[83],[39,18265,18266,18269],{"style":5503},[39,18267],{"className":18268,"style":935},[212],[39,18270],{"className":18271,"style":5511},[5510],[39,18273,18274,18277],{"style":5514},[39,18275],{"className":18276,"style":935},[212],[39,18278,18280],{"className":18279},[83],[39,18281,546],{"className":18282},[83],[39,18284,673],{"className":18285},[672],[39,18287,18289],{"className":18288},[200],[39,18290,18293],{"className":18291,"style":18292},[204],"height:0.7693em;",[39,18294],{},[39,18296],{"className":18297},[263,5463],[39,18299],{"className":18300,"style":231},[230],[39,18302,793],{"className":18303},[235],[39,18305],{"className":18306,"style":231},[230],[39,18308,18310,18313,18316,18319,18322,18325,18328,18331,18334,18337,18340,18343],{"className":18309},[74],[39,18311],{"className":18312,"style":245},[78],[39,18314,18152],{"className":18315},[83],[39,18317,155],{"className":18318},[252],[39,18320],{"className":18321,"style":10787},[230],[39,18323],{"className":18324,"style":256},[230],[39,18326,4230],{"className":18327,"style":461},[83,84],[39,18329,136],{"className":18330},[182],[39,18332,17359],{"className":18333},[83],[39,18335,161],{"className":18336},[263],[39,18338],{"className":18339,"style":231},[230],[39,18341,793],{"className":18342},[235],[39,18344],{"className":18345,"style":231},[230],[39,18347,18349,18352,18355,18358,18361,18364,18367,18370,18373],{"className":18348},[74],[39,18350],{"className":18351,"style":245},[78],[39,18353,17739],{"className":18354},[83],[39,18356,4230],{"className":18357,"style":461},[83,84],[39,18359,136],{"className":18360},[182],[39,18362,546],{"className":18363},[83],[39,18365,161],{"className":18366},[263],[39,18368],{"className":18369,"style":231},[230],[39,18371,793],{"className":18372},[235],[39,18374],{"className":18375,"style":231},[230],[39,18377,18379,18382],{"className":18378},[74],[39,18380],{"className":18381,"style":2473},[78],[39,18383,18181],{"className":18384},[83],[11,18386,18387],{},"用矩阵形式验证，转移矩阵与奖励向量为",[39,18389,18391],{"className":18390,"translate":43},[746],[39,18392,18394,18483],{"className":18393,"translate":43},[42],[39,18395,18397],{"className":18396},[47],[49,18398,18399],{"xmlns":51,"display":755},[53,18400,18401,18480],{},[56,18402,18403,18405,18407,18445,18447,18449,18451,18453,18478],{},[59,18404,131],{},[133,18406,793],{},[56,18408,18409,18411,18443],{},[133,18410,136],{"fence":70},[759,18412,18415,18429],{"rowspacing":18413,"columnalign":18414,"columnspacing":6891},"0.16em","center 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className":19321},[1223],[39,19323,19325,19646],{"className":19324},[196,649],[39,19326,19328,19643],{"className":19327},[200],[39,19329,19331,19410],{"className":19330,"style":19284},[204],[39,19332,19333,19336],{"style":19287},[39,19334],{"className":19335,"style":19291},[212],[39,19337,19339,19342,19345,19348,19351,19354,19357,19360,19363,19366,19369,19372,19407],{"className":19338},[83],[39,19340],{"className":19341},[83],[39,19343],{"className":19344,"style":231},[230],[39,19346,793],{"className":19347},[235],[39,19349],{"className":19350,"style":231},[230],[39,19352,136],{"className":19353},[182],[39,19355,14373],{"className":19356,"style":14420},[83,84],[39,19358],{"className":19359,"style":461},[230],[39,19361,5378],{"className":19362},[465],[39,19364],{"className":19365,"style":461},[230],[39,19367,17739],{"className":19368},[83],[39,19370,131],{"className":19371,"style":178},[83,84],[39,19373,19375,19378],{"className":19374},[263],[39,19376,161],{"className":19377},[263],[39,19379,19381],{"className":19380},[192],[39,19382,19384],{"className":19383},[196],[39,19385,19387],{"className":19386},[200],[39,19388,19390],{"className":19389,"style":14940},[204],[39,19391,19392,19395],{"style":1027},[39,19393],{"className":19394,"style":213},[212],[39,19396,19398],{"className":19397},[217,218,219,220],[39,19399,19401,19404],{"className":19400},[83,220],[39,19402,5378],{"className":19403},[83,220],[39,19405,546],{"className":19406},[83,220],[39,19408,401],{"className":19409,"style":438},[83,84],[39,19411,19412,19415],{"style":19300},[39,19413],{"className":19414,"style":19291},[212],[39,19416,19418,19421,19424,19427,19430,19493,19496,19562,19565,19568,19571,19637,19640],{"className":19417},[83],[39,19419],{"className":19420},[83],[39,19422],{"className":19423,"style":231},[230],[39,19425,793],{"className":19426},[235],[39,19428],{"className":19429,"style":231},[230],[39,19431,19433,19436,19490],{"className":19432},[83],[39,19434],{"className":19435},[182,5463],[39,19437,19439],{"className":19438},[5362],[39,19440,19442,19481],{"className":19441},[196,649],[39,19443,19445,19478],{"className":19444},[200],[39,19446,19448,19459,19467],{"className":19447,"style":18240},[204],[39,19449,19450,19453],{"style":5479},[39,19451],{"className":19452,"style":935},[212],[39,19454,19456],{"className":19455},[83],[39,19457,19199],{"className":19458},[83],[39,19460,19461,19464],{"style":5503},[39,19462],{"className":19463,"style":935},[212],[39,19465],{"className":19466,"style":5511},[5510],[39,19468,19469,19472],{"style":5514},[39,19470],{"className":19471,"style":935},[212],[39,19473,19475],{"className":19474},[83],[39,19476,546],{"className":19477},[83],[39,19479,673],{"className":19480},[672],[39,19482,19484],{"className":19483},[200],[39,19485,19488],{"className":19486,"style":19487},[204],"height:0.686em;",[39,19489],{},[39,19491],{"className":19492},[263,5463],[39,19494],{"className":19495,"style":256},[230],[39,19497,19499,19505,19556],{"className":19498},[2842],[39,19500,19502],{"className":19501,"style":4592},[182,4591],[39,19503,136],{"className":19504},[18520,219],[39,19506,19508],{"className":19507},[83],[39,19509,19511],{"className":19510},[759],[39,19512,19514],{"className":19513},[18530],[39,19515,19517,19548],{"className":19516},[196,649],[39,19518,19520,19545],{"className":19519},[200],[39,19521,19523,19534],{"className":19522,"style":18540},[204],[39,19524,19525,19528],{"style":18543},[39,19526],{"className":19527,"style":935},[212],[39,19529,19531],{"className":19530},[83],[39,19532,546],{"className":19533},[83],[39,19535,19536,19539],{"style":18555},[39,19537],{"className":19538,"style":935},[212],[39,19540,19542],{"className":19541},[83],[39,19543,17739],{"className":19544},[83],[39,19546,673],{"className":19547},[672],[39,19549,19551],{"className":19550},[200],[39,19552,19554],{"className":19553,"style":18574},[204],[39,19555],{},[39,19557,19559],{"className":19558,"style":4592},[263,4591],[39,19560,161],{"className":19561},[18520,219],[39,19563],{"className":19564,"style":231},[230],[39,19566,793],{"className":19567},[235],[39,19569],{"className":19570,"style":231},[230],[39,19572,19574,19580,19631],{"className":19573},[2842],[39,19575,19577],{"className":19576,"style":4592},[182,4591],[39,19578,136],{"className":19579},[18520,219],[39,19581,19583],{"className":19582},[83],[39,19584,19586],{"className":19585},[759],[39,19587,19589],{"className":19588},[18530],[39,19590,19592,19623],{"className":19591},[196,649],[39,19593,19595,19620],{"className":19594},[200],[39,19596,19598,19609],{"className":19597,"style":18540},[204],[39,19599,19600,19603],{"style":18543},[39,19601],{"className":19602,"style":935},[212],[39,19604,19606],{"className":19605},[83],[39,19607,18152],{"className":19608},[83],[39,19610,19611,19614],{"style":18555},[39,19612],{"className":19613,"style":935},[212],[39,19615,19617],{"className":19616},[83],[39,19618,19248],{"className":19619},[83],[39,19621,673],{"className":19622},[672],[39,19624,19626],{"className":19625},[200],[39,19627,19629],{"className":19628,"style":18574},[204],[39,19630],{},[39,19632,19634],{"className":19633,"style":4592},[263,4591],[39,19635,161],{"className":19636},[18520,219],[39,19638],{"className":19639,"style":256},[230],[39,19641,2903],{"className":19642},[83],[39,19644,673],{"className":19645},[672],[39,19647,19649],{"className":19648},[200],[39,19650,19652],{"className":19651,"style":19316},[204],[39,19653],{},[11,19655,19656,19657,19740],{},"这个例子说明，即使状态每两步就回到原处，只要 ",[39,19658,19660,19682],{"className":19659,"translate":43},[42],[39,19661,19663],{"className":19662},[47],[49,19664,19665],{"xmlns":51},[53,19666,19667,19679],{},[56,19668,19669,19675,19677],{},[138,19670,19671,19673],{},[59,19672,531],{},[539,19674,17359],{},[133,19676,1693],{},[539,19678,546],{},[63,19680,19681],{"encoding":65},"\\gamma^2\u003C1",[39,19683,19685,19731],{"className":19684,"ariaHidden":70},[69],[39,19686,19688,19692,19722,19725,19728],{"className":19687},[74],[39,19689],{"className":19690,"style":19691},[78],"height:1.0085em;vertical-align:-0.1944em;",[39,19693,19695,19698],{"className":19694},[83],[39,19696,531],{"className":19697,"style":565},[83,84],[39,19699,19701],{"className":19700},[192],[39,19702,19704],{"className":19703},[196],[39,19705,19707],{"className":19706},[200],[39,19708,19711],{"className":19709,"style":19710},[204],"height:0.8141em;",[39,19712,19713,19716],{"style":208},[39,19714],{"className":19715,"style":213},[212],[39,19717,19719],{"className":19718},[217,218,219,220],[39,19720,17359],{"className":19721},[83,220],[39,19723],{"className":19724,"style":231},[230],[39,19726,1693],{"className":19727},[235],[39,19729],{"className":19730,"style":231},[230],[39,19732,19734,19737],{"className":19733},[74],[39,19735],{"className":19736,"style":2473},[78],[39,19738,546],{"className":19739},[83],"，折扣回报的期望仍然是有限的。值函数不是把无限多项直接相加，而是通过解线性方程组把无限和压缩成一个有限量。",[26,19742,19744],{"id":19743},"_25-方程与回报定义的等价性","2.5 方程与回报定义的等价性",[11,19746,19747],{},"贝尔曼方程不仅是值函数的必要条件，也是充分条件。下面的命题说明，任何有界解都必然等于由折扣回报定义的值函数。",[31,19749,19752],{"id":19750,"title":19751},"proposition-policy-verification","命题 2.3（策略评估的验证定理）",[11,19753,37,19754,16486,19822,14627,19908,19989,19990,20048],{},[39,19755,19757,19782],{"className":19756,"translate":43},[42],[39,19758,19760],{"className":19759},[47],[49,19761,19762],{"xmlns":51},[53,19763,19764,19779],{},[56,19765,19766,19768,19770,19773,19775,19777],{},[59,19767,4230],{},[133,19769,534],{},[59,19771,19772],{},"B",[133,19774,136],{"stretchy":135},[59,19776,61],{},[133,19778,161],{"stretchy":135},[63,19780,19781],{"encoding":65},"V\\in B(S)",[39,19783,19785,19803],{"className":19784,"ariaHidden":70},[69],[39,19786,19788,19791,19794,19797,19800],{"className":19787},[74],[39,19789],{"className":19790,"style":13061},[78],[39,19792,4230],{"className":19793,"style":461},[83,84],[39,19795],{"className":19796,"style":231},[230],[39,19798,534],{"className":19799},[235],[39,19801],{"className":19802,"style":231},[230],[39,19804,19806,19809,19813,19816,19819],{"className":19805},[74],[39,19807],{"className":19808,"style":245},[78],[39,19810,19772],{"className":19811,"style":19812},[83,84],"margin-right:0.0502em;",[39,19814,136],{"className":19815},[182],[39,19817,61],{"className":19818,"style":85},[83,84],[39,19820,161],{"className":19821},[263],[39,19823,19825,19849],{"className":19824,"translate":43},[42],[39,19826,19828],{"className":19827},[47],[49,19829,19830],{"xmlns":51},[53,19831,19832,19846],{},[56,19833,19834,19836,19838,19844],{},[59,19835,4230],{},[133,19837,793],{},[138,19839,19840,19842],{},[59,19841,11772],{"mathvariant":617},[59,19843,2098],{},[59,19845,4230],{},[63,19847,19848],{"encoding":65},"V=\\mathcal{T}^\\pi V",[39,19850,19852,19870],{"className":19851,"ariaHidden":70},[69],[39,19853,19855,19858,19861,19864,19867],{"className":19854},[74],[39,19856],{"className":19857,"style":79},[78],[39,19859,4230],{"className":19860,"style":461},[83,84],[39,19862],{"className":19863,"style":231},[230],[39,19865,793],{"className":19866},[235],[39,19868],{"className":19869,"style":231},[230],[39,19871,19873,19876,19905],{"className":19872},[74],[39,19874],{"className":19875,"style":79},[78],[39,19877,19879,19882],{"className":19878},[83],[39,19880,11772],{"className":19881,"style":11793},[83,641],[39,19883,19885],{"className":19884},[192],[39,19886,19888],{"className":19887},[196],[39,19889,19891],{"className":19890},[200],[39,19892,19894],{"className":19893,"style":6410},[204],[39,19895,19896,19899],{"style":208},[39,19897],{"className":19898,"style":213},[212],[39,19900,19902],{"className":19901},[217,218,219,220],[39,19903,2098],{"className":19904,"style":2114},[83,84,220],[39,19906,4230],{"className":19907,"style":461},[83,84],[39,19909,19911,19933],{"className":19910,"translate":43},[42],[39,19912,19914],{"className":19913},[47],[49,19915,19916],{"xmlns":51},[53,19917,19918,19930],{},[56,19919,19920,19922,19924],{},[59,19921,4230],{},[133,19923,793],{},[138,19925,19926,19928],{},[59,19927,4230],{},[59,19929,2098],{},[63,19931,19932],{"encoding":65},"V=V^\\pi",[39,19934,19936,19954],{"className":19935,"ariaHidden":70},[69],[39,19937,19939,19942,19945,19948,19951],{"className":19938},[74],[39,19940],{"className":19941,"style":79},[78],[39,19943,4230],{"className":19944,"style":461},[83,84],[39,19946],{"className":19947,"style":231},[230],[39,19949,793],{"className":19950},[235],[39,19952],{"className":19953,"style":231},[230],[39,19955,19957,19960],{"className":19956},[74],[39,19958],{"className":19959,"style":79},[78],[39,19961,19963,19966],{"className":19962},[83],[39,19964,4230],{"className":19965,"style":461},[83,84],[39,19967,19969],{"className":19968},[192],[39,19970,19972],{"className":19971},[196],[39,19973,19975],{"className":19974},[200],[39,19976,19978],{"className":19977,"style":6410},[204],[39,19979,19980,19983],{"style":208},[39,19981],{"className":19982,"style":213},[212],[39,19984,19986],{"className":19985},[217,218,219,220],[39,19987,2098],{"className":19988,"style":2114},[83,84,220],"。特别地，",[39,19991,19993,20010],{"className":19992,"translate":43},[42],[39,19994,19996],{"className":19995},[47],[49,19997,19998],{"xmlns":51},[53,19999,20000,20008],{},[56,20001,20002],{},[138,20003,20004,20006],{},[59,20005,11772],{"mathvariant":617},[59,20007,2098],{},[63,20009,11777],{"encoding":65},[39,20011,20013],{"className":20012,"ariaHidden":70},[69],[39,20014,20016,20019],{"className":20015},[74],[39,20017],{"className":20018,"style":79},[78],[39,20020,20022,20025],{"className":20021},[83],[39,20023,11772],{"className":20024,"style":11793},[83,641],[39,20026,20028],{"className":20027},[192],[39,20029,20031],{"className":20030},[196],[39,20032,20034],{"className":20033},[200],[39,20035,20037],{"className":20036,"style":6410},[204],[39,20038,20039,20042],{"style":208},[39,20040],{"className":20041,"style":213},[212],[39,20043,20045],{"className":20044},[217,218,219,220],[39,20046,2098],{"className":20047,"style":2114},[83,84,220]," 在有界函数空间中至多有一个不动点。",[11,20050,20051,20053,20054,20139],{},[1489,20052,5595],{}," 设 ",[39,20055,20057,20080],{"className":20056,"translate":43},[42],[39,20058,20060],{"className":20059},[47],[49,20061,20062],{"xmlns":51},[53,20063,20064,20078],{},[56,20065,20066,20068,20070,20076],{},[59,20067,4230],{},[133,20069,793],{},[138,20071,20072,20074],{},[59,20073,11772],{"mathvariant":617},[59,20075,2098],{},[59,20077,4230],{},[63,20079,19848],{"encoding":65},[39,20081,20083,20101],{"className":20082,"ariaHidden":70},[69],[39,20084,20086,20089,20092,20095,20098],{"className":20085},[74],[39,20087],{"className":20088,"style":79},[78],[39,20090,4230],{"className":20091,"style":461},[83,84],[39,20093],{"className":20094,"style":231},[230],[39,20096,793],{"className":20097},[235],[39,20099],{"className":20100,"style":231},[230],[39,20102,20104,20107,20136],{"className":20103},[74],[39,20105],{"className":20106,"style":79},[78],[39,20108,20110,20113],{"className":20109},[83],[39,20111,11772],{"className":20112,"style":11793},[83,641],[39,20114,20116],{"className":20115},[192],[39,20117,20119],{"className":20118},[196],[39,20120,20122],{"className":20121},[200],[39,20123,20125],{"className":20124,"style":6410},[204],[39,20126,20127,20130],{"style":208},[39,20128],{"className":20129,"style":213},[212],[39,20131,20133],{"className":20132},[217,218,219,220],[39,20134,2098],{"className":20135,"style":2114},[83,84,220],[39,20137,4230],{"className":20138,"style":461},[83,84],"。反复代入得",[39,20141,20143],{"className":20142,"translate":43},[746],[39,20144,20146,20234],{"className":20145,"translate":43},[42],[39,20147,20149],{"className":20148},[47],[49,20150,20151],{"xmlns":51,"display":755},[53,20152,20153,20231],{},[56,20154,20155,20157,20159,20179,20185,20187,20193,20199,20205,20207,20213,20215,20221,20227,20229],{},[59,20156,4230],{},[133,20158,793],{},[3187,20160,20161,20163,20171],{},[133,20162,3191],{},[56,20164,20165,20167,20169],{},[59,20166,3196],{},[133,20168,793],{},[539,20170,541],{},[56,20172,20173,20175,20177],{},[59,20174,12962],{},[133,20176,5378],{},[539,20178,546],{},[138,20180,20181,20183],{},[59,20182,531],{},[59,20184,3196],{},[133,20186,136],{"stretchy":135},[138,20188,20189,20191],{},[59,20190,131],{},[59,20192,2098],{},[138,20194,20195,20197],{},[133,20196,161],{"stretchy":135},[59,20198,3196],{},[138,20200,20201,20203],{},[59,20202,401],{},[59,20204,2098],{},[133,20206,788],{},[138,20208,20209,20211],{},[59,20210,531],{},[59,20212,12962],{},[133,20214,136],{"stretchy":135},[138,20216,20217,20219],{},[59,20218,131],{},[59,20220,2098],{},[138,20222,20223,20225],{},[133,20224,161],{"stretchy":135},[59,20226,12962],{},[59,20228,4230],{},[59,20230,2903],{"mathvariant":145},[63,20232,20233],{"encoding":65},"V=\\sum_{k=0}^{n-1}\\gamma^k(P^\\pi)^k r^\\pi\n+\\gamma^n(P^\\pi)^n V.",[39,20235,20237,20255,20467],{"className":20236,"ariaHidden":70},[69],[39,20238,20240,20243,20246,20249,20252],{"className":20239},[74],[39,20241],{"className":20242,"style":79},[78],[39,20244,4230],{"className":20245,"style":461},[83,84],[39,20247],{"className":20248,"style":231},[230],[39,20250,793],{"className":20251},[235],[39,20253],{"className":20254,"style":231},[230],[39,20256,20258,20262,20336,20339,20368,20371,20400,20429,20458,20461,20464],{"className":20257},[74],[39,20259],{"className":20260,"style":20261},[78],"height:3.1032em;vertical-align:-1.3021em;",[39,20263,20265],{"className":20264},[1744,3294],[39,20266,20268,20328],{"className":20267},[196,649],[39,20269,20271,20325],{"className":20270},[200],[39,20272,20275,20295,20305],{"className":20273,"style":20274},[204],"height:1.8011em;",[39,20276,20277,20280],{"style":3307},[39,20278],{"className":20279,"style":3311},[212],[39,20281,20283],{"className":20282},[217,218,219,220],[39,20284,20286,20289,20292],{"className":20285},[83,220],[39,20287,3196],{"className":20288,"style":3321},[83,84,220],[39,20290,793],{"className":20291},[235,220],[39,20293,541],{"className":20294},[83,220],[39,20296,20297,20300],{"style":3330},[39,20298],{"className":20299,"style":3311},[212],[39,20301,20302],{},[39,20303,3191],{"className":20304},[1744,3339,3340],[39,20306,20307,20310],{"style":3343},[39,20308],{"className":20309,"style":3311},[212],[39,20311,20313],{"className":20312},[217,218,219,220],[39,20314,20316,20319,20322],{"className":20315},[83,220],[39,20317,12962],{"className":20318},[83,84,220],[39,20320,5378],{"className":20321},[465,220],[39,20323,546],{"className":20324},[83,220],[39,20326,673],{"className":20327},[672],[39,20329,20331],{"className":20330},[200],[39,20332,20334],{"className":20333,"style":3365},[204],[39,20335],{},[39,20337],{"className":20338,"style":256},[230],[39,20340,20342,20345],{"className":20341},[83],[39,20343,531],{"className":20344,"style":565},[83,84],[39,20346,20348],{"className":20347},[192],[39,20349,20351],{"className":20350},[196],[39,20352,20354],{"className":20353},[200],[39,20355,20357],{"className":20356,"style":3389},[204],[39,20358,20359,20362],{"style":1027},[39,20360],{"className":20361,"style":213},[212],[39,20363,20365],{"className":20364},[217,218,219,220],[39,20366,3196],{"className":20367,"style":3321},[83,84,220],[39,20369,136],{"className":20370},[182],[39,20372,20374,20377],{"className":20373},[83],[39,20375,131],{"className":20376,"style":178},[83,84],[39,20378,20380],{"className":20379},[192],[39,20381,20383],{"className":20382},[196],[39,20384,20386],{"className":20385},[200],[39,20387,20389],{"className":20388,"style":4421},[204],[39,20390,20391,20394],{"style":1027},[39,20392],{"className":20393,"style":213},[212],[39,20395,20397],{"className":20396},[217,218,219,220],[39,20398,2098],{"className":20399,"style":2114},[83,84,220],[39,20401,20403,20406],{"className":20402},[263],[39,20404,161],{"className":20405},[263],[39,20407,20409],{"className":20408},[192],[39,20410,20412],{"className":20411},[196],[39,20413,20415],{"className":20414},[200],[39,20416,20418],{"className":20417,"style":3389},[204],[39,20419,20420,20423],{"style":1027},[39,20421],{"className":20422,"style":213},[212],[39,20424,20426],{"className":20425},[217,218,219,220],[39,20427,3196],{"className":20428,"style":3321},[83,84,220],[39,20430,20432,20435],{"className":20431},[83],[39,20433,401],{"className":20434,"style":438},[83,84],[39,20436,20438],{"className":20437},[192],[39,20439,20441],{"className":20440},[196],[39,20442,20444],{"className":20443},[200],[39,20445,20447],{"className":20446,"style":4421},[204],[39,20448,20449,20452],{"style":1027},[39,20450],{"className":20451,"style":213},[212],[39,20453,20455],{"className":20454},[217,218,219,220],[39,20456,2098],{"className":20457,"style":2114},[83,84,220],[39,20459],{"className":20460,"style":461},[230],[39,20462,788],{"className":20463},[465],[39,20465],{"className":20466,"style":461},[230],[39,20468,20470,20473,20502,20505,20534,20563,20566],{"className":20469},[74],[39,20471],{"className":20472,"style":245},[78],[39,20474,20476,20479],{"className":20475},[83],[39,20477,531],{"className":20478,"style":565},[83,84],[39,20480,20482],{"className":20481},[192],[39,20483,20485],{"className":20484},[196],[39,20486,20488],{"className":20487},[200],[39,20489,20491],{"className":20490,"style":4421},[204],[39,20492,20493,20496],{"style":1027},[39,20494],{"className":20495,"style":213},[212],[39,20497,20499],{"className":20498},[217,218,219,220],[39,20500,12962],{"className":20501},[83,84,220],[39,20503,136],{"className":20504},[182],[39,20506,20508,20511],{"className":20507},[83],[39,20509,131],{"className":20510,"style":178},[83,84],[39,20512,20514],{"className":20513},[192],[39,20515,20517],{"className":20516},[196],[39,20518,20520],{"className":20519},[200],[39,20521,20523],{"className":20522,"style":4421},[204],[39,20524,20525,20528],{"style":1027},[39,20526],{"className":20527,"style":213},[212],[39,20529,20531],{"className":20530},[217,218,219,220],[39,20532,2098],{"className":20533,"style":2114},[83,84,220],[39,20535,20537,20540],{"className":20536},[263],[39,20538,161],{"className":20539},[263],[39,20541,20543],{"className":20542},[192],[39,20544,20546],{"className":20545},[196],[39,20547,20549],{"className":20548},[200],[39,20550,20552],{"className":20551,"style":4421},[204],[39,20553,20554,20557],{"style":1027},[39,20555],{"className":20556,"style":213},[212],[39,20558,20560],{"className":20559},[217,218,219,220],[39,20561,12962],{"className":20562},[83,84,220],[39,20564,4230],{"className":20565,"style":461},[83,84],[39,20567,2903],{"className":20568},[83],[11,20570,20571,20572,20800,20801,20829],{},"由于 ",[39,20573,20575,20623],{"className":20574,"translate":43},[42],[39,20576,20578],{"className":20577},[47],[49,20579,20580],{"xmlns":51},[53,20581,20582,20620],{},[56,20583,20584,20586,20588,20594,20600,20602,20608,20610,20612,20614],{},[59,20585,15389],{"mathvariant":145},[133,20587,136],{"stretchy":135},[138,20589,20590,20592],{},[59,20591,131],{},[59,20593,2098],{},[138,20595,20596,20598],{},[133,20597,161],{"stretchy":135},[59,20599,12962],{},[59,20601,4230],{},[613,20603,20604,20606],{},[59,20605,15389],{"mathvariant":145},[59,20607,1696],{"mathvariant":145},[133,20609,1824],{},[59,20611,15389],{"mathvariant":145},[59,20613,4230],{},[613,20615,20616,20618],{},[59,20617,15389],{"mathvariant":145},[59,20619,1696],{"mathvariant":145},[63,20621,20622],{"encoding":65},"\\|(P^\\pi)^n V\\|_\\infty\\le\\|V\\|_\\infty",[39,20624,20626,20748],{"className":20625,"ariaHidden":70},[69],[39,20627,20629,20632,20635,20638,20667,20696,20699,20739,20742,20745],{"className":20628},[74],[39,20630],{"className":20631,"style":245},[78],[39,20633,15389],{"className":20634},[83],[39,20636,136],{"className":20637},[182],[39,20639,20641,20644],{"className":20640},[83],[39,20642,131],{"className":20643,"style":178},[83,84],[39,20645,20647],{"className":20646},[192],[39,20648,20650],{"className":20649},[196],[39,20651,20653],{"className":20652},[200],[39,20654,20656],{"className":20655,"style":6410},[204],[39,20657,20658,20661],{"style":208},[39,20659],{"className":20660,"style":213},[212],[39,20662,20664],{"className":20663},[217,218,219,220],[39,20665,2098],{"className":20666,"style":2114},[83,84,220],[39,20668,20670,20673],{"className":20669},[263],[39,20671,161],{"className":20672},[263],[39,20674,20676],{"className":20675},[192],[39,20677,20679],{"className":20678},[196],[39,20680,20682],{"className":20681},[200],[39,20683,20685],{"className":20684,"style":6410},[204],[39,20686,20687,20690],{"style":208},[39,20688],{"className":20689,"style":213},[212],[39,20691,20693],{"className":20692},[217,218,219,220],[39,20694,12962],{"className":20695},[83,84,220],[39,20697,4230],{"className":20698,"style":461},[83,84],[39,20700,20702,20705],{"className":20701},[83],[39,20703,15389],{"className":20704},[83],[39,20706,20708],{"className":20707},[192],[39,20709,20711,20731],{"className":20710},[196,649],[39,20712,20714,20728],{"className":20713},[200],[39,20715,20717],{"className":20716,"style":1728},[204],[39,20718,20719,20722],{"style":966},[39,20720],{"className":20721,"style":213},[212],[39,20723,20725],{"className":20724},[217,218,219,220],[39,20726,1696],{"className":20727},[83,220],[39,20729,673],{"className":20730},[672],[39,20732,20734],{"className":20733},[200],[39,20735,20737],{"className":20736,"style":680},[204],[39,20738],{},[39,20740],{"className":20741,"style":231},[230],[39,20743,1824],{"className":20744},[235],[39,20746],{"className":20747,"style":231},[230],[39,20749,20751,20754,20757,20760],{"className":20750},[74],[39,20752],{"className":20753,"style":245},[78],[39,20755,15389],{"className":20756},[83],[39,20758,4230],{"className":20759,"style":461},[83,84],[39,20761,20763,20766],{"className":20762},[83],[39,20764,15389],{"className":20765},[83],[39,20767,20769],{"className":20768},[192],[39,20770,20772,20792],{"className":20771},[196,649],[39,20773,20775,20789],{"className":20774},[200],[39,20776,20778],{"className":20777,"style":1728},[204],[39,20779,20780,20783],{"style":966},[39,20781],{"className":20782,"style":213},[212],[39,20784,20786],{"className":20785},[217,218,219,220],[39,20787,1696],{"className":20788},[83,220],[39,20790,673],{"className":20791},[672],[39,20793,20795],{"className":20794},[200],[39,20796,20798],{"className":20797,"style":680},[204],[39,20799],{},"，最后一项在 sup 范数下趋于 ",[39,20802,20804,20817],{"className":20803,"translate":43},[42],[39,20805,20807],{"className":20806},[47],[49,20808,20809],{"xmlns":51},[53,20810,20811,20815],{},[56,20812,20813],{},[539,20814,541],{},[63,20816,541],{"encoding":65},[39,20818,20820],{"className":20819,"ariaHidden":70},[69],[39,20821,20823,20826],{"className":20822},[74],[39,20824],{"className":20825,"style":2473},[78],[39,20827,541],{"className":20828},[83],"。另一方面，",[39,20831,20833],{"className":20832,"translate":43},[746],[39,20834,20836,20888],{"className":20835,"translate":43},[42],[39,20837,20839],{"className":20838},[47],[49,20840,20841],{"xmlns":51,"display":755},[53,20842,20843,20885],{},[56,20844,20845,20859,20865,20867,20873,20879],{},[3187,20846,20847,20849,20857],{},[133,20848,3191],{},[56,20850,20851,20853,20855],{},[59,20852,3196],{},[133,20854,793],{},[539,20856,541],{},[59,20858,1696],{"mathvariant":145},[138,20860,20861,20863],{},[59,20862,531],{},[59,20864,3196],{},[133,20866,136],{"stretchy":135},[138,20868,20869,20871],{},[59,20870,131],{},[59,20872,2098],{},[138,20874,20875,20877],{},[133,20876,161],{"stretchy":135},[59,20878,3196],{},[138,20880,20881,20883],{},[59,20882,401],{},[59,20884,2098],{},[63,20886,20887],{"encoding":65},"\\sum_{k=0}^{\\infty}\\gamma^k(P^\\pi)^k r^\\pi",[39,20889,20891],{"className":20890,"ariaHidden":70},[69],[39,20892,20894,20897,20964,20967,20996,20999,21028,21057],{"className":20893},[74],[39,20895],{"className":20896,"style":3290},[78],[39,20898,20900],{"className":20899},[1744,3294],[39,20901,20903,20956],{"className":20902},[196,649],[39,20904,20906,20953],{"className":20905},[200],[39,20907,20909,20929,20939],{"className":20908,"style":3304},[204],[39,20910,20911,20914],{"style":3307},[39,20912],{"className":20913,"style":3311},[212],[39,20915,20917],{"className":20916},[217,218,219,220],[39,20918,20920,20923,20926],{"className":20919},[83,220],[39,20921,3196],{"className":20922,"style":3321},[83,84,220],[39,20924,793],{"className":20925},[235,220],[39,20927,541],{"className":20928},[83,220],[39,20930,20931,20934],{"style":3330},[39,20932],{"className":20933,"style":3311},[212],[39,20935,20936],{},[39,20937,3191],{"className":20938},[1744,3339,3340],[39,20940,20941,20944],{"style":3343},[39,20942],{"className":20943,"style":3311},[212],[39,20945,20947],{"className":20946},[217,218,219,220],[39,20948,20950],{"className":20949},[83,220],[39,20951,1696],{"className":20952},[83,220],[39,20954,673],{"className":20955},[672],[39,20957,20959],{"className":20958},[200],[39,20960,20962],{"className":20961,"style":3365},[204],[39,20963],{},[39,20965],{"className":20966,"style":256},[230],[39,20968,20970,20973],{"className":20969},[83],[39,20971,531],{"className":20972,"style":565},[83,84],[39,20974,20976],{"className":20975},[192],[39,20977,20979],{"className":20978},[196],[39,20980,20982],{"className":20981},[200],[39,20983,20985],{"className":20984,"style":3389},[204],[39,20986,20987,20990],{"style":1027},[39,20988],{"className":20989,"style":213},[212],[39,20991,20993],{"className":20992},[217,218,219,220],[39,20994,3196],{"className":20995,"style":3321},[83,84,220],[39,20997,136],{"className":20998},[182],[39,21000,21002,21005],{"className":21001},[83],[39,21003,131],{"className":21004,"style":178},[83,84],[39,21006,21008],{"className":21007},[192],[39,21009,21011],{"className":21010},[196],[39,21012,21014],{"className":21013},[200],[39,21015,21017],{"className":21016,"style":4421},[204],[39,21018,21019,21022],{"style":1027},[39,21020],{"className":21021,"style":213},[212],[39,21023,21025],{"className":21024},[217,218,219,220],[39,21026,2098],{"className":21027,"style":2114},[83,84,220],[39,21029,21031,21034],{"className":21030},[263],[39,21032,161],{"className":21033},[263],[39,21035,21037],{"className":21036},[192],[39,21038,21040],{"className":21039},[196],[39,21041,21043],{"className":21042},[200],[39,21044,21046],{"className":21045,"style":3389},[204],[39,21047,21048,21051],{"style":1027},[39,21049],{"className":21050,"style":213},[212],[39,21052,21054],{"className":21053},[217,218,219,220],[39,21055,3196],{"className":21056,"style":3321},[83,84,220],[39,21058,21060,21063],{"className":21059},[83],[39,21061,401],{"className":21062,"style":438},[83,84],[39,21064,21066],{"className":21065},[192],[39,21067,21069],{"className":21068},[196],[39,21070,21072],{"className":21071},[200],[39,21073,21075],{"className":21074,"style":4421},[204],[39,21076,21077,21080],{"style":1027},[39,21078],{"className":21079,"style":213},[212],[39,21081,21083],{"className":21082},[217,218,219,220],[39,21084,2098],{"className":21085,"style":2114},[83,84,220],[11,21087,21088,21089,21147,21148,21228],{},"正是折扣回报的期望，即 ",[39,21090,21092,21109],{"className":21091,"translate":43},[42],[39,21093,21095],{"className":21094},[47],[49,21096,21097],{"xmlns":51},[53,21098,21099,21107],{},[56,21100,21101],{},[138,21102,21103,21105],{},[59,21104,4230],{},[59,21106,2098],{},[63,21108,6382],{"encoding":65},[39,21110,21112],{"className":21111,"ariaHidden":70},[69],[39,21113,21115,21118],{"className":21114},[74],[39,21116],{"className":21117,"style":79},[78],[39,21119,21121,21124],{"className":21120},[83],[39,21122,4230],{"className":21123,"style":461},[83,84],[39,21125,21127],{"className":21126},[192],[39,21128,21130],{"className":21129},[196],[39,21131,21133],{"className":21132},[200],[39,21134,21136],{"className":21135,"style":6410},[204],[39,21137,21138,21141],{"style":208},[39,21139],{"className":21140,"style":213},[212],[39,21142,21144],{"className":21143},[217,218,219,220],[39,21145,2098],{"className":21146,"style":2114},[83,84,220],"。故 ",[39,21149,21151,21172],{"className":21150,"translate":43},[42],[39,21152,21154],{"className":21153},[47],[49,21155,21156],{"xmlns":51},[53,21157,21158,21170],{},[56,21159,21160,21162,21164],{},[59,21161,4230],{},[133,21163,793],{},[138,21165,21166,21168],{},[59,21167,4230],{},[59,21169,2098],{},[63,21171,19932],{"encoding":65},[39,21173,21175,21193],{"className":21174,"ariaHidden":70},[69],[39,21176,21178,21181,21184,21187,21190],{"className":21177},[74],[39,21179],{"className":21180,"style":79},[78],[39,21182,4230],{"className":21183,"style":461},[83,84],[39,21185],{"className":21186,"style":231},[230],[39,21188,793],{"className":21189},[235],[39,21191],{"className":21192,"style":231},[230],[39,21194,21196,21199],{"className":21195},[74],[39,21197],{"className":21198,"style":79},[78],[39,21200,21202,21205],{"className":21201},[83],[39,21203,4230],{"className":21204,"style":461},[83,84],[39,21206,21208],{"className":21207},[192],[39,21209,21211],{"className":21210},[196],[39,21212,21214],{"className":21213},[200],[39,21215,21217],{"className":21216,"style":6410},[204],[39,21218,21219,21222],{"style":208},[39,21220],{"className":21221,"style":213},[212],[39,21223,21225],{"className":21224},[217,218,219,220],[39,21226,2098],{"className":21227,"style":2114},[83,84,220],"。证毕。",[11,21230,21231,21232,21290,21291,21295,21296,21346],{},"这个证明不依赖矩阵的有限维结构，只用到 ",[39,21233,21235,21252],{"className":21234,"translate":43},[42],[39,21236,21238],{"className":21237},[47],[49,21239,21240],{"xmlns":51},[53,21241,21242,21250],{},[56,21243,21244],{},[138,21245,21246,21248],{},[59,21247,131],{},[59,21249,2098],{},[63,21251,15306],{"encoding":65},[39,21253,21255],{"className":21254,"ariaHidden":70},[69],[39,21256,21258,21261],{"className":21257},[74],[39,21259],{"className":21260,"style":79},[78],[39,21262,21264,21267],{"className":21263},[83],[39,21265,131],{"className":21266,"style":178},[83,84],[39,21268,21270],{"className":21269},[192],[39,21271,21273],{"className":21272},[196],[39,21274,21276],{"className":21275},[200],[39,21277,21279],{"className":21278,"style":6410},[204],[39,21280,21281,21284],{"style":208},[39,21282],{"className":21283,"style":213},[212],[39,21285,21287],{"className":21286},[217,218,219,220],[39,21288,2098],{"className":21289,"style":2114},[83,84,220]," 是随机核这一事实。因此它可以直接推广到一般状态空间。",[158,21292,21294],{"href":21293},"#proposition-policy-verification","命题 2.3"," 也说明了贝尔曼方程的一个常见误读：它并不是「用未来价值近似当前价值」，而是在精确地刻画值函数。只要 ",[39,21297,21299,21316],{"className":21298,"translate":43},[42],[39,21300,21302],{"className":21301},[47],[49,21303,21304],{"xmlns":51},[53,21305,21306,21314],{},[56,21307,21308,21310,21312],{},[59,21309,531],{},[133,21311,1693],{},[539,21313,546],{},[63,21315,17296],{"encoding":65},[39,21317,21319,21337],{"className":21318,"ariaHidden":70},[69],[39,21320,21322,21325,21328,21331,21334],{"className":21321},[74],[39,21323],{"className":21324,"style":561},[78],[39,21326,531],{"className":21327,"style":565},[83,84],[39,21329],{"className":21330,"style":231},[230],[39,21332,1693],{"className":21333},[235],[39,21335],{"className":21336,"style":231},[230],[39,21338,21340,21343],{"className":21339},[74],[39,21341],{"className":21342,"style":2473},[78],[39,21344,546],{"className":21345},[83]," 且奖励有界，满足方程的函数就是值函数本身。",[21,21348,21350],{"id":21349},"三贝尔曼算子与压缩映射","三、贝尔曼算子与压缩映射",[26,21352,21354],{"id":21353},"_31-有界函数空间","3.1 有界函数空间",[11,21356,21357,21358,21402,21403,21431],{},"记 ",[39,21359,21361,21381],{"className":21360,"translate":43},[42],[39,21362,21364],{"className":21363},[47],[49,21365,21366],{"xmlns":51},[53,21367,21368,21378],{},[56,21369,21370,21372,21374,21376],{},[59,21371,19772],{},[133,21373,136],{"stretchy":135},[59,21375,61],{},[133,21377,161],{"stretchy":135},[63,21379,21380],{"encoding":65},"B(S)",[39,21382,21384],{"className":21383,"ariaHidden":70},[69],[39,21385,21387,21390,21393,21396,21399],{"className":21386},[74],[39,21388],{"className":21389,"style":245},[78],[39,21391,19772],{"className":21392,"style":19812},[83,84],[39,21394,136],{"className":21395},[182],[39,21397,61],{"className":21398,"style":85},[83,84],[39,21400,161],{"className":21401},[263]," 为 ",[39,21404,21406,21419],{"className":21405,"translate":43},[42],[39,21407,21409],{"className":21408},[47],[49,21410,21411],{"xmlns":51},[53,21412,21413,21417],{},[56,21414,21415],{},[59,21416,61],{},[63,21418,61],{"encoding":65},[39,21420,21422],{"className":21421,"ariaHidden":70},[69],[39,21423,21425,21428],{"className":21424},[74],[39,21426],{"className":21427,"style":79},[78],[39,21429,61],{"className":21430,"style":85},[83,84]," 上全体有界实值函数构成的空间，并赋予 sup 范数",[39,21433,21435],{"className":21434,"translate":43},[746],[39,21436,21438,21493],{"className":21437,"translate":43},[42],[39,21439,21441],{"className":21440},[47],[49,21442,21443],{"xmlns":51,"display":755},[53,21444,21445,21490],{},[56,21446,21447,21449,21451,21457,21459,21476,21478,21480,21482,21484,21486,21488],{},[59,21448,15389],{"mathvariant":145},[59,21450,4230],{},[613,21452,21453,21455],{},[59,21454,15389],{"mathvariant":145},[59,21456,1696],{"mathvariant":145},[133,21458,793],{},[6556,21460,21461,21468],{},[56,21462,21463,21466],{},[59,21464,21465],{},"sup",[133,21467,1690],{},[56,21469,21470,21472,21474],{},[59,21471,142],{},[133,21473,534],{},[59,21475,61],{},[59,21477,150],{"mathvariant":145},[59,21479,4230],{},[133,21481,136],{"stretchy":135},[59,21483,142],{},[133,21485,161],{"stretchy":135},[59,21487,150],{"mathvariant":145},[59,21489,2903],{"mathvariant":145},[63,21491,21492],{"encoding":65},"\\|V\\|_\\infty=\\sup_{s\\in S}|V(s)|.",[39,21494,21496,21557],{"className":21495,"ariaHidden":70},[69],[39,21497,21499,21502,21505,21508,21548,21551,21554],{"className":21498},[74],[39,21500],{"className":21501,"style":245},[78],[39,21503,15389],{"className":21504},[83],[39,21506,4230],{"className":21507,"style":461},[83,84],[39,21509,21511,21514],{"className":21510},[83],[39,21512,15389],{"className":21513},[83],[39,21515,21517],{"className":21516},[192],[39,21518,21520,21540],{"className":21519},[196,649],[39,21521,21523,21537],{"className":21522},[200],[39,21524,21526],{"className":21525,"style":1728},[204],[39,21527,21528,21531],{"style":966},[39,21529],{"className":21530,"style":213},[212],[39,21532,21534],{"className":21533},[217,218,219,220],[39,21535,1696],{"className":21536},[83,220],[39,21538,673],{"className":21539},[672],[39,21541,21543],{"className":21542},[200],[39,21544,21546],{"className":21545,"style":680},[204],[39,21547],{},[39,21549],{"className":21550,"style":231},[230],[39,21552,793],{"className":21553},[235],[39,21555],{"className":21556,"style":231},[230],[39,21558,21560,21564,21620,21623,21626,21629,21632,21635,21638],{"className":21559},[74],[39,21561],{"className":21562,"style":21563},[78],"height:1.7161em;vertical-align:-0.9661em;",[39,21565,21567],{"className":21566},[1744,3294],[39,21568,21570,21611],{"className":21569},[196,649],[39,21571,21573,21608],{"className":21572},[200],[39,21574,21576,21597],{"className":21575,"style":290},[204],[39,21577,21579,21582],{"style":21578},"top:-2.1612em;margin-left:0em;",[39,21580],{"className":21581,"style":935},[212],[39,21583,21585],{"className":21584},[217,218,219,220],[39,21586,21588,21591,21594],{"className":21587},[83,220],[39,21589,142],{"className":21590},[83,84,220],[39,21592,534],{"className":21593},[235,220],[39,21595,61],{"className":21596,"style":85},[83,84,220],[39,21598,21600,21603],{"style":21599},"top:-3em;",[39,21601],{"className":21602,"style":935},[212],[39,21604,21605],{},[39,21606,21465],{"className":21607},[1744],[39,21609,673],{"className":21610},[672],[39,21612,21614],{"className":21613},[200],[39,21615,21618],{"className":21616,"style":21617},[204],"height:0.9661em;",[39,21619],{},[39,21621],{"className":21622,"style":256},[230],[39,21624,150],{"className":21625},[83],[39,21627,4230],{"className":21628,"style":461},[83,84],[39,21630,136],{"className":21631},[182],[39,21633,142],{"className":21634},[83,84],[39,21636,161],{"className":21637},[263],[39,21639,21641],{"className":21640},[83],"∣.",[11,21643,12932,21644,21672,21673,21803],{},[39,21645,21647,21660],{"className":21646,"translate":43},[42],[39,21648,21650],{"className":21649},[47],[49,21651,21652],{"xmlns":51},[53,21653,21654,21658],{},[56,21655,21656],{},[59,21657,61],{},[63,21659,61],{"encoding":65},[39,21661,21663],{"className":21662,"ariaHidden":70},[69],[39,21664,21666,21669],{"className":21665},[74],[39,21667],{"className":21668,"style":79},[78],[39,21670,61],{"className":21671,"style":85},[83,84]," 为有限集时，上确界就是最大值。",[39,21674,21676,21712],{"className":21675,"translate":43},[42],[39,21677,21679],{"className":21678},[47],[49,21680,21681],{"xmlns":51},[53,21682,21683,21709],{},[56,21684,21685,21687,21689,21691,21693,21695,21697,21699,21701,21707],{},[133,21686,136],{"stretchy":135},[59,21688,19772],{},[133,21690,136],{"stretchy":135},[59,21692,61],{},[133,21694,161],{"stretchy":135},[133,21696,155],{"separator":70},[59,21698,15389],{"mathvariant":145},[133,21700,8885],{},[613,21702,21703,21705],{},[59,21704,15389],{"mathvariant":145},[59,21706,1696],{"mathvariant":145},[133,21708,161],{"stretchy":135},[63,21710,21711],{"encoding":65},"(B(S),\\|\\cdot\\|_\\infty)",[39,21713,21715,21754],{"className":21714,"ariaHidden":70},[69],[39,21716,21718,21721,21724,21727,21730,21733,21736,21739,21742,21745,21748,21751],{"className":21717},[74],[39,21719],{"className":21720,"style":245},[78],[39,21722,136],{"className":21723},[182],[39,21725,19772],{"className":21726,"style":19812},[83,84],[39,21728,136],{"className":21729},[182],[39,21731,61],{"className":21732,"style":85},[83,84],[39,21734,161],{"className":21735},[263],[39,21737,155],{"className":21738},[252],[39,21740],{"className":21741,"style":256},[230],[39,21743,15389],{"className":21744},[83],[39,21746],{"className":21747,"style":461},[230],[39,21749,8885],{"className":21750},[465],[39,21752],{"className":21753,"style":461},[230],[39,21755,21757,21760,21800],{"className":21756},[74],[39,21758],{"className":21759,"style":245},[78],[39,21761,21763,21766],{"className":21762},[83],[39,21764,15389],{"className":21765},[83],[39,21767,21769],{"className":21768},[192],[39,21770,21772,21792],{"className":21771},[196,649],[39,21773,21775,21789],{"className":21774},[200],[39,21776,21778],{"className":21777,"style":1728},[204],[39,21779,21780,21783],{"style":966},[39,21781],{"className":21782,"style":213},[212],[39,21784,21786],{"className":21785},[217,218,219,220],[39,21787,1696],{"className":21788},[83,220],[39,21790,673],{"className":21791},[672],[39,21793,21795],{"className":21794},[200],[39,21796,21798],{"className":21797,"style":680},[204],[39,21799],{},[39,21801,161],{"className":21802},[263]," 是完备的 Banach 空间。这个完备性是 Banach 不动点定理能够使用的关键。",[11,21805,21806,21807,21865,21866,21927,21928,21971,21972,22030,22031,22089],{},"贝尔曼期望算子 ",[39,21808,21810,21827],{"className":21809,"translate":43},[42],[39,21811,21813],{"className":21812},[47],[49,21814,21815],{"xmlns":51},[53,21816,21817,21825],{},[56,21818,21819],{},[138,21820,21821,21823],{},[59,21822,11772],{"mathvariant":617},[59,21824,2098],{},[63,21826,11777],{"encoding":65},[39,21828,21830],{"className":21829,"ariaHidden":70},[69],[39,21831,21833,21836],{"className":21832},[74],[39,21834],{"className":21835,"style":79},[78],[39,21837,21839,21842],{"className":21838},[83],[39,21840,11772],{"className":21841,"style":11793},[83,641],[39,21843,21845],{"className":21844},[192],[39,21846,21848],{"className":21847},[196],[39,21849,21851],{"className":21850},[200],[39,21852,21854],{"className":21853,"style":6410},[204],[39,21855,21856,21859],{"style":208},[39,21857],{"className":21858,"style":213},[212],[39,21860,21862],{"className":21861},[217,218,219,220],[39,21863,2098],{"className":21864,"style":2114},[83,84,220]," 与贝尔曼最优算子 ",[39,21867,21869,21888],{"className":21868,"translate":43},[42],[39,21870,21872],{"className":21871},[47],[49,21873,21874],{"xmlns":51},[53,21875,21876,21885],{},[56,21877,21878],{},[138,21879,21880,21882],{},[59,21881,11772],{"mathvariant":617},[133,21883,21884],{},"∗",[63,21886,21887],{"encoding":65},"\\mathcal{T}^*",[39,21889,21891],{"className":21890,"ariaHidden":70},[69],[39,21892,21894,21898],{"className":21893},[74],[39,21895],{"className":21896,"style":21897},[78],"height:0.6887em;",[39,21899,21901,21904],{"className":21900},[83],[39,21902,11772],{"className":21903,"style":11793},[83,641],[39,21905,21907],{"className":21906},[192],[39,21908,21910],{"className":21909},[196],[39,21911,21913],{"className":21912},[200],[39,21914,21916],{"className":21915,"style":21897},[204],[39,21917,21918,21921],{"style":208},[39,21919],{"className":21920,"style":213},[212],[39,21922,21924],{"className":21923},[217,218,219,220],[39,21925,21884],{"className":21926},[465,220]," 都将 ",[39,21929,21931,21950],{"className":21930,"translate":43},[42],[39,21932,21934],{"className":21933},[47],[49,21935,21936],{"xmlns":51},[53,21937,21938,21948],{},[56,21939,21940,21942,21944,21946],{},[59,21941,19772],{},[133,21943,136],{"stretchy":135},[59,21945,61],{},[133,21947,161],{"stretchy":135},[63,21949,21380],{"encoding":65},[39,21951,21953],{"className":21952,"ariaHidden":70},[69],[39,21954,21956,21959,21962,21965,21968],{"className":21955},[74],[39,21957],{"className":21958,"style":245},[78],[39,21960,19772],{"className":21961,"style":19812},[83,84],[39,21963,136],{"className":21964},[182],[39,21966,61],{"className":21967,"style":85},[83,84],[39,21969,161],{"className":21970},[263]," 映射到自身。",[39,21973,21975,21992],{"className":21974,"translate":43},[42],[39,21976,21978],{"className":21977},[47],[49,21979,21980],{"xmlns":51},[53,21981,21982,21990],{},[56,21983,21984],{},[138,21985,21986,21988],{},[59,21987,11772],{"mathvariant":617},[59,21989,2098],{},[63,21991,11777],{"encoding":65},[39,21993,21995],{"className":21994,"ariaHidden":70},[69],[39,21996,21998,22001],{"className":21997},[74],[39,21999],{"className":22000,"style":79},[78],[39,22002,22004,22007],{"className":22003},[83],[39,22005,11772],{"className":22006,"style":11793},[83,641],[39,22008,22010],{"className":22009},[192],[39,22011,22013],{"className":22012},[196],[39,22014,22016],{"className":22015},[200],[39,22017,22019],{"className":22018,"style":6410},[204],[39,22020,22021,22024],{"style":208},[39,22022],{"className":22023,"style":213},[212],[39,22025,22027],{"className":22026},[217,218,219,220],[39,22028,2098],{"className":22029,"style":2114},[83,84,220]," 的定义已在 2.2 节给出，",[39,22032,22034,22051],{"className":22033,"translate":43},[42],[39,22035,22037],{"className":22036},[47],[49,22038,22039],{"xmlns":51},[53,22040,22041,22049],{},[56,22042,22043],{},[138,22044,22045,22047],{},[59,22046,11772],{"mathvariant":617},[133,22048,21884],{},[63,22050,21887],{"encoding":65},[39,22052,22054],{"className":22053,"ariaHidden":70},[69],[39,22055,22057,22060],{"className":22056},[74],[39,22058],{"className":22059,"style":21897},[78],[39,22061,22063,22066],{"className":22062},[83],[39,22064,11772],{"className":22065,"style":11793},[83,641],[39,22067,22069],{"className":22068},[192],[39,22070,22072],{"className":22071},[196],[39,22073,22075],{"className":22074},[200],[39,22076,22078],{"className":22077,"style":21897},[204],[39,22079,22080,22083],{"style":208},[39,22081],{"className":22082,"style":213},[212],[39,22084,22086],{"className":22085},[217,218,219,220],[39,22087,21884],{"className":22088},[465,220]," 定义为",[39,22091,22093],{"className":22092,"translate":43},[746],[39,22094,22096,22212],{"className":22095,"translate":43},[42],[39,22097,22099],{"className":22098},[47],[49,22100,22101],{"xmlns":51,"display":755},[53,22102,22103,22209],{},[56,22104,22105,22107,22113,22115,22117,22119,22121,22123,22125,22135,22145,22147,22149,22155,22157,22159,22161,22163,22165,22207],{},[133,22106,136],{"stretchy":135},[138,22108,22109,22111],{},[59,22110,11772],{"mathvariant":617},[133,22112,21884],{},[59,22114,4230],{},[133,22116,161],{"stretchy":135},[133,22118,136],{"stretchy":135},[59,22120,142],{},[133,22122,161],{"stretchy":135},[133,22124,793],{},[6556,22126,22127,22133],{},[56,22128,22129,22131],{},[59,22130,1687],{},[133,22132,1690],{},[59,22134,158],{},[6556,22136,22137,22139],{},[133,22138,3191],{},[138,22140,22141,22143],{},[59,22142,142],{},[133,22144,147],{"mathvariant":145,"lspace":146,"rspace":146},[59,22146,131],{},[133,22148,136],{"stretchy":135},[138,22150,22151,22153],{},[59,22152,142],{},[133,22154,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,22156,150],{},[59,22158,142],{},[133,22160,155],{"separator":70},[59,22162,158],{},[133,22164,161],{"stretchy":135},[56,22166,22167,22169,22171,22173,22175,22177,22179,22181,22187,22189,22191,22193,22195,22197,22203,22205],{},[133,22168,537],{"fence":70},[59,22170,401],{},[133,22172,136],{"stretchy":135},[59,22174,142],{},[133,22176,155],{"separator":70},[59,22178,158],{},[133,22180,155],{"separator":70},[138,22182,22183,22185],{},[59,22184,142],{},[133,22186,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,22188,161],{"stretchy":135},[133,22190,788],{},[59,22192,531],{},[59,22194,4230],{},[133,22196,136],{"stretchy":135},[138,22198,22199,22201],{},[59,22200,142],{},[133,22202,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,22204,161],{"stretchy":135},[133,22206,4280],{"fence":70},[59,22208,2903],{"mathvariant":145},[63,22210,22211],{"encoding":65},"(\\mathcal{T}^*V)(s)\n=\\max_{a}\\sum_{s'}P(s'\\mid s,a)\n\\left[r(s,a,s')+\\gamma V(s')\\right].",[39,22213,22215,22278,22463],{"className":22214,"ariaHidden":70},[69],[39,22216,22218,22221,22224,22254,22257,22260,22263,22266,22269,22272,22275],{"className":22217},[74],[39,22219],{"className":22220,"style":245},[78],[39,22222,136],{"className":22223},[182],[39,22225,22227,22230],{"className":22226},[83],[39,22228,11772],{"className":22229,"style":11793},[83,641],[39,22231,22233],{"className":22232},[192],[39,22234,22236],{"className":22235},[196],[39,22237,22239],{"className":22238},[200],[39,22240,22243],{"className":22241,"style":22242},[204],"height:0.7387em;",[39,22244,22245,22248],{"style":1027},[39,22246],{"className":22247,"style":213},[212],[39,22249,22251],{"className":22250},[217,218,219,220],[39,22252,21884],{"className":22253},[465,220],[39,22255,4230],{"className":22256,"style":461},[83,84],[39,22258,161],{"className":22259},[263],[39,22261,136],{"className":22262},[182],[39,22264,142],{"className":22265},[83,84],[39,22267,161],{"className":22268},[263],[39,22270],{"className":22271,"style":231},[230],[39,22273,793],{"className":22274},[235],[39,22276],{"className":22277,"style":231},[230],[39,22279,22281,22285,22334,22337,22413,22416,22419,22422,22454,22457,22460],{"className":22280},[74],[39,22282],{"className":22283,"style":22284},[78],"height:2.344em;vertical-align:-1.294em;",[39,22286,22288],{"className":22287},[1744,3294],[39,22289,22291,22325],{"className":22290},[196,649],[39,22292,22294,22322],{"className":22293},[200],[39,22295,22297,22312],{"className":22296,"style":290},[204],[39,22298,22300,22303],{"style":22299},"top:-2.4em;margin-left:0em;",[39,22301],{"className":22302,"style":935},[212],[39,22304,22306],{"className":22305},[217,218,219,220],[39,22307,22309],{"className":22308},[83,220],[39,22310,158],{"className":22311},[83,84,220],[39,22313,22314,22317],{"style":21599},[39,22315],{"className":22316,"style":935},[212],[39,22318,22319],{},[39,22320,1687],{"className":22321},[1744],[39,22323,673],{"className":22324},[672],[39,22326,22328],{"className":22327},[200],[39,22329,22332],{"className":22330,"style":22331},[204],"height:0.7em;",[39,22333],{},[39,22335],{"className":22336,"style":256},[230],[39,22338,22340],{"className":22339},[1744,3294],[39,22341,22343,22405],{"className":22342},[196,649],[39,22344,22346,22402],{"className":22345},[200],[39,22347,22349,22392],{"className":22348,"style":6670},[204],[39,22350,22351,22354],{"style":7097},[39,22352],{"className":22353,"style":3311},[212],[39,22355,22357],{"className":22356},[217,218,219,220],[39,22358,22360],{"className":22359},[83,220],[39,22361,22363,22366],{"className":22362},[83,220],[39,22364,142],{"className":22365},[83,84,220],[39,22367,22369],{"className":22368},[192],[39,22370,22372],{"className":22371},[196],[39,22373,22375],{"className":22374},[200],[39,22376,22378],{"className":22377,"style":3997},[204],[39,22379,22380,22383],{"style":4000},[39,22381],{"className":22382,"style":4004},[212],[39,22384,22386],{"className":22385},[217,4008,4009,220],[39,22387,22389],{"className":22388},[83,220],[39,22390,147],{"className":22391},[83,220],[39,22393,22394,22397],{"style":3330},[39,22395],{"className":22396,"style":3311},[212],[39,22398,22399],{},[39,22400,3191],{"className":22401},[1744,3339,3340],[39,22403,673],{"className":22404},[672],[39,22406,22408],{"className":22407},[200],[39,22409,22411],{"className":22410,"style":7158},[204],[39,22412],{},[39,22414],{"className":22415,"style":256},[230],[39,22417,131],{"className":22418,"style":178},[83,84],[39,22420,136],{"className":22421},[182],[39,22423,22425,22428],{"className":22424},[83],[39,22426,142],{"className":22427},[83,84],[39,22429,22431],{"className":22430},[192],[39,22432,22434],{"className":22433},[196],[39,22435,22437],{"className":22436},[200],[39,22438,22440],{"className":22439,"style":1024},[204],[39,22441,22442,22445],{"style":1027},[39,22443],{"className":22444,"style":213},[212],[39,22446,22448],{"className":22447},[217,218,219,220],[39,22449,22451],{"className":22450},[83,220],[39,22452,147],{"className":22453},[83,220],[39,22455],{"className":22456,"style":231},[230],[39,22458,150],{"className":22459},[235],[39,22461],{"className":22462,"style":231},[230],[39,22464,22466,22470,22473,22476,22479,22482,22485,22488,22609,22612],{"className":22465},[74],[39,22467],{"className":22468,"style":22469},[78],"height:1.0519em;vertical-align:-0.25em;",[39,22471,142],{"className":22472},[83,84],[39,22474,155],{"className":22475},[252],[39,22477],{"className":22478,"style":256},[230],[39,22480,158],{"className":22481},[83,84],[39,22483,161],{"className":22484},[263],[39,22486],{"className":22487,"style":256},[230],[39,22489,22491,22494,22497,22500,22503,22506,22509,22512,22515,22518,22550,22553,22556,22559,22562,22565,22568,22571,22603,22606],{"className":22490},[2842],[39,22492,537],{"className":22493,"style":4592},[182,4591],[39,22495,401],{"className":22496,"style":438},[83,84],[39,22498,136],{"className":22499},[182],[39,22501,142],{"className":22502},[83,84],[39,22504,155],{"className":22505},[252],[39,22507],{"className":22508,"style":256},[230],[39,22510,158],{"className":22511},[83,84],[39,22513,155],{"className":22514},[252],[39,22516],{"className":22517,"style":256},[230],[39,22519,22521,22524],{"className":22520},[83],[39,22522,142],{"className":22523},[83,84],[39,22525,22527],{"className":22526},[192],[39,22528,22530],{"className":22529},[196],[39,22531,22533],{"className":22532},[200],[39,22534,22536],{"className":22535,"style":1024},[204],[39,22537,22538,22541],{"style":1027},[39,22539],{"className":22540,"style":213},[212],[39,22542,22544],{"className":22543},[217,218,219,220],[39,22545,22547],{"className":22546},[83,220],[39,22548,147],{"className":22549},[83,220],[39,22551,161],{"className":22552},[263],[39,22554],{"className":22555,"style":461},[230],[39,22557,788],{"className":22558},[465],[39,22560],{"className":22561,"style":461},[230],[39,22563,531],{"className":22564,"style":565},[83,84],[39,22566,4230],{"className":22567,"style":461},[83,84],[39,22569,136],{"className":22570},[182],[39,22572,22574,22577],{"className":22573},[83],[39,22575,142],{"className":22576},[83,84],[39,22578,22580],{"className":22579},[192],[39,22581,22583],{"className":22582},[196],[39,22584,22586],{"className":22585},[200],[39,22587,22589],{"className":22588,"style":1024},[204],[39,22590,22591,22594],{"style":1027},[39,22592],{"className":22593,"style":213},[212],[39,22595,22597],{"className":22596},[217,218,219,220],[39,22598,22600],{"className":22599},[83,220],[39,22601,147],{"className":22602},[83,220],[39,22604,161],{"className":22605},[263],[39,22607,4280],{"className":22608,"style":4592},[263,4591],[39,22610],{"className":22611,"style":256},[230],[39,22613,2903],{"className":22614},[83],[11,22616,22617,22618,22676],{},"以 ",[39,22619,22621,22638],{"className":22620,"translate":43},[42],[39,22622,22624],{"className":22623},[47],[49,22625,22626],{"xmlns":51},[53,22627,22628,22636],{},[56,22629,22630],{},[138,22631,22632,22634],{},[59,22633,11772],{"mathvariant":617},[59,22635,2098],{},[63,22637,11777],{"encoding":65},[39,22639,22641],{"className":22640,"ariaHidden":70},[69],[39,22642,22644,22647],{"className":22643},[74],[39,22645],{"className":22646,"style":79},[78],[39,22648,22650,22653],{"className":22649},[83],[39,22651,11772],{"className":22652,"style":11793},[83,641],[39,22654,22656],{"className":22655},[192],[39,22657,22659],{"className":22658},[196],[39,22660,22662],{"className":22661},[200],[39,22663,22665],{"className":22664,"style":6410},[204],[39,22666,22667,22670],{"style":208},[39,22668],{"className":22669,"style":213},[212],[39,22671,22673],{"className":22672},[217,218,219,220],[39,22674,2098],{"className":22675,"style":2114},[83,84,220]," 为例，",[39,22678,22680],{"className":22679,"translate":43},[746],[39,22681,22683,22853],{"className":22682,"translate":43},[42],[39,22684,22686],{"className":22685},[47],[49,22687,22688],{"xmlns":51,"display":755},[53,22689,22690,22850],{},[759,22691,22692,22808],{"rowspacing":761,"columnalign":762,"columnspacing":146},[764,22693,22694,22722],{},[767,22695,22696],{},[770,22697,22698],{"scriptlevel":541,"displaystyle":70},[56,22699,22700,22702,22704,22710,22712,22714,22716,22718,22720],{},[59,22701,150],{"mathvariant":145},[133,22703,136],{"stretchy":135},[138,22705,22706,22708],{},[59,22707,11772],{"mathvariant":617},[59,22709,2098],{},[59,22711,4230],{},[133,22713,161],{"stretchy":135},[133,22715,136],{"stretchy":135},[59,22717,142],{},[133,22719,161],{"stretchy":135},[59,22721,150],{"mathvariant":145},[767,22723,22724],{},[770,22725,22726],{"scriptlevel":541,"displaystyle":70},[56,22727,22728,22730,22732,22738,22740,22742,22744,22746,22748,22750,22760,22762,22764,22770,22772,22774,22776,22778,22780],{},[56,22729],{},[133,22731,1824],{},[6556,22733,22734,22736],{},[133,22735,3191],{},[59,22737,158],{},[59,22739,2098],{},[133,22741,136],{"stretchy":135},[59,22743,158],{},[133,22745,150],{},[59,22747,142],{},[133,22749,161],{"stretchy":135},[6556,22751,22752,22754],{},[133,22753,3191],{},[138,22755,22756,22758],{},[59,22757,142],{},[133,22759,147],{"mathvariant":145,"lspace":146,"rspace":146},[59,22761,131],{},[133,22763,136],{"stretchy":135},[138,22765,22766,22768],{},[59,22767,142],{},[133,22769,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,22771,150],{},[59,22773,142],{},[133,22775,155],{"separator":70},[59,22777,158],{},[133,22779,161],{"stretchy":135},[56,22781,22782,22784,22792,22794,22796,22798,22800,22806],{},[133,22783,136],{"fence":70},[613,22785,22786,22788,22790],{},[59,22787,422],{},[59,22789,1687],{},[133,22791,1690],{},[133,22793,788],{},[59,22795,531],{},[59,22797,15389],{"mathvariant":145},[59,22799,4230],{},[613,22801,22802,22804],{},[59,22803,15389],{"mathvariant":145},[59,22805,1696],{"mathvariant":145},[133,22807,161],{"fence":70},[764,22809,22810,22816],{},[767,22811,22812],{},[770,22813,22814],{"scriptlevel":541,"displaystyle":70},[56,22815],{},[767,22817,22818],{},[770,22819,22820],{"scriptlevel":541,"displaystyle":70},[56,22821,22822,22824,22826,22834,22836,22838,22840,22842,22848],{},[56,22823],{},[133,22825,793],{},[613,22827,22828,22830,22832],{},[59,22829,422],{},[59,22831,1687],{},[133,22833,1690],{},[133,22835,788],{},[59,22837,531],{},[59,22839,15389],{"mathvariant":145},[59,22841,4230],{},[613,22843,22844,22846],{},[59,22845,15389],{"mathvariant":145},[59,22847,1696],{"mathvariant":145},[133,22849,155],{"separator":70},[63,22851,22852],{"encoding":65},"\\begin{aligned}\n|(\\mathcal{T}^\\pi V)(s)|\n&\\le\\sum_{a}\\pi(a\\mid s)\\sum_{s'}P(s'\\mid s,a)\n\\left(R_{\\max}+\\gamma\\|V\\|_\\infty\\right)\\\\\n&=R_{\\max}+\\gamma\\|V\\|_\\infty,\n\\end{aligned}",[39,22854,22856],{"className":22855,"ariaHidden":70},[69],[39,22857,22859,22862],{"className":22858},[74],[39,22860],{"className":22861,"style":6954},[78],[39,22863,22865],{"className":22864},[83],[39,22866,22868,22960],{"className":22867},[759],[39,22869,22871],{"className":22870},[918],[39,22872,22874,22952],{"className":22873},[196,649],[39,22875,22877,22949],{"className":22876},[200],[39,22878,22880,22941],{"className":22879,"style":6973},[204],[39,22881,22882,22885],{"style":6976},[39,22883],{"className":22884,"style":3311},[212],[39,22886,22888,22891,22894,22923,22926,22929,22932,22935,22938],{"className":22887},[83],[39,22889,150],{"className":22890},[83],[39,22892,136],{"className":22893},[182],[39,22895,22897,22900],{"className":22896},[83],[39,22898,11772],{"className":22899,"style":11793},[83,641],[39,22901,22903],{"className":22902},[192],[39,22904,22906],{"className":22905},[196],[39,22907,22909],{"className":22908},[200],[39,22910,22912],{"className":22911,"style":4421},[204],[39,22913,22914,22917],{"style":1027},[39,22915],{"className":22916,"style":213},[212],[39,22918,22920],{"className":22919},[217,218,219,220],[39,22921,2098],{"className":22922,"style":2114},[83,84,220],[39,22924,4230],{"className":22925,"style":461},[83,84],[39,22927,161],{"className":22928},[263],[39,22930,136],{"className":22931},[182],[39,22933,142],{"className":22934},[83,84],[39,22936,161],{"className":22937},[263],[39,22939,150],{"className":22940},[83],[39,22942,22943,22946],{"style":7032},[39,22944],{"className":22945,"style":3311},[212],[39,22947],{"className":22948},[83],[39,22950,673],{"className":22951},[672],[39,22953,22955],{"className":22954},[200],[39,22956,22958],{"className":22957,"style":7048},[204],[39,22959],{},[39,22961,22963],{"className":22962},[1223],[39,22964,22966,23468],{"className":22965},[196,649],[39,22967,22969,23465],{"className":22968},[200],[39,22970,22972,23332],{"className":22971,"style":6973},[204],[39,22973,22974,22977],{"style":6976},[39,22975],{"className":22976,"style":3311},[212],[39,22978,22980,22983,22986,22989,22992,23039,23042,23045,23048,23051,23054,23057,23060,23063,23066,23069,23145,23148,23151,23154,23186,23189,23192,23195,23198,23201,23204,23207,23210,23213],{"className":22979},[83],[39,22981],{"className":22982},[83],[39,22984],{"className":22985,"style":231},[230],[39,22987,1824],{"className":22988},[235],[39,22990],{"className":22991,"style":231},[230],[39,22993,22995],{"className":22994},[1744,3294],[39,22996,22998,23031],{"className":22997},[196,649],[39,22999,23001,23028],{"className":23000},[200],[39,23002,23004,23018],{"className":23003,"style":6670},[204],[39,23005,23006,23009],{"style":6673},[39,23007],{"className":23008,"style":3311},[212],[39,23010,23012],{"className":23011},[217,218,219,220],[39,23013,23015],{"className":23014},[83,220],[39,23016,158],{"className":23017},[83,84,220],[39,23019,23020,23023],{"style":3330},[39,23021],{"className":23022,"style":3311},[212],[39,23024,23025],{},[39,23026,3191],{"className":23027},[1744,3339,3340],[39,23029,673],{"className":23030},[672],[39,23032,23034],{"className":23033},[200],[39,23035,23037],{"className":23036,"style":6705},[204],[39,23038],{},[39,23040],{"className":23041,"style":256},[230],[39,23043,2098],{"className":23044,"style":2114},[83,84],[39,23046,136],{"className":23047},[182],[39,23049,158],{"className":23050},[83,84],[39,23052],{"className":23053,"style":231},[230],[39,23055,150],{"className":23056},[235],[39,23058],{"className":23059,"style":231},[230],[39,23061,142],{"className":23062},[83,84],[39,23064,161],{"className":23065},[263],[39,23067],{"className":23068,"style":256},[230],[39,23070,23072],{"className":23071},[1744,3294],[39,23073,23075,23137],{"className":23074},[196,649],[39,23076,23078,23134],{"className":23077},[200],[39,23079,23081,23124],{"className":23080,"style":6670},[204],[39,23082,23083,23086],{"style":7097},[39,23084],{"className":23085,"style":3311},[212],[39,23087,23089],{"className":23088},[217,218,219,220],[39,23090,23092],{"className":23091},[83,220],[39,23093,23095,23098],{"className":23094},[83,220],[39,23096,142],{"className":23097},[83,84,220],[39,23099,23101],{"className":23100},[192],[39,23102,23104],{"className":23103},[196],[39,23105,23107],{"className":23106},[200],[39,23108,23110],{"className":23109,"style":3997},[204],[39,23111,23112,23115],{"style":4000},[39,23113],{"className":23114,"style":4004},[212],[39,23116,23118],{"className":23117},[217,4008,4009,220],[39,23119,23121],{"className":23120},[83,220],[39,23122,147],{"className":23123},[83,220],[39,23125,23126,23129],{"style":3330},[39,23127],{"className":23128,"style":3311},[212],[39,23130,23131],{},[39,23132,3191],{"className":23133},[1744,3339,3340],[39,23135,673],{"className":23136},[672],[39,23138,23140],{"className":23139},[200],[39,23141,23143],{"className":23142,"style":7158},[204],[39,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V\\|_\\infty\u003C\\infty",[39,23514,23516,23606],{"className":23515,"ariaHidden":70},[69],[39,23517,23519,23522,23525,23554,23557,23597,23600,23603],{"className":23518},[74],[39,23520],{"className":23521,"style":245},[78],[39,23523,15389],{"className":23524},[83],[39,23526,23528,23531],{"className":23527},[83],[39,23529,11772],{"className":23530,"style":11793},[83,641],[39,23532,23534],{"className":23533},[192],[39,23535,23537],{"className":23536},[196],[39,23538,23540],{"className":23539},[200],[39,23541,23543],{"className":23542,"style":6410},[204],[39,23544,23545,23548],{"style":208},[39,23546],{"className":23547,"style":213},[212],[39,23549,23551],{"className":23550},[217,218,219,220],[39,23552,2098],{"className":23553,"style":2114},[83,84,220],[39,23555,4230],{"className":23556,"style":461},[83,84],[39,23558,23560,23563],{"className":23559},[83],[39,23561,15389],{"className":23562},[83],[39,23564,23566],{"className":23565},[192],[39,23567,23569,23589],{"className":23568},[196,649],[39,23570,23572,23586],{"className":23571},[200],[39,23573,23575],{"className":23574,"style":1728},[204],[39,23576,23577,23580],{"style":966},[39,23578],{"className":23579,"style":213},[212],[39,23581,23583],{"className":23582},[217,218,219,220],[39,23584,1696],{"className":23585},[83,220],[39,23587,673],{"className":23588},[672],[39,23590,23592],{"className":23591},[200],[39,23593,23595],{"className":23594,"style":680},[204],[39,23596],{},[39,23598],{"className":23599,"style":231},[230],[39,23601,1693],{"className":23602},[235],[39,23604],{"className":23605,"style":231},[230],[39,23607,23609,23612],{"className":23608},[74],[39,23610],{"className":23611,"style":290},[78],[39,23613,1696],{"className":23614},[83],"。对 ",[39,23617,23619,23636],{"className":23618,"translate":43},[42],[39,23620,23622],{"className":23621},[47],[49,23623,23624],{"xmlns":51},[53,23625,23626,23634],{},[56,23627,23628],{},[138,23629,23630,23632],{},[59,23631,11772],{"mathvariant":617},[133,23633,21884],{},[63,23635,21887],{"encoding":65},[39,23637,23639],{"className":23638,"ariaHidden":70},[69],[39,23640,23642,23645],{"className":23641},[74],[39,23643],{"className":23644,"style":21897},[78],[39,23646,23648,23651],{"className":23647},[83],[39,23649,11772],{"className":23650,"style":11793},[83,641],[39,23652,23654],{"className":23653},[192],[39,23655,23657],{"className":23656},[196],[39,23658,23660],{"className":23659},[200],[39,23661,23663],{"className":23662,"style":21897},[204],[39,23664,23665,23668],{"style":208},[39,23666],{"className":23667,"style":213},[212],[39,23669,23671],{"className":23670},[217,218,219,220],[39,23672,21884],{"className":23673},[465,220]," 有同样的界。因此算子迭代不会离开这个空间。",[26,23676,23678],{"id":23677},"_32-单调性与常数平移","3.2 单调性与常数平移",[31,23680,23683,23819,24060,24116,24573],{"id":23681,"title":23682},"proposition-monotonicity","命题 3.1（单调性与常数平移）",[11,23684,37,23685,23765,23766,23818],{},[39,23686,23688,23717],{"className":23687,"translate":43},[42],[39,23689,23691],{"className":23690},[47],[49,23692,23693],{"xmlns":51},[53,23694,23695,23714],{},[56,23696,23697,23699,23701,23704,23706,23708,23710,23712],{},[59,23698,4230],{},[133,23700,155],{"separator":70},[59,23702,23703],{},"W",[133,23705,534],{},[59,23707,19772],{},[133,23709,136],{"stretchy":135},[59,23711,61],{},[133,23713,161],{"stretchy":135},[63,23715,23716],{"encoding":65},"V,W\\in B(S)",[39,23718,23720,23747],{"className":23719,"ariaHidden":70},[69],[39,23721,23723,23726,23729,23732,23735,23738,23741,23744],{"className":23722},[74],[39,23724],{"className":23725,"style":6452},[78],[39,23727,4230],{"className":23728,"style":461},[83,84],[39,23730,155],{"className":23731},[252],[39,23733],{"className":23734,"style":256},[230],[39,23736,23703],{"className":23737,"style":178},[83,84],[39,23739],{"className":23740,"style":231},[230],[39,23742,534],{"className":23743},[235],[39,23745],{"className":23746,"style":231},[230],[39,23748,23750,23753,23756,23759,23762],{"className":23749},[74],[39,23751],{"className":23752,"style":245},[78],[39,23754,19772],{"className":23755,"style":19812},[83,84],[39,23757,136],{"className":23758},[182],[39,23760,61],{"className":23761,"style":85},[83,84],[39,23763,161],{"className":23764},[263]," 且 ",[39,23767,23769,23787],{"className":23768,"translate":43},[42],[39,23770,23772],{"className":23771},[47],[49,23773,23774],{"xmlns":51},[53,23775,23776,23784],{},[56,23777,23778,23780,23782],{},[59,23779,4230],{},[133,23781,1824],{},[59,23783,23703],{},[63,23785,23786],{"encoding":65},"V\\le W",[39,23788,23790,23809],{"className":23789,"ariaHidden":70},[69],[39,23791,23793,23797,23800,23803,23806],{"className":23792},[74],[39,23794],{"className":23795,"style":23796},[78],"height:0.8193em;vertical-align:-0.136em;",[39,23798,4230],{"className":23799,"style":461},[83,84],[39,23801],{"className":23802,"style":231},[230],[39,23804,1824],{"className":23805},[235],[39,23807],{"className":23808,"style":231},[230],[39,23810,23812,23815],{"className":23811},[74],[39,23813],{"className":23814,"style":79},[78],[39,23816,23703],{"className":23817,"style":178},[83,84]," 逐点成立。则",[39,23820,23822],{"className":23821,"translate":43},[746],[39,23823,23825,23879],{"className":23824,"translate":43},[42],[39,23826,23828],{"className":23827},[47],[49,23829,23830],{"xmlns":51,"display":755},[53,23831,23832,23876],{},[56,23833,23834,23840,23842,23844,23850,23852,23854,23856,23862,23864,23866,23872,23874],{},[138,23835,23836,23838],{},[59,23837,11772],{"mathvariant":617},[59,23839,2098],{},[59,23841,4230],{},[133,23843,1824],{},[138,23845,23846,23848],{},[59,23847,11772],{"mathvariant":617},[59,23849,2098],{},[59,23851,23703],{},[133,23853,155],{"separator":70},[230,23855],{"width":10251},[138,23857,23858,23860],{},[59,23859,11772],{"mathvariant":617},[133,23861,21884],{},[59,23863,4230],{},[133,23865,1824],{},[138,23867,23868,23870],{},[59,23869,11772],{"mathvariant":617},[133,23871,21884],{},[59,23873,23703],{},[59,23875,2903],{"mathvariant":145},[63,23877,23878],{"encoding":65},"\\mathcal{T}^\\pi V\\le\\mathcal{T}^\\pi W,\n\\qquad\n\\mathcal{T}^* V\\le\\mathcal{T}^* W.",[39,23880,23882,23930,24019],{"className":23881,"ariaHidden":70},[69],[39,23883,23885,23889,23918,23921,23924,23927],{"className":23884},[74],[39,23886],{"className":23887,"style":23888},[78],"height:0.8504em;vertical-align:-0.136em;",[39,23890,23892,23895],{"className":23891},[83],[39,23893,11772],{"className":23894,"style":11793},[83,641],[39,23896,23898],{"className":23897},[192],[39,23899,23901],{"className":23900},[196],[39,23902,23904],{"className":23903},[200],[39,23905,23907],{"className":23906,"style":4421},[204],[39,23908,23909,23912],{"style":1027},[39,23910],{"className":23911,"style":213},[212],[39,23913,23915],{"className":23914},[217,218,219,220],[39,23916,2098],{"className":23917,"style":2114},[83,84,220],[39,23919,4230],{"className":23920,"style":461},[83,84],[39,23922],{"className":23923,"style":231},[230],[39,23925,1824],{"className":23926},[235],[39,23928],{"className":23929,"style":231},[230],[39,23931,23933,23937,23966,23969,23972,23975,23978,24007,24010,24013,24016],{"className":23932},[74],[39,23934],{"className":23935,"style":23936},[78],"height:0.9331em;vertical-align:-0.1944em;",[39,23938,23940,23943],{"className":23939},[83],[39,23941,11772],{"className":23942,"style":11793},[83,641],[39,23944,23946],{"className":23945},[192],[39,23947,23949],{"className":23948},[196],[39,23950,23952],{"className":23951},[200],[39,23953,23955],{"className":23954,"style":4421},[204],[39,23956,23957,23960],{"style":1027},[39,23958],{"className":23959,"style":213},[212],[39,23961,23963],{"className":23962},[217,218,219,220],[39,23964,2098],{"className":23965,"style":2114},[83,84,220],[39,23967,23703],{"className":23968,"style":178},[83,84],[39,23970,155],{"className":23971},[252],[39,23973],{"className":23974,"style":10787},[230],[39,23976],{"className":23977,"style":256},[230],[39,23979,23981,23984],{"className":23980},[83],[39,23982,11772],{"className":23983,"style":11793},[83,641],[39,23985,23987],{"className":23986},[192],[39,23988,23990],{"className":23989},[196],[39,23991,23993],{"className":23992},[200],[39,23994,23996],{"className":23995,"style":22242},[204],[39,23997,23998,24001],{"style":1027},[39,23999],{"className":24000,"style":213},[212],[39,24002,24004],{"className":24003},[217,218,219,220],[39,24005,21884],{"className":24006},[465,220],[39,24008,4230],{"className":24009,"style":461},[83,84],[39,24011],{"className":24012,"style":231},[230],[39,24014,1824],{"className":24015},[235],[39,24017],{"className":24018,"style":231},[230],[39,24020,24022,24025,24054,24057],{"className":24021},[74],[39,24023],{"className":24024,"style":22242},[78],[39,24026,24028,24031],{"className":24027},[83],[39,24029,11772],{"className":24030,"style":11793},[83,641],[39,24032,24034],{"className":24033},[192],[39,24035,24037],{"className":24036},[196],[39,24038,24040],{"className":24039},[200],[39,24041,24043],{"className":24042,"style":22242},[204],[39,24044,24045,24048],{"style":1027},[39,24046],{"className":24047,"style":213},[212],[39,24049,24051],{"className":24050},[217,218,219,220],[39,24052,21884],{"className":24053},[465,220],[39,24055,23703],{"className":24056,"style":178},[83,84],[39,24058,2903],{"className":24059},[83],[11,24061,24062,24063,742],{},"此外，对任意常数 ",[39,24064,24066,24085],{"className":24065,"translate":43},[42],[39,24067,24069],{"className":24068},[47],[49,24070,24071],{"xmlns":51},[53,24072,24073,24082],{},[56,24074,24075,24078,24080],{},[59,24076,24077],{},"c",[133,24079,534],{},[59,24081,422],{"mathvariant":421},[63,24083,24084],{"encoding":65},"c\\in\\mathbb{R}",[39,24086,24088,24107],{"className":24087,"ariaHidden":70},[69],[39,24089,24091,24095,24098,24101,24104],{"className":24090},[74],[39,24092],{"className":24093,"style":24094},[78],"height:0.5782em;vertical-align:-0.0391em;",[39,24096,24077],{"className":24097},[83,84],[39,24099],{"className":24100,"style":231},[230],[39,24102,534],{"className":24103},[235],[39,24105],{"className":24106,"style":231},[230],[39,24108,24110,24113],{"className":24109},[74],[39,24111],{"className":24112,"style":511},[78],[39,24114,422],{"className":24115},[83,515],[39,24117,24119],{"className":24118,"translate":43},[746],[39,24120,24122,24243],{"className":24121,"translate":43},[42],[39,24123,24125],{"className":24124},[47],[49,24126,24127],{"xmlns":51,"display":755},[53,24128,24129,24240],{},[759,24130,24131,24186],{"rowspacing":761,"columnalign":762,"columnspacing":146},[764,24132,24133,24158],{},[767,24134,24135],{},[770,24136,24137],{"scriptlevel":541,"displaystyle":70},[56,24138,24139,24145,24147,24149,24151,24153,24156],{},[138,24140,24141,24143],{},[59,24142,11772],{"mathvariant":617},[59,24144,2098],{},[133,24146,136],{"stretchy":135},[59,24148,4230],{},[133,24150,788],{},[59,24152,24077],{},[539,24154,546],{"mathvariant":24155},"bold",[133,24157,161],{"stretchy":135},[767,24159,24160],{},[770,24161,24162],{"scriptlevel":541,"displaystyle":70},[56,24163,24164,24166,24168,24174,24176,24178,24180,24182,24184],{},[56,24165],{},[133,24167,793],{},[138,24169,24170,24172],{},[59,24171,11772],{"mathvariant":617},[59,24173,2098],{},[59,24175,4230],{},[133,24177,788],{},[59,24179,531],{},[59,24181,24077],{},[539,24183,546],{"mathvariant":24155},[133,24185,155],{"separator":70},[764,24187,24188,24212],{},[767,24189,24190],{},[770,24191,24192],{"scriptlevel":541,"displaystyle":70},[56,24193,24194,24200,24202,24204,24206,24208,24210],{},[138,24195,24196,24198],{},[59,24197,11772],{"mathvariant":617},[133,24199,21884],{},[133,24201,136],{"stretchy":135},[59,24203,4230],{},[133,24205,788],{},[59,24207,24077],{},[539,24209,546],{"mathvariant":24155},[133,24211,161],{"stretchy":135},[767,24213,24214],{},[770,24215,24216],{"scriptlevel":541,"displaystyle":70},[56,24217,24218,24220,24222,24228,24230,24232,24234,24236,24238],{},[56,24219],{},[133,24221,793],{},[138,24223,24224,24226],{},[59,24225,11772],{"mathvariant":617},[133,24227,21884],{},[59,24229,4230],{},[133,24231,788],{},[59,24233,531],{},[59,24235,24077],{},[539,24237,546],{"mathvariant":24155},[133,24239,155],{"separator":70},[63,24241,24242],{"encoding":65},"\\begin{aligned}\n\\mathcal{T}^\\pi(V+c\\mathbf{1})\n&=\\mathcal{T}^\\pi V+\\gamma c\\mathbf{1},\\\\\n\\mathcal{T}^*(V+c\\mathbf{1})\n&=\\mathcal{T}^* V+\\gamma c\\mathbf{1},\n\\end{aligned}",[39,24244,24246],{"className":24245,"ariaHidden":70},[69],[39,24247,24249,24252],{"className":24248},[74],[39,24250],{"className":24251,"style":4374},[78],[39,24253,24255],{"className":24254},[83],[39,24256,24258,24404],{"className":24257},[759],[39,24259,24261],{"className":24260},[918],[39,24262,24264,24396],{"className":24263},[196,649],[39,24265,24267,24393],{"className":24266},[200],[39,24268,24270,24332],{"className":24269,"style":4393},[204],[39,24271,24272,24275],{"style":4396},[39,24273],{"className":24274,"style":935},[212],[39,24276,24278,24307,24310,24313,24316,24319,24322,24325,24329],{"className":24277},[83],[39,24279,24281,24284],{"className":24280},[83],[39,24282,11772],{"className":24283,"style":11793},[83,641],[39,24285,24287],{"className":24286},[192],[39,24288,24290],{"className":24289},[196],[39,24291,24293],{"className":24292},[200],[39,24294,24296],{"className":24295,"style":4421},[204],[39,24297,24298,24301],{"style":1027},[39,24299],{"className":24300,"style":213},[212],[39,24302,24304],{"className":24303},[217,218,219,220],[39,24305,2098],{"className":24306,"style":2114},[83,84,220],[39,24308,136],{"className":24309},[182],[39,24311,4230],{"className":24312,"style":461},[83,84],[39,24314],{"className":24315,"style":461},[230],[39,24317,788],{"className":24318},[465],[39,24320],{"className":24321,"style":461},[230],[39,24323,24077],{"className":24324},[83,84],[39,24326,546],{"className":24327},[83,24328],"mathbf",[39,24330,161],{"className":24331},[263],[39,24333,24334,24337],{"style":4444},[39,24335],{"className":24336,"style":935},[212],[39,24338,24340,24369,24372,24375,24378,24381,24384,24387,24390],{"className":24339},[83],[39,24341,24343,24346],{"className":24342},[83],[39,24344,11772],{"className":24345,"style":11793},[83,641],[39,24347,24349],{"className":24348},[192],[39,24350,24352],{"className":24351},[196],[39,24353,24355],{"className":24354},[200],[39,24356,24358],{"className":24357,"style":22242},[204],[39,24359,24360,24363],{"style":1027},[39,24361],{"className":24362,"style":213},[212],[39,24364,24366],{"className":24365},[217,218,219,220],[39,24367,21884],{"className":24368},[465,220],[39,24370,136],{"className":24371},[182],[39,24373,4230],{"className":24374,"style":461},[83,84],[39,24376],{"className":24377,"style":461},[230],[39,24379,788],{"className":24380},[465],[39,24382],{"className":24383,"style":461},[230],[39,24385,24077],{"className":24386},[83,84],[39,24388,546],{"className":24389},[83,24328],[39,24391,161],{"className":24392},[263],[39,24394,673],{"className":24395},[672],[39,24397,24399],{"className":24398},[200],[39,24400,24402],{"className":24401,"style":4507},[204],[39,24403],{},[39,24405,24407],{"className":24406},[1223],[39,24408,24410,24565],{"className":24409},[196,649],[39,24411,24413,24562],{"className":24412},[200],[39,24414,24416,24489],{"className":24415,"style":4393},[204],[39,24417,24418,24421],{"style":4396},[39,24419],{"className":24420,"style":935},[212],[39,24422,24424,24427,24430,24433,24436,24465,24468,24471,24474,24477,24480,24483,24486],{"className":24423},[83],[39,24425],{"className":24426},[83],[39,24428],{"className":24429,"style":231},[230],[39,24431,793],{"className":24432},[235],[39,24434],{"className":24435,"style":231},[230],[39,24437,24439,24442],{"className":24438},[83],[39,24440,11772],{"className":24441,"style":11793},[83,641],[39,24443,24445],{"className":24444},[192],[39,24446,24448],{"className":24447},[196],[39,24449,24451],{"className":24450},[200],[39,24452,24454],{"className":24453,"style":4421},[204],[39,24455,24456,24459],{"style":1027},[39,24457],{"className":24458,"style":213},[212],[39,24460,24462],{"className":24461},[217,218,219,220],[39,24463,2098],{"className":24464,"style":2114},[83,84,220],[39,24466,4230],{"className":24467,"style":461},[83,84],[39,24469],{"className":24470,"style":461},[230],[39,24472,788],{"className":24473},[465],[39,24475],{"className":24476,"style":461},[230],[39,24478,531],{"className":24479,"style":565},[83,84],[39,24481,24077],{"className":24482},[83,84],[39,24484,546],{"className":24485},[83,24328],[39,24487,155],{"className":24488},[252],[39,24490,24491,24494],{"style":4444},[39,24492],{"className":24493,"style":935},[212],[39,24495,24497,24500,24503,24506,24509,24538,24541,24544,24547,24550,24553,24556,24559],{"className":24496},[83],[39,24498],{"className":24499},[83],[39,24501],{"className":24502,"style":231},[230],[39,24504,793],{"className":24505},[235],[39,24507],{"className":24508,"style":231},[230],[39,24510,24512,24515],{"className":24511},[83],[39,24513,11772],{"className":24514,"style":11793},[83,641],[39,24516,24518],{"className":24517},[192],[39,24519,24521],{"className":24520},[196],[39,24522,24524],{"className":24523},[200],[39,24525,24527],{"className":24526,"style":22242},[204],[39,24528,24529,24532],{"style":1027},[39,24530],{"className":24531,"style":213},[212],[39,24533,24535],{"className":24534},[217,218,219,220],[39,24536,21884],{"className":24537},[465,220],[39,24539,4230],{"className":24540,"style":461},[83,84],[39,24542],{"className":24543,"style":461},[230],[39,24545,788],{"className":24546},[465],[39,24548],{"className":24549,"style":461},[230],[39,24551,531],{"className":24552,"style":565},[83,84],[39,24554,24077],{"className":24555},[83,84],[39,24557,546],{"className":24558},[83,24328],[39,24560,155],{"className":24561},[252],[39,24563,673],{"className":24564},[672],[39,24566,24568],{"className":24567},[200],[39,24569,24571],{"className":24570,"style":4507},[204],[39,24572],{},[11,24574,24575,24576,24605,24606,24634],{},"其中 ",[39,24577,24579,24593],{"className":24578,"translate":43},[42],[39,24580,24582],{"className":24581},[47],[49,24583,24584],{"xmlns":51},[53,24585,24586,24590],{},[56,24587,24588],{},[539,24589,546],{"mathvariant":24155},[63,24591,24592],{"encoding":65},"\\mathbf{1}",[39,24594,24596],{"className":24595,"ariaHidden":70},[69],[39,24597,24599,24602],{"className":24598},[74],[39,24600],{"className":24601,"style":2473},[78],[39,24603,546],{"className":24604},[83,24328]," 表示恒等于 ",[39,24607,24609,24622],{"className":24608,"translate":43},[42],[39,24610,24612],{"className":24611},[47],[49,24613,24614],{"xmlns":51},[53,24615,24616,24620],{},[56,24617,24618],{},[539,24619,546],{},[63,24621,546],{"encoding":65},[39,24623,24625],{"className":24624,"ariaHidden":70},[69],[39,24626,24628,24631],{"className":24627},[74],[39,24629],{"className":24630,"style":2473},[78],[39,24632,546],{"className":24633},[83]," 的函数。",[11,24636,24637,24639,24640,24690,24691,742],{},[1489,24638,5595],{}," 若 ",[39,24641,24643,24660],{"className":24642,"translate":43},[42],[39,24644,24646],{"className":24645},[47],[49,24647,24648],{"xmlns":51},[53,24649,24650,24658],{},[56,24651,24652,24654,24656],{},[59,24653,4230],{},[133,24655,1824],{},[59,24657,23703],{},[63,24659,23786],{"encoding":65},[39,24661,24663,24681],{"className":24662,"ariaHidden":70},[69],[39,24664,24666,24669,24672,24675,24678],{"className":24665},[74],[39,24667],{"className":24668,"style":23796},[78],[39,24670,4230],{"className":24671,"style":461},[83,84],[39,24673],{"className":24674,"style":231},[230],[39,24676,1824],{"className":24677},[235],[39,24679],{"className":24680,"style":231},[230],[39,24682,24684,24687],{"className":24683},[74],[39,24685],{"className":24686,"style":79},[78],[39,24688,23703],{"className":24689,"style":178},[83,84],"，则对每个 ",[39,24692,24694,24723],{"className":24693,"translate":43},[42],[39,24695,24697],{"className":24696},[47],[49,24698,24699],{"xmlns":51},[53,24700,24701,24721],{},[56,24702,24703,24705,24707,24709,24711,24713,24719],{},[133,24704,136],{"stretchy":135},[59,24706,142],{},[133,24708,155],{"separator":70},[59,24710,158],{},[133,24712,155],{"separator":70},[138,24714,24715,24717],{},[59,24716,142],{},[133,24718,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,24720,161],{"stretchy":135},[63,24722,2010],{"encoding":65},[39,24724,24726],{"className":24725,"ariaHidden":70},[69],[39,24727,24729,24732,24735,24738,24741,24744,24747,24750,24753,24785],{"className":24728},[74],[39,24730],{"className":24731,"style":174},[78],[39,24733,136],{"className":24734},[182],[39,24736,142],{"className":24737},[83,84],[39,24739,155],{"className":24740},[252],[39,24742],{"className":24743,"style":256},[230],[39,24745,158],{"className":24746},[83,84],[39,24748,155],{"className":24749},[252],[39,24751],{"className":24752,"style":256},[230],[39,24754,24756,24759],{"className":24755},[83],[39,24757,142],{"className":24758},[83,84],[39,24760,24762],{"className":24761},[192],[39,24763,24765],{"className":24764},[196],[39,24766,24768],{"className":24767},[200],[39,24769,24771],{"className":24770,"style":205},[204],[39,24772,24773,24776],{"style":208},[39,24774],{"className":24775,"style":213},[212],[39,24777,24779],{"className":24778},[217,218,219,220],[39,24780,24782],{"className":24781},[83,220],[39,24783,147],{"className":24784},[83,220],[39,24786,161],{"className":24787},[263],[39,24789,24791],{"className":24790,"translate":43},[746],[39,24792,24794,24882],{"className":24793,"translate":43},[42],[39,24795,24797],{"className":24796},[47],[49,24798,24799],{"xmlns":51,"display":755},[53,24800,24801,24879],{},[56,24802,24803,24805,24807,24809,24811,24813,24815,24821,24823,24825,24827,24829,24831,24837,24839,24841,24843,24845,24847,24849,24851,24853,24859,24861,24863,24865,24867,24869,24875,24877],{},[59,24804,401],{},[133,24806,136],{"stretchy":135},[59,24808,142],{},[133,24810,155],{"separator":70},[59,24812,158],{},[133,24814,155],{"separator":70},[138,24816,24817,24819],{},[59,24818,142],{},[133,24820,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,24822,161],{"stretchy":135},[133,24824,788],{},[59,24826,531],{},[59,24828,4230],{},[133,24830,136],{"stretchy":135},[138,24832,24833,24835],{},[59,24834,142],{},[133,24836,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,24838,161],{"stretchy":135},[133,24840,1824],{},[59,24842,401],{},[133,24844,136],{"stretchy":135},[59,24846,142],{},[133,24848,155],{"separator":70},[59,24850,158],{},[133,24852,155],{"separator":70},[138,24854,24855,24857],{},[59,24856,142],{},[133,24858,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,24860,161],{"stretchy":135},[133,24862,788],{},[59,24864,531],{},[59,24866,23703],{},[133,24868,136],{"stretchy":135},[138,24870,24871,24873],{},[59,24872,142],{},[133,24874,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,24876,161],{"stretchy":135},[59,24878,2903],{"mathvariant":145},[63,24880,24881],{"encoding":65},"r(s,a,s')+\\gamma V(s')\\le r(s,a,s')+\\gamma W(s').",[39,24883,24885,24959,25018,25092],{"className":24884,"ariaHidden":70},[69],[39,24886,24888,24891,24894,24897,24900,24903,24906,24909,24912,24915,24947,24950,24953,24956],{"className":24887},[74],[39,24889],{"className":24890,"style":22469},[78],[39,24892,401],{"className":24893,"style":438},[83,84],[39,24895,136],{"className":24896},[182],[39,24898,142],{"className":24899},[83,84],[39,24901,155],{"className":24902},[252],[39,24904],{"className":24905,"style":256},[230],[39,24907,158],{"className":24908},[83,84],[39,24910,155],{"className":24911},[252],[39,24913],{"className":24914,"style":256},[230],[39,24916,24918,24921],{"className":24917},[83],[39,24919,142],{"className":24920},[83,84],[39,24922,24924],{"className":24923},[192],[39,24925,24927],{"className":24926},[196],[39,24928,24930],{"className":24929},[200],[39,24931,24933],{"className":24932,"style":1024},[204],[39,24934,24935,24938],{"style":1027},[39,24936],{"className":24937,"style":213},[212],[39,24939,24941],{"className":24940},[217,218,219,220],[39,24942,24944],{"className":24943},[83,220],[39,24945,147],{"className":24946},[83,220],[39,24948,161],{"className":24949},[263],[39,24951],{"className":24952,"style":461},[230],[39,24954,788],{"className":24955},[465],[39,24957],{"className":24958,"style":461},[230],[39,24960,24962,24965,24968,24971,24974,25006,25009,25012,25015],{"className":24961},[74],[39,24963],{"className":24964,"style":22469},[78],[39,24966,531],{"className":24967,"style":565},[83,84],[39,24969,4230],{"className":24970,"style":461},[83,84],[39,24972,136],{"className":24973},[182],[39,24975,24977,24980],{"className":24976},[83],[39,24978,142],{"className":24979},[83,84],[39,24981,24983],{"className":24982},[192],[39,24984,24986],{"className":24985},[196],[39,24987,24989],{"className":24988},[200],[39,24990,24992],{"className":24991,"style":1024},[204],[39,24993,24994,24997],{"style":1027},[39,24995],{"className":24996,"style":213},[212],[39,24998,25000],{"className":24999},[217,218,219,220],[39,25001,25003],{"className":25002},[83,220],[39,25004,147],{"className":25005},[83,220],[39,25007,161],{"className":25008},[263],[39,25010],{"className":25011,"style":231},[230],[39,25013,1824],{"className":25014},[235],[39,25016],{"className":25017,"style":231},[230],[39,25019,25021,25024,25027,25030,25033,25036,25039,25042,25045,25048,25080,25083,25086,25089],{"className":25020},[74],[39,25022],{"className":25023,"style":22469},[78],[39,25025,401],{"className":25026,"style":438},[83,84],[39,25028,136],{"className":25029},[182],[39,25031,142],{"className":25032},[83,84],[39,25034,155],{"className":25035},[252],[39,25037],{"className":25038,"style":256},[230],[39,25040,158],{"className":25041},[83,84],[39,25043,155],{"className":25044},[252],[39,25046],{"className":25047,"style":256},[230],[39,25049,25051,25054],{"className":25050},[83],[39,25052,142],{"className":25053},[83,84],[39,25055,25057],{"className":25056},[192],[39,25058,25060],{"className":25059},[196],[39,25061,25063],{"className":25062},[200],[39,25064,25066],{"className":25065,"style":1024},[204],[39,25067,25068,25071],{"style":1027},[39,25069],{"className":25070,"style":213},[212],[39,25072,25074],{"className":25073},[217,218,219,220],[39,25075,25077],{"className":25076},[83,220],[39,25078,147],{"className":25079},[83,220],[39,25081,161],{"className":25082},[263],[39,25084],{"className":25085,"style":461},[230],[39,25087,788],{"className":25088},[465],[39,25090],{"className":25091,"style":461},[230],[39,25093,25095,25098,25101,25104,25107,25139,25142],{"className":25094},[74],[39,25096],{"className":25097,"style":22469},[78],[39,25099,531],{"className":25100,"style":565},[83,84],[39,25102,23703],{"className":25103,"style":178},[83,84],[39,25105,136],{"className":25106},[182],[39,25108,25110,25113],{"className":25109},[83],[39,25111,142],{"className":25112},[83,84],[39,25114,25116],{"className":25115},[192],[39,25117,25119],{"className":25118},[196],[39,25120,25122],{"className":25121},[200],[39,25123,25125],{"className":25124,"style":1024},[204],[39,25126,25127,25130],{"style":1027},[39,25128],{"className":25129,"style":213},[212],[39,25131,25133],{"className":25132},[217,218,219,220],[39,25134,25136],{"className":25135},[83,220],[39,25137,147],{"className":25138},[83,220],[39,25140,161],{"className":25141},[263],[39,25143,2903],{"className":25144},[83],[11,25146,20571,25147,23765,25236,25370,25371,23615,25492,25550,25551,1607],{},[39,25148,25150,25179],{"className":25149,"translate":43},[42],[39,25151,25153],{"className":25152},[47],[49,25154,25155],{"xmlns":51},[53,25156,25157,25176],{},[56,25158,25159,25161,25163,25165,25167,25169,25171,25174],{},[59,25160,2098],{},[133,25162,136],{"stretchy":135},[59,25164,158],{},[133,25166,150],{},[59,25168,142],{},[133,25170,161],{"stretchy":135},[133,25172,25173],{},"≥",[539,25175,541],{},[63,25177,25178],{"encoding":65},"\\pi(a\\mid s)\\ge0",[39,25180,25182,25206,25227],{"className":25181,"ariaHidden":70},[69],[39,25183,25185,25188,25191,25194,25197,25200,25203],{"className":25184},[74],[39,25186],{"className":25187,"style":245},[78],[39,25189,2098],{"className":25190,"style":2114},[83,84],[39,25192,136],{"className":25193},[182],[39,25195,158],{"className":25196},[83,84],[39,25198],{"className":25199,"style":231},[230],[39,25201,150],{"className":25202},[235],[39,25204],{"className":25205,"style":231},[230],[39,25207,25209,25212,25215,25218,25221,25224],{"className":25208},[74],[39,25210],{"className":25211,"style":245},[78],[39,25213,142],{"className":25214},[83,84],[39,25216,161],{"className":25217},[263],[39,25219],{"className":25220,"style":231},[230],[39,25222,25173],{"className":25223},[235],[39,25225],{"className":25226,"style":231},[230],[39,25228,25230,25233],{"className":25229},[74],[39,25231],{"className":25232,"style":2473},[78],[39,25234,541],{"className":25235},[83],[39,25237,25239,25275],{"className":25238,"translate":43},[42],[39,25240,25242],{"className":25241},[47],[49,25243,25244],{"xmlns":51},[53,25245,25246,25272],{},[56,25247,25248,25250,25252,25258,25260,25262,25264,25266,25268,25270],{},[59,25249,131],{},[133,25251,136],{"stretchy":135},[138,25253,25254,25256],{},[59,25255,142],{},[133,25257,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,25259,150],{},[59,25261,142],{},[133,25263,155],{"separator":70},[59,25265,158],{},[133,25267,161],{"stretchy":135},[133,25269,25173],{},[539,25271,541],{},[63,25273,25274],{"encoding":65},"P(s'\\mid s,a)\\ge0",[39,25276,25278,25331,25361],{"className":25277,"ariaHidden":70},[69],[39,25279,25281,25284,25287,25290,25322,25325,25328],{"className":25280},[74],[39,25282],{"className":25283,"style":174},[78],[39,25285,131],{"className":25286,"style":178},[83,84],[39,25288,136],{"className":25289},[182],[39,25291,25293,25296],{"className":25292},[83],[39,25294,142],{"className":25295},[83,84],[39,25297,25299],{"className":25298},[192],[39,25300,25302],{"className":25301},[196],[39,25303,25305],{"className":25304},[200],[39,25306,25308],{"className":25307,"style":205},[204],[39,25309,25310,25313],{"style":208},[39,25311],{"className":25312,"style":213},[212],[39,25314,25316],{"className":25315},[217,218,219,220],[39,25317,25319],{"className":25318},[83,220],[39,25320,147],{"className":25321},[83,220],[39,25323],{"className":25324,"style":231},[230],[39,25326,150],{"className":25327},[235],[39,25329],{"className":25330,"style":231},[230],[39,25332,25334,25337,25340,25343,25346,25349,25352,25355,25358],{"className":25333},[74],[39,25335],{"className":25336,"style":245},[78],[39,25338,142],{"className":25339},[83,84],[39,25341,155],{"className":25342},[252],[39,25344],{"className":25345,"style":256},[230],[39,25347,158],{"className":25348},[83,84],[39,25350,161],{"className":25351},[263],[39,25353],{"className":25354,"style":231},[230],[39,25356,25173],{"className":25357},[235],[39,25359],{"className":25360,"style":231},[230],[39,25362,25364,25367],{"className":25363},[74],[39,25365],{"className":25366,"style":2473},[78],[39,25368,541],{"className":25369},[83],"，加权求和保持不等号，故 ",[39,25372,25374,25404],{"className":25373,"translate":43},[42],[39,25375,25377],{"className":25376},[47],[49,25378,25379],{"xmlns":51},[53,25380,25381,25401],{},[56,25382,25383,25389,25391,25393,25399],{},[138,25384,25385,25387],{},[59,25386,11772],{"mathvariant":617},[59,25388,2098],{},[59,25390,4230],{},[133,25392,1824],{},[138,25394,25395,25397],{},[59,25396,11772],{"mathvariant":617},[59,25398,2098],{},[59,25400,23703],{},[63,25402,25403],{"encoding":65},"\\mathcal{T}^\\pi V\\le\\mathcal{T}^\\pi W",[39,25405,25407,25454],{"className":25406,"ariaHidden":70},[69],[39,25408,25410,25413,25442,25445,25448,25451],{"className":25409},[74],[39,25411],{"className":25412,"style":23796},[78],[39,25414,25416,25419],{"className":25415},[83],[39,25417,11772],{"className":25418,"style":11793},[83,641],[39,25420,25422],{"className":25421},[192],[39,25423,25425],{"className":25424},[196],[39,25426,25428],{"className":25427},[200],[39,25429,25431],{"className":25430,"style":6410},[204],[39,25432,25433,25436],{"style":208},[39,25434],{"className":25435,"style":213},[212],[39,25437,25439],{"className":25438},[217,218,219,220],[39,25440,2098],{"className":25441,"style":2114},[83,84,220],[39,25443,4230],{"className":25444,"style":461},[83,84],[39,25446],{"className":25447,"style":231},[230],[39,25449,1824],{"className":25450},[235],[39,25452],{"className":25453,"style":231},[230],[39,25455,25457,25460,25489],{"className":25456},[74],[39,25458],{"className":25459,"style":79},[78],[39,25461,25463,25466],{"className":25462},[83],[39,25464,11772],{"className":25465,"style":11793},[83,641],[39,25467,25469],{"className":25468},[192],[39,25470,25472],{"className":25471},[196],[39,25473,25475],{"className":25474},[200],[39,25476,25478],{"className":25477,"style":6410},[204],[39,25479,25480,25483],{"style":208},[39,25481],{"className":25482,"style":213},[212],[39,25484,25486],{"className":25485},[217,218,219,220],[39,25487,2098],{"className":25488,"style":2114},[83,84,220],[39,25490,23703],{"className":25491,"style":178},[83,84],[39,25493,25495,25512],{"className":25494,"translate":43},[42],[39,25496,25498],{"className":25497},[47],[49,25499,25500],{"xmlns":51},[53,25501,25502,25510],{},[56,25503,25504],{},[138,25505,25506,25508],{},[59,25507,11772],{"mathvariant":617},[133,25509,21884],{},[63,25511,21887],{"encoding":65},[39,25513,25515],{"className":25514,"ariaHidden":70},[69],[39,25516,25518,25521],{"className":25517},[74],[39,25519],{"className":25520,"style":21897},[78],[39,25522,25524,25527],{"className":25523},[83],[39,25525,11772],{"className":25526,"style":11793},[83,641],[39,25528,25530],{"className":25529},[192],[39,25531,25533],{"className":25532},[196],[39,25534,25536],{"className":25535},[200],[39,25537,25539],{"className":25538,"style":21897},[204],[39,25540,25541,25544],{"style":208},[39,25542],{"className":25543,"style":213},[212],[39,25545,25547],{"className":25546},[217,218,219,220],[39,25548,21884],{"className":25549},[465,220],"，逐点取最大值也保持逐点序，故 ",[39,25552,25554,25584],{"className":25553,"translate":43},[42],[39,25555,25557],{"className":25556},[47],[49,25558,25559],{"xmlns":51},[53,25560,25561,25581],{},[56,25562,25563,25569,25571,25573,25579],{},[138,25564,25565,25567],{},[59,25566,11772],{"mathvariant":617},[133,25568,21884],{},[59,25570,4230],{},[133,25572,1824],{},[138,25574,25575,25577],{},[59,25576,11772],{"mathvariant":617},[133,25578,21884],{},[59,25580,23703],{},[63,25582,25583],{"encoding":65},"\\mathcal{T}^*V\\le\\mathcal{T}^*W",[39,25585,25587,25635],{"className":25586,"ariaHidden":70},[69],[39,25588,25590,25594,25623,25626,25629,25632],{"className":25589},[74],[39,25591],{"className":25592,"style":25593},[78],"height:0.8247em;vertical-align:-0.136em;",[39,25595,25597,25600],{"className":25596},[83],[39,25598,11772],{"className":25599,"style":11793},[83,641],[39,25601,25603],{"className":25602},[192],[39,25604,25606],{"className":25605},[196],[39,25607,25609],{"className":25608},[200],[39,25610,25612],{"className":25611,"style":21897},[204],[39,25613,25614,25617],{"style":208},[39,25615],{"className":25616,"style":213},[212],[39,25618,25620],{"className":25619},[217,218,219,220],[39,25621,21884],{"className":25622},[465,220],[39,25624,4230],{"className":25625,"style":461},[83,84],[39,25627],{"className":25628,"style":231},[230],[39,25630,1824],{"className":25631},[235],[39,25633],{"className":25634,"style":231},[230],[39,25636,25638,25641,25670],{"className":25637},[74],[39,25639],{"className":25640,"style":21897},[78],[39,25642,25644,25647],{"className":25643},[83],[39,25645,11772],{"className":25646,"style":11793},[83,641],[39,25648,25650],{"className":25649},[192],[39,25651,25653],{"className":25652},[196],[39,25654,25656],{"className":25655},[200],[39,25657,25659],{"className":25658,"style":21897},[204],[39,25660,25661,25664],{"style":208},[39,25662],{"className":25663,"style":213},[212],[39,25665,25667],{"className":25666},[217,218,219,220],[39,25668,21884],{"className":25669},[465,220],[39,25671,23703],{"className":25672,"style":178},[83,84],[11,25674,25675],{},"常数平移式由直接计算得到：",[39,25677,25679],{"className":25678,"translate":43},[746],[39,25680,25682,25861],{"className":25681,"translate":43},[42],[39,25683,25685],{"className":25684},[47],[49,25686,25687],{"xmlns":51,"display":755},[53,25688,25689,25858],{},[759,25690,25691,25823],{"rowspacing":761,"columnalign":762,"columnspacing":146},[764,25692,25693,25717],{},[767,25694,25695],{},[770,25696,25697],{"scriptlevel":541,"displaystyle":70},[56,25698,25699,25705,25707,25709,25711,25713,25715],{},[138,25700,25701,25703],{},[59,25702,11772],{"mathvariant":617},[59,25704,2098],{},[133,25706,136],{"stretchy":135},[59,25708,4230],{},[133,25710,788],{},[59,25712,24077],{},[539,25714,546],{"mathvariant":24155},[133,25716,161],{"stretchy":135},[767,25718,25719],{},[770,25720,25721],{"scriptlevel":541,"displaystyle":70},[56,25722,25723,25725,25727,25733,25735,25737,25739,25741,25743,25745,25755,25757,25759,25765,25767,25769,25771,25773,25775],{},[56,25724],{},[133,25726,793],{},[6556,25728,25729,25731],{},[133,25730,3191],{},[59,25732,158],{},[59,25734,2098],{},[133,25736,136],{"stretchy":135},[59,25738,158],{},[133,25740,150],{},[59,25742,142],{},[133,25744,161],{"stretchy":135},[6556,25746,25747,25749],{},[133,25748,3191],{},[138,25750,25751,25753],{},[59,25752,142],{},[133,25754,147],{"mathvariant":145,"lspace":146,"rspace":146},[59,25756,131],{},[133,25758,136],{"stretchy":135},[138,25760,25761,25763],{},[59,25762,142],{},[133,25764,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,25766,150],{},[59,25768,142],{},[133,25770,155],{"separator":70},[59,25772,158],{},[133,25774,161],{"stretchy":135},[56,25776,25777,25779,25781,25783,25785,25787,25789,25791,25797,25799,25801,25803,25805,25807,25813,25815,25817,25819,25821],{},[133,25778,537],{"fence":70},[59,25780,401],{},[133,25782,136],{"stretchy":135},[59,25784,142],{},[133,25786,155],{"separator":70},[59,25788,158],{},[133,25790,155],{"separator":70},[138,25792,25793,25795],{},[59,25794,142],{},[133,25796,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,25798,161],{"stretchy":135},[133,25800,788],{},[59,25802,531],{},[59,25804,4230],{},[133,25806,136],{"stretchy":135},[138,25808,25809,25811],{},[59,25810,142],{},[133,25812,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,25814,161],{"stretchy":135},[133,25816,788],{},[59,25818,531],{},[59,25820,24077],{},[133,25822,4280],{"fence":70},[764,25824,25825,25831],{},[767,25826,25827],{},[770,25828,25829],{"scriptlevel":541,"displaystyle":70},[56,25830],{},[767,25832,25833],{},[770,25834,25835],{"scriptlevel":541,"displaystyle":70},[56,25836,25837,25839,25841,25847,25849,25851,25853,25855],{},[56,25838],{},[133,25840,793],{},[138,25842,25843,25845],{},[59,25844,11772],{"mathvariant":617},[59,25846,2098],{},[59,25848,4230],{},[133,25850,788],{},[59,25852,531],{},[59,25854,24077],{},[539,25856,25857],{"mathvariant":24155},"1.",[63,25859,25860],{"encoding":65},"\\begin{aligned}\n\\mathcal{T}^\\pi(V+c\\mathbf{1})\n&=\\sum_{a}\\pi(a\\mid s)\\sum_{s'}P(s'\\mid s,a)\n\\left[r(s,a,s')+\\gamma V(s')+\\gamma c\\right]\\\\\n&=\\mathcal{T}^\\pi V+\\gamma c\\mathbf{1}.\n\\end{aligned}",[39,25862,25864],{"className":25863,"ariaHidden":70},[69],[39,25865,25867,25870],{"className":25866},[74],[39,25868],{"className":25869,"style":6954},[78],[39,25871,25873],{"className":25872},[83],[39,25874,25876,25968],{"className":25875},[759],[39,25877,25879],{"className":25878},[918],[39,25880,25882,25960],{"className":25881},[196,649],[39,25883,25885,25957],{"className":25884},[200],[39,25886,25888,25949],{"className":25887,"style":6973},[204],[39,25889,25890,25893],{"style":6976},[39,25891],{"className":25892,"style":3311},[212],[39,25894,25896,25925,25928,25931,25934,25937,25940,25943,25946],{"className":25895},[83],[39,25897,25899,25902],{"className":25898},[83],[39,25900,11772],{"className":25901,"style":11793},[83,641],[39,25903,25905],{"className":25904},[192],[39,25906,25908],{"className":25907},[196],[39,25909,25911],{"className":25910},[200],[39,25912,25914],{"className":25913,"style":4421},[204],[39,25915,25916,25919],{"style":1027},[39,25917],{"className":25918,"style":213},[212],[39,25920,25922],{"className":25921},[217,218,219,220],[39,25923,2098],{"className":25924,"style":2114},[83,84,220],[39,25926,136],{"className":25927},[182],[39,25929,4230],{"className":25930,"style":461},[83,84],[39,25932],{"className":25933,"style":461},[230],[39,25935,788],{"className":25936},[465],[39,25938],{"className":25939,"style":461},[230],[39,25941,24077],{"className":25942},[83,84],[39,25944,546],{"className":25945},[83,24328],[39,25947,161],{"className":25948},[263],[39,25950,25951,25954],{"style":7032},[39,25952],{"className":25953,"style":3311},[212],[39,25955],{"className":25956},[83],[39,25958,673],{"className":25959},[672],[39,25961,25963],{"className":25962},[200],[39,25964,25966],{"className":25965,"style":7048},[204],[39,25967],{},[39,25969,25971],{"className":25970},[1223],[39,25972,25974,26433],{"className":25973},[196,649],[39,25975,25977,26430],{"className":25976},[200],[39,25978,25980,26357],{"className":25979,"style":6973},[204],[39,25981,25982,25985],{"style":6976},[39,25983],{"className":25984,"style":3311},[212],[39,25986,25988,25991,25994,25997,26000,26047,26050,26053,26056,26059,26062,26065,26068,26071,26074,26077,26153,26156,26159,26162,26194,26197,26200,26203,26206,26209,26212,26215,26218,26221],{"className":25987},[83],[39,25989],{"className":25990},[83],[39,25992],{"className":25993,"style":231},[230],[39,25995,793],{"className":25996},[235],[39,25998],{"className":25999,"style":231},[230],[39,26001,26003],{"className":26002},[1744,3294],[39,26004,26006,26039],{"className":26005},[196,649],[39,26007,26009,26036],{"className":26008},[200],[39,26010,26012,26026],{"className":26011,"style":6670},[204],[39,26013,26014,26017],{"style":6673},[39,26015],{"className":26016,"style":3311},[212],[39,26018,26020],{"className":26019},[217,218,219,220],[39,26021,26023],{"className":26022},[83,220],[39,26024,158],{"className":26025},[83,84,220],[39,26027,26028,26031],{"style":3330},[39,26029],{"className":26030,"style":3311},[212],[39,26032,26033],{},[39,26034,3191],{"className":26035},[1744,3339,3340],[39,26037,673],{"className":26038},[672],[39,26040,26042],{"className":26041},[200],[39,26043,26045],{"className":26044,"style":6705},[204],[39,26046],{},[39,26048],{"className":26049,"style":256},[230],[39,26051,2098],{"className":26052,"style":2114},[83,84],[39,26054,136],{"className":26055},[182],[39,26057,158],{"className":26058},[83,84],[39,26060],{"className":26061,"style":231},[230],[39,26063,150],{"className":26064},[235],[39,26066],{"className":26067,"style":231},[230],[39,26069,142],{"className":26070},[83,84],[39,26072,161],{"className":26073},[263],[39,26075],{"className":26076,"style":256},[230],[39,26078,26080],{"className":26079},[1744,3294],[39,26081,26083,26145],{"className":26082},[196,649],[39,26084,26086,26142],{"className":26085},[200],[39,26087,26089,26132],{"className":26088,"style":6670},[204],[39,26090,26091,26094],{"style":7097},[39,26092],{"className":26093,"style":3311},[212],[39,26095,26097],{"className":26096},[217,218,219,220],[39,26098,26100],{"className":26099},[83,220],[39,26101,26103,26106],{"className":26102},[83,220],[39,26104,142],{"className":26105},[83,84,220],[39,26107,26109],{"className":26108},[192],[39,26110,26112],{"className":26111},[196],[39,26113,26115],{"className":26114},[200],[39,26116,26118],{"className":26117,"style":3997},[204],[39,26119,26120,26123],{"style":4000},[39,26121],{"className":26122,"style":4004},[212],[39,26124,26126],{"className":26125},[217,4008,4009,220],[39,26127,26129],{"className":26128},[83,220],[39,26130,147],{"className":26131},[83,220],[39,26133,26134,26137],{"style":3330},[39,26135],{"className":26136,"style":3311},[212],[39,26138,26139],{},[39,26140,3191],{"className":26141},[1744,3339,3340],[39,26143,673],{"className":26144},[672],[39,26146,26148],{"className":26147},[200],[39,26149,26151],{"className":26150,"style":7158},[204],[39,26152],{},[39,26154],{"className":26155,"style":256},[230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",[39,26445,26447,26464],{"className":26446,"translate":43},[42],[39,26448,26450],{"className":26449},[47],[49,26451,26452],{"xmlns":51},[53,26453,26454,26462],{},[56,26455,26456],{},[138,26457,26458,26460],{},[59,26459,11772],{"mathvariant":617},[133,26461,21884],{},[63,26463,21887],{"encoding":65},[39,26465,26467],{"className":26466,"ariaHidden":70},[69],[39,26468,26470,26473],{"className":26469},[74],[39,26471],{"className":26472,"style":21897},[78],[39,26474,26476,26479],{"className":26475},[83],[39,26477,11772],{"className":26478,"style":11793},[83,641],[39,26480,26482],{"className":26481},[192],[39,26483,26485],{"className":26484},[196],[39,26486,26488],{"className":26487},[200],[39,26489,26491],{"className":26490,"style":21897},[204],[39,26492,26493,26496],{"style":208},[39,26494],{"className":26495,"style":213},[212],[39,26497,26499],{"className":26498},[217,218,219,220],[39,26500,21884],{"className":26501},[465,220],"，常数 ",[39,26504,26506,26522],{"className":26505,"translate":43},[42],[39,26507,26509],{"className":26508},[47],[49,26510,26511],{"xmlns":51},[53,26512,26513,26519],{},[56,26514,26515,26517],{},[59,26516,531],{},[59,26518,24077],{},[63,26520,26521],{"encoding":65},"\\gamma c",[39,26523,26525],{"className":26524,"ariaHidden":70},[69],[39,26526,26528,26531,26534],{"className":26527},[74],[39,26529],{"className":26530,"style":2654},[78],[39,26532,531],{"className":26533,"style":565},[83,84],[39,26535,24077],{"className":26536},[83,84]," 与动作 ",[39,26539,26541,26554],{"className":26540,"translate":43},[42],[39,26542,26544],{"className":26543},[47],[49,26545,26546],{"xmlns":51},[53,26547,26548,26552],{},[56,26549,26550],{},[59,26551,158],{},[63,26553,158],{"encoding":65},[39,26555,26557],{"className":26556,"ariaHidden":70},[69],[39,26558,26560,26563],{"className":26559},[74],[39,26561],{"className":26562,"style":290},[78],[39,26564,158],{"className":26565},[83,84]," 无关，可以从最大值中提出。证毕。",[11,26568,26569,26570,26599,26600,26628],{},"这两个性质看起来简单，却足以推出压缩性。单调性说明算子保持函数之间的序关系，常数平移式说明算子对整体平移的响应恰好按 ",[39,26571,26573,26587],{"className":26572,"translate":43},[42],[39,26574,26576],{"className":26575},[47],[49,26577,26578],{"xmlns":51},[53,26579,26580,26584],{},[56,26581,26582],{},[59,26583,531],{},[63,26585,26586],{"encoding":65},"\\gamma",[39,26588,26590],{"className":26589,"ariaHidden":70},[69],[39,26591,26593,26596],{"className":26592},[74],[39,26594],{"className":26595,"style":2654},[78],[39,26597,531],{"className":26598,"style":565},[83,84]," 缩小。二者结合，就得到「两个函数经过算子作用后距离至多变为原来的 ",[39,26601,26603,26616],{"className":26602,"translate":43},[42],[39,26604,26606],{"className":26605},[47],[49,26607,26608],{"xmlns":51},[53,26609,26610,26614],{},[56,26611,26612],{},[59,26613,531],{},[63,26615,26586],{"encoding":65},[39,26617,26619],{"className":26618,"ariaHidden":70},[69],[39,26620,26622,26625],{"className":26621},[74],[39,26623],{"className":26624,"style":2654},[78],[39,26626,531],{"className":26627,"style":565},[83,84]," 倍」这一结论。",[26,26630,26632],{"id":26631},"_33-压缩映射","3.3 压缩映射",[31,26634,26637,27062],{"id":26635,"title":26636},"theorem-contraction","定理 3.2（贝尔曼算子的压缩性）",[11,26638,37,26639,14627,26711,1639,26769,26827,26828,26871,26872,26900,26901,742],{},[39,26640,26642,26663],{"className":26641,"translate":43},[42],[39,26643,26645],{"className":26644},[47],[49,26646,26647],{"xmlns":51},[53,26648,26649,26661],{},[56,26650,26651,26653,26655,26657,26659],{},[539,26652,541],{},[133,26654,1824],{},[59,26656,531],{},[133,26658,1693],{},[539,26660,546],{},[63,26662,14577],{"encoding":65},[39,26664,26666,26684,26702],{"className":26665,"ariaHidden":70},[69],[39,26667,26669,26672,26675,26678,26681],{"className":26668},[74],[39,26670],{"className":26671,"style":14587},[78],[39,26673,541],{"className":26674},[83],[39,26676],{"className":26677,"style":231},[230],[39,26679,1824],{"className":26680},[235],[39,26682],{"className":26683,"style":231},[230],[39,26685,26687,26690,26693,26696,26699],{"className":26686},[74],[39,26688],{"className":26689,"style":561},[78],[39,26691,531],{"className":26692,"style":565},[83,84],[39,26694],{"className":26695,"style":231},[230],[39,26697,1693],{"className":26698},[235],[39,26700],{"className":26701,"style":231},[230],[39,26703,26705,26708],{"className":26704},[74],[39,26706],{"className":26707,"style":2473},[78],[39,26709,546],{"className":26710},[83],[39,26712,26714,26731],{"className":26713,"translate":43},[42],[39,26715,26717],{"className":26716},[47],[49,26718,26719],{"xmlns":51},[53,26720,26721,26729],{},[56,26722,26723],{},[138,26724,26725,26727],{},[59,26726,11772],{"mathvariant":617},[59,26728,2098],{},[63,26730,11777],{"encoding":65},[39,26732,26734],{"className":26733,"ariaHidden":70},[69],[39,26735,26737,26740],{"className":26736},[74],[39,26738],{"className":26739,"style":79},[78],[39,26741,26743,26746],{"className":26742},[83],[39,26744,11772],{"className":26745,"style":11793},[83,641],[39,26747,26749],{"className":26748},[192],[39,26750,26752],{"className":26751},[196],[39,26753,26755],{"className":26754},[200],[39,26756,26758],{"className":26757,"style":6410},[204],[39,26759,26760,26763],{"style":208},[39,26761],{"className":26762,"style":213},[212],[39,26764,26766],{"className":26765},[217,218,219,220],[39,26767,2098],{"className":26768,"style":2114},[83,84,220],[39,26770,26772,26789],{"className":26771,"translate":43},[42],[39,26773,26775],{"className":26774},[47],[49,26776,26777],{"xmlns":51},[53,26778,26779,26787],{},[56,26780,26781],{},[138,26782,26783,26785],{},[59,26784,11772],{"mathvariant":617},[133,26786,21884],{},[63,26788,21887],{"encoding":65},[39,26790,26792],{"className":26791,"ariaHidden":70},[69],[39,26793,26795,26798],{"className":26794},[74],[39,26796],{"className":26797,"style":21897},[78],[39,26799,26801,26804],{"className":26800},[83],[39,26802,11772],{"className":26803,"style":11793},[83,641],[39,26805,26807],{"className":26806},[192],[39,26808,26810],{"className":26809},[196],[39,26811,26813],{"className":26812},[200],[39,26814,26816],{"className":26815,"style":21897},[204],[39,26817,26818,26821],{"style":208},[39,26819],{"className":26820,"style":213},[212],[39,26822,26824],{"className":26823},[217,218,219,220],[39,26825,21884],{"className":26826},[465,220]," 都是 ",[39,26829,26831,26850],{"className":26830,"translate":43},[42],[39,26832,26834],{"className":26833},[47],[49,26835,26836],{"xmlns":51},[53,26837,26838,26848],{},[56,26839,26840,26842,26844,26846],{},[59,26841,19772],{},[133,26843,136],{"stretchy":135},[59,26845,61],{},[133,26847,161],{"stretchy":135},[63,26849,21380],{"encoding":65},[39,26851,26853],{"className":26852,"ariaHidden":70},[69],[39,26854,26856,26859,26862,26865,26868],{"className":26855},[74],[39,26857],{"className":26858,"style":245},[78],[39,26860,19772],{"className":26861,"style":19812},[83,84],[39,26863,136],{"className":26864},[182],[39,26866,61],{"className":26867,"style":85},[83,84],[39,26869,161],{"className":26870},[263]," 上的 ",[39,26873,26875,26888],{"className":26874,"translate":43},[42],[39,26876,26878],{"className":26877},[47],[49,26879,26880],{"xmlns":51},[53,26881,26882,26886],{},[56,26883,26884],{},[59,26885,531],{},[63,26887,26586],{"encoding":65},[39,26889,26891],{"className":26890,"ariaHidden":70},[69],[39,26892,26894,26897],{"className":26893},[74],[39,26895],{"className":26896,"style":2654},[78],[39,26898,531],{"className":26899,"style":565},[83,84],"-压缩映射，即对任意 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c\\mathbf{1}\n\\le\\mathcal{T}V_1\n\\le\\mathcal{T}V_2+\\gamma c\\mathbf{1}.",[39,27984,27986,28044,28068,28126,28184],{"className":27985,"ariaHidden":70},[69],[39,27987,27989,27992,27995,28035,28038,28041],{"className":27988},[74],[39,27990],{"className":27991,"style":634},[78],[39,27993,11772],{"className":27994,"style":11793},[83,641],[39,27996,27998,28001],{"className":27997},[83],[39,27999,4230],{"className":28000,"style":461},[83,84],[39,28002,28004],{"className":28003},[192],[39,28005,28007,28027],{"className":28006},[196,649],[39,28008,28010,28024],{"className":28009},[200],[39,28011,28013],{"className":28012,"style":963},[204],[39,28014,28015,28018],{"style":13607},[39,28016],{"className":28017,"style":213},[212],[39,28019,28021],{"className":28020},[217,218,219,220],[39,28022,17359],{"className":28023},[83,220],[39,28025,673],{"className":28026},[672],[39,28028,28030],{"className":28029},[200],[39,28031,28033],{"className":28032,"style":680},[204],[39,28034],{},[39,28036],{"className":28037,"style":461},[230],[39,28039,5378],{"className":28040},[465],[39,28042],{"className":28043,"style":461},[230],[39,28045,28047,28050,28053,28056,28059,28062,28065],{"className":28046},[74],[39,28048],{"className":28049,"style":17578},[78],[39,28051,531],{"className":28052,"style":565},[83,84],[39,28054,24077],{"className":28055},[83,84],[39,28057,546],{"className":28058},[83,24328],[39,28060],{"className":28061,"style":231},[230],[39,28063,1824],{"className":28064},[235],[39,28066],{"className":28067,"style":231},[230],[39,28069,28071,28074,28077,28117,28120,28123],{"className":28070},[74],[39,28072],{"className":28073,"style":634},[78],[39,28075,11772],{"className":28076,"style":11793},[83,641],[39,28078,28080,28083],{"className":28079},[83],[39,28081,4230],{"className":28082,"style":461},[83,84],[39,28084,28086],{"className":28085},[192],[39,28087,28089,28109],{"className":28088},[196,649],[39,28090,28092,28106],{"className":28091},[200],[39,28093,28095],{"className":28094,"style":963},[204],[39,28096,28097,28100],{"style":13607},[39,28098],{"className":28099,"style":213},[212],[39,28101,28103],{"className":28102},[217,218,219,220],[39,28104,546],{"className":28105},[83,220],[39,28107,673],{"className":28108},[672],[39,28110,28112],{"className":28111},[200],[39,28113,28115],{"className":28114,"style":680},[204],[39,28116],{},[39,28118],{"className":28119,"style":231},[230],[39,28121,1824],{"className":28122},[235],[39,28124],{"className":28125,"style":231},[230],[39,28127,28129,28132,28135,28175,28178,28181],{"className":28128},[74],[39,28130],{"className":28131,"style":634},[78],[39,28133,11772],{"className":28134,"style":11793},[83,641],[39,28136,28138,28141],{"className":28137},[83],[39,28139,4230],{"className":28140,"style":461},[83,84],[39,28142,28144],{"className":28143},[192],[39,28145,28147,28167],{"className":28146},[196,649],[39,28148,28150,28164],{"className":28149},[200],[39,28151,28153],{"className":28152,"style":963},[204],[39,28154,28155,28158],{"style":13607},[39,28156],{"className":28157,"style":213},[212],[39,28159,28161],{"className":28160},[217,218,219,220],[39,28162,17359],{"className":28163},[83,220],[39,28165,673],{"className":28166},[672],[39,28168,28170],{"className":28169},[200],[39,28171,28173],{"className":28172,"style":680},[204],[39,28174],{},[39,28176],{"className":28177,"style":461},[230],[39,28179,788],{"className":28180},[465],[39,28182],{"className":28183,"style":461},[230],[39,28185,28187,28190,28193,28196,28199],{"className":28186},[74],[39,28188],{"className":28189,"style":17578},[78],[39,28191,531],{"className":28192,"style":565},[83,84],[39,28194,24077],{"className":28195},[83,84],[39,28197,546],{"className":28198},[83,24328],[39,28200,2903],{"className":28201},[83],[11,28203,28204,28205,742],{},"因此对一切 ",[39,28206,28208,28221],{"className":28207,"translate":43},[42],[39,28209,28211],{"className":28210},[47],[49,28212,28213],{"xmlns":51},[53,28214,28215,28219],{},[56,28216,28217],{},[59,28218,142],{},[63,28220,142],{"encoding":65},[39,28222,28224],{"className":28223,"ariaHidden":70},[69],[39,28225,28227,28230],{"className":28226},[74],[39,28228],{"className":28229,"style":290},[78],[39,28231,142],{"className":28232},[83,84],[39,28234,28236],{"className":28235,"translate":43},[746],[39,28237,28239,28337],{"className":28238,"translate":43},[42],[39,28240,28242],{"className":28241},[47],[49,28243,28244],{"xmlns":51,"display":755},[53,28245,28246,28334],{},[759,28247,28248,28308],{"rowspacing":761,"columnalign":762,"columnspacing":146},[764,28249,28250,28256],{},[767,28251,28252],{},[770,28253,28254],{"scriptlevel":541,"displaystyle":70},[56,28255],{},[767,28257,28258],{},[770,28259,28260],{"scriptlevel":541,"displaystyle":70},[56,28261,28262,28264],{},[56,28263],{},[56,28265,28266,28268,28270,28272,28278,28280,28282,28284,28286,28288,28290,28292,28298,28300,28302,28304,28306],{},[133,28267,150],{"fence":70},[133,28269,136],{"stretchy":135},[59,28271,11772],{"mathvariant":617},[613,28273,28274,28276],{},[59,28275,4230],{},[539,28277,546],{},[133,28279,161],{"stretchy":135},[133,28281,136],{"stretchy":135},[59,28283,142],{},[133,28285,161],{"stretchy":135},[133,28287,5378],{},[133,28289,136],{"stretchy":135},[59,28291,11772],{"mathvariant":617},[613,28293,28294,28296],{},[59,28295,4230],{},[539,28297,17359],{},[133,28299,161],{"stretchy":135},[133,28301,136],{"stretchy":135},[59,28303,142],{},[133,28305,161],{"stretchy":135},[133,28307,150],{"fence":70},[764,28309,28310,28316],{},[767,28311,28312],{},[770,28313,28314],{"scriptlevel":541,"displaystyle":70},[56,28315],{},[767,28317,28318],{},[770,28319,28320],{"scriptlevel":541,"displaystyle":70},[56,28321,28322,28324,28326,28328,28330,28332],{},[56,28323],{},[230,28325],{"width":6891},[133,28327,1824],{},[59,28329,531],{},[59,28331,24077],{},[59,28333,2903],{"mathvariant":145},[63,28335,28336],{"encoding":65},"\\begin{aligned}\n&\\left|(\\mathcal{T}V_1)(s)-(\\mathcal{T}V_2)(s)\\right|\\\\\n&\\quad\\le\\gamma c.\n\\end{aligned}",[39,28338,28340],{"className":28339,"ariaHidden":70},[69],[39,28341,28343,28346],{"className":28342},[74],[39,28344],{"className":28345,"style":4374},[78],[39,28347,28349],{"className":28348},[83],[39,28350,28352,28393],{"className":28351},[759],[39,28353,28355],{"className":28354},[918],[39,28356,28358,28385],{"className":28357},[196,649],[39,28359,28361,28382],{"className":28360},[200],[39,28362,28364,28374],{"className":28363,"style":4393},[204],[39,28365,28367,28371],{"style":28366},"top:-3.6em;",[39,28368],{"className":28369,"style":28370},[212],"height:2.84em;",[39,28372],{"className":28373},[83],[39,28375,28376,28379],{"style":5834},[39,28377],{"className":28378,"style":28370},[212],[39,28380],{"className":28381},[83],[39,28383,673],{"className":28384},[672],[39,28386,28388],{"className":28387},[200],[39,28389,28391],{"className":28390,"style":4507},[204],[39,28392],{},[39,28394,28396],{"className":28395},[1223],[39,28397,28399,28588],{"className":28398},[196,649],[39,28400,28402,28585],{"className":28401},[200],[39,28403,28405,28553],{"className":28404,"style":4393},[204],[39,28406,28407,28410],{"style":4396},[39,28408],{"className":28409,"style":935},[212],[39,28411,28413,28416,28419],{"className":28412},[83],[39,28414],{"className":28415},[83],[39,28417],{"className":28418,"style":256},[230],[39,28420,28422,28425,28428,28431,28471,28474,28477,28480,28483,28486,28489,28492,28495,28498,28538,28541,28544,28547,28550],{"className":28421},[2842],[39,28423,150],{"className":28424,"style":4592},[182,4591],[39,28426,136],{"className":28427},[182],[39,28429,11772],{"className":28430,"style":11793},[83,641],[39,28432,28434,28437],{"className":28433},[83],[39,28435,4230],{"className":28436,"style":461},[83,84],[39,28438,28440],{"className":28439},[192],[39,28441,28443,28463],{"className":28442},[196,649],[39,28444,28446,28460],{"className":28445},[200],[39,28447,28449],{"className":28448,"style":963},[204],[39,28450,28451,28454],{"style":13607},[39,28452],{"className":28453,"style":213},[212],[39,28455,28457],{"className":28456},[217,218,219,220],[39,28458,546],{"className":28459},[83,220],[39,28461,673],{"className":28462},[672],[39,28464,28466],{"className":28465},[200],[39,28467,28469],{"className":28468,"style":680},[204],[39,28470],{},[39,28472,161],{"className":28473},[263],[39,28475,136],{"className":28476},[182],[39,28478,142],{"className":28479},[83,84],[39,28481,161],{"className":28482},[263],[39,28484],{"className":28485,"style":461},[230],[39,28487,5378],{"className":28488},[465],[39,28490],{"className":28491,"style":461},[230],[39,28493,136],{"className":28494},[182],[39,28496,11772],{"className":28497,"style":11793},[83,641],[39,28499,28501,28504],{"className":28500},[83],[39,28502,4230],{"className":28503,"style":461},[83,84],[39,28505,28507],{"className":28506},[192],[39,28508,28510,28530],{"className":28509},[196,649],[39,28511,28513,28527],{"className":28512},[200],[39,28514,28516],{"className":28515,"style":963},[204],[39,28517,28518,28521],{"style":13607},[39,28519],{"className":28520,"style":213},[212],[39,28522,28524],{"className":28523},[217,218,219,220],[39,28525,17359],{"className":28526},[83,220],[39,28528,673],{"className":28529},[672],[39,28531,28533],{"className":28532},[200],[39,28534,28536],{"className":28535,"style":680},[204],[39,28537],{},[39,28539,161],{"className":28540},[263],[39,28542,136],{"className":28543},[182],[39,28545,142],{"className":28546},[83,84],[39,28548,161],{"className":28549},[263],[39,28551,150],{"className":28552,"style":4592},[263,4591],[39,28554,28555,28558],{"style":4444},[39,28556],{"className":28557,"style":935},[212],[39,28559,28561,28564,28567,28570,28573,28576,28579,28582],{"className":28560},[83],[39,28562],{"className":28563},[83],[39,28565],{"className":28566,"style":7240},[230],[39,28568],{"className":28569,"style":231},[230],[39,28571,1824],{"className":28572},[235],[39,28574],{"className":28575,"style":231},[230],[39,28577,531],{"className":28578,"style":565},[83,84],[39,28580,24077],{"className":28581},[83,84],[39,28583,2903],{"className":28584},[83],[39,28586,673],{"className":28587},[672],[39,28589,28591],{"className":28590},[200],[39,28592,28594],{"className":28593,"style":4507},[204],[39,28595],{},[11,28597,26443,28598,28626],{},[39,28599,28601,28614],{"className":28600,"translate":43},[42],[39,28602,28604],{"className":28603},[47],[49,28605,28606],{"xmlns":51},[53,28607,28608,28612],{},[56,28609,28610],{},[59,28611,142],{},[63,28613,142],{"encoding":65},[39,28615,28617],{"className":28616,"ariaHidden":70},[69],[39,28618,28620,28623],{"className":28619},[74],[39,28621],{"className":28622,"style":290},[78],[39,28624,142],{"className":28625},[83,84]," 取上确界即得结论。证毕。",[11,28628,28629,28630,28658,28659,28709,28710,28739,28740,28769,28770,742,28799,28828,28829,28915],{},"压缩性的直观含义是：无论两个价值函数相差多大，经过一次贝尔曼更新后，它们的最大差异最多变成原来的 ",[39,28631,28633,28646],{"className":28632,"translate":43},[42],[39,28634,28636],{"className":28635},[47],[49,28637,28638],{"xmlns":51},[53,28639,28640,28644],{},[56,28641,28642],{},[59,28643,531],{},[63,28645,26586],{"encoding":65},[39,28647,28649],{"className":28648,"ariaHidden":70},[69],[39,28650,28652,28655],{"className":28651},[74],[39,28653],{"className":28654,"style":2654},[78],[39,28656,531],{"className":28657,"style":565},[83,84]," 倍。若 ",[39,28660,28662,28679],{"className":28661,"translate":43},[42],[39,28663,28665],{"className":28664},[47],[49,28666,28667],{"xmlns":51},[53,28668,28669,28677],{},[56,28670,28671,28673,28675],{},[59,28672,531],{},[133,28674,793],{},[539,28676,17739],{},[63,28678,17742],{"encoding":65},[39,28680,28682,28700],{"className":28681,"ariaHidden":70},[69],[39,28683,28685,28688,28691,28694,28697],{"className":28684},[74],[39,28686],{"className":28687,"style":2654},[78],[39,28689,531],{"className":28690,"style":565},[83,84],[39,28692],{"className":28693,"style":231},[230],[39,28695,793],{"className":28696},[235],[39,28698],{"className":28699,"style":231},[230],[39,28701,28703,28706],{"className":28702},[74],[39,28704],{"className":28705,"style":2473},[78],[39,28707,17739],{"className":28708},[83],"，初始最大差异为 ",[39,28711,28713,28727],{"className":28712,"translate":43},[42],[39,28714,28716],{"className":28715},[47],[49,28717,28718],{"xmlns":51},[53,28719,28720,28725],{},[56,28721,28722],{},[539,28723,28724],{},"100",[63,28726,28724],{"encoding":65},[39,28728,28730],{"className":28729,"ariaHidden":70},[69],[39,28731,28733,28736],{"className":28732},[74],[39,28734],{"className":28735,"style":2473},[78],[39,28737,28724],{"className":28738},[83],"，则一次更新后差异不超过 ",[39,28741,28743,28757],{"className":28742,"translate":43},[42],[39,28744,28746],{"className":28745},[47],[49,28747,28748],{"xmlns":51},[53,28749,28750,28755],{},[56,28751,28752],{},[539,28753,28754],{},"90",[63,28756,28754],{"encoding":65},[39,28758,28760],{"className":28759,"ariaHidden":70},[69],[39,28761,28763,28766],{"className":28762},[74],[39,28764],{"className":28765,"style":2473},[78],[39,28767,28754],{"className":28768},[83],"，两次后不超过 ",[39,28771,28773,28787],{"className":28772,"translate":43},[42],[39,28774,28776],{"className":28775},[47],[49,28777,28778],{"xmlns":51},[53,28779,28780,28785],{},[56,28781,28782],{},[539,28783,28784],{},"81",[63,28786,28784],{"encoding":65},[39,28788,28790],{"className":28789,"ariaHidden":70},[69],[39,28791,28793,28796],{"className":28792},[74],[39,28794],{"className":28795,"style":2473},[78],[39,28797,28784],{"className":28798},[83],[39,28800,28802,28815],{"className":28801,"translate":43},[42],[39,28803,28805],{"className":28804},[47],[49,28806,28807],{"xmlns":51},[53,28808,28809,28813],{},[56,28810,28811],{},[59,28812,3196],{},[63,28814,3196],{"encoding":65},[39,28816,28818],{"className":28817,"ariaHidden":70},[69],[39,28819,28821,28825],{"className":28820},[74],[39,28822],{"className":28823,"style":28824},[78],"height:0.6944em;",[39,28826,3196],{"className":28827,"style":3321},[83,84]," 次后不超过 ",[39,28830,28832,28854],{"className":28831,"translate":43},[42],[39,28833,28835],{"className":28834},[47],[49,28836,28837],{"xmlns":51},[53,28838,28839,28851],{},[56,28840,28841,28847,28849],{},[138,28842,28843,28845],{},[539,28844,17739],{},[59,28846,3196],{},[133,28848,8885],{},[539,28850,28724],{},[63,28852,28853],{"encoding":65},"0.9^k\\cdot100",[39,28855,28857,28906],{"className":28856,"ariaHidden":70},[69],[39,28858,28860,28863,28867,28897,28900,28903],{"className":28859},[74],[39,28861],{"className":28862,"style":15577},[78],[39,28864,28866],{"className":28865},[83],"0.",[39,28868,28870,28874],{"className":28869},[83],[39,28871,28873],{"className":28872},[83],"9",[39,28875,28877],{"className":28876},[192],[39,28878,28880],{"className":28879},[196],[39,28881,28883],{"className":28882},[200],[39,28884,28886],{"className":28885,"style":15577},[204],[39,28887,28888,28891],{"style":208},[39,28889],{"className":28890,"style":213},[212],[39,28892,28894],{"className":28893},[217,218,219,220],[39,28895,3196],{"className":28896,"style":3321},[83,84,220],[39,28898],{"className":28899,"style":461},[230],[39,28901,8885],{"className":28902},[465],[39,28904],{"className":28905,"style":461},[230],[39,28907,28909,28912],{"className":28908},[74],[39,28910],{"className":28911,"style":2473},[78],[39,28913,28724],{"className":28914},[83],"。这正是后文几何收敛性的来源。",[11,28917,28918,28919,28969,28970,29021,29022,29393],{},"需要强调的是，压缩性依赖于 ",[39,28920,28922,28939],{"className":28921,"translate":43},[42],[39,28923,28925],{"className":28924},[47],[49,28926,28927],{"xmlns":51},[53,28928,28929,28937],{},[56,28930,28931,28933,28935],{},[59,28932,531],{},[133,28934,1693],{},[539,28936,546],{},[63,28938,17296],{"encoding":65},[39,28940,28942,28960],{"className":28941,"ariaHidden":70},[69],[39,28943,28945,28948,28951,28954,28957],{"className":28944},[74],[39,28946],{"className":28947,"style":561},[78],[39,28949,531],{"className":28950,"style":565},[83,84],[39,28952],{"className":28953,"style":231},[230],[39,28955,1693],{"className":28956},[235],[39,28958],{"className":28959,"style":231},[230],[39,28961,28963,28966],{"className":28962},[74],[39,28964],{"className":28965,"style":2473},[78],[39,28967,546],{"className":28968},[83],"。当 ",[39,28971,28973,28991],{"className":28972,"translate":43},[42],[39,28974,28976],{"className":28975},[47],[49,28977,28978],{"xmlns":51},[53,28979,28980,28988],{},[56,28981,28982,28984,28986],{},[59,28983,531],{},[133,28985,793],{},[539,28987,546],{},[63,28989,28990],{"encoding":65},"\\gamma=1",[39,28992,28994,29012],{"className":28993,"ariaHidden":70},[69],[39,28995,28997,29000,29003,29006,29009],{"className":28996},[74],[39,28998],{"className":28999,"style":2654},[78],[39,29001,531],{"className":29002,"style":565},[83,84],[39,29004],{"className":29005,"style":231},[230],[39,29007,793],{"className":29008},[235],[39,29010],{"className":29011,"style":231},[230],[39,29013,29015,29018],{"className":29014},[74],[39,29016],{"className":29017,"style":2473},[78],[39,29019,546],{"className":29020},[83]," 时，贝尔曼算子通常只是非扩张的，即只能得到 ",[39,29023,29025,29087],{"className":29024,"translate":43},[42],[39,29026,29028],{"className":29027},[47],[49,29029,29030],{"xmlns":51},[53,29031,29032,29084],{},[56,29033,29034,29036,29038,29044,29046,29048,29054,29060,29062,29064,29070,29072,29078],{},[59,29035,15389],{"mathvariant":145},[59,29037,11772],{"mathvariant":617},[613,29039,29040,29042],{},[59,29041,4230],{},[539,29043,546],{},[133,29045,5378],{},[59,29047,11772],{"mathvariant":617},[613,29049,29050,29052],{},[59,29051,4230],{},[539,29053,17359],{},[613,29055,29056,29058],{},[59,29057,15389],{"mathvariant":145},[59,29059,1696],{"mathvariant":145},[133,29061,1824],{},[59,29063,15389],{"mathvariant":145},[613,29065,29066,29068],{},[59,29067,4230],{},[539,29069,546],{},[133,29071,5378],{},[613,29073,29074,29076],{},[59,29075,4230],{},[539,29077,17359],{},[613,29079,29080,29082],{},[59,29081,15389],{"mathvariant":145},[59,29083,1696],{"mathvariant":145},[63,29085,29086],{"encoding":65},"\\|\\mathcal{T}V_1-\\mathcal{T}V_2\\|_\\infty\\le\\|V_1-V_2\\|_\\infty",[39,29088,29090,29151,29249,29307],{"className":29089,"ariaHidden":70},[69],[39,29091,29093,29096,29099,29102,29142,29145,29148],{"className":29092},[74],[39,29094],{"className":29095,"style":245},[78],[39,29097,15389],{"className":29098},[83],[39,29100,11772],{"className":29101,"style":11793},[83,641],[39,29103,29105,29108],{"className":29104},[83],[39,29106,4230],{"className":29107,"style":461},[83,84],[39,29109,29111],{"className":29110},[192],[39,29112,29114,29134],{"className":29113},[196,649],[39,29115,29117,29131],{"className":29116},[200],[39,29118,29120],{"className":29119,"style":963},[204],[39,29121,29122,29125],{"style":13607},[39,29123],{"className":29124,"style":213},[212],[39,29126,29128],{"className":29127},[217,218,219,220],[39,29129,546],{"className":29130},[83,220],[39,29132,673],{"className":29133},[672],[39,29135,29137],{"className":29136},[200],[39,29138,29140],{"className":29139,"style":680},[204],[39,29141],{},[39,29143],{"className":29144,"style":461},[230],[39,29146,5378],{"className":29147},[465],[39,29149],{"className":29150,"style":461},[230],[39,29152,29154,29157,29160,29200,29240,29243,29246],{"className":29153},[74],[39,29155],{"className":29156,"style":245},[78],[39,29158,11772],{"className":29159,"style":11793},[83,641],[39,29161,29163,29166],{"className":29162},[83],[39,29164,4230],{"className":29165,"style":461},[83,84],[39,29167,29169],{"className":29168},[192],[39,29170,29172,29192],{"className":29171},[196,649],[39,29173,29175,29189],{"className":29174},[200],[39,29176,29178],{"className":29177,"style":963},[204],[39,29179,29180,29183],{"style":13607},[39,29181],{"className":29182,"style":213},[212],[39,29184,29186],{"className":29185},[217,218,219,220],[39,29187,17359],{"className":29188},[83,220],[39,29190,673],{"className":29191},[672],[39,29193,29195],{"className":29194},[200],[39,29196,29198],{"className":29197,"style":680},[204],[39,29199],{},[39,29201,29203,29206],{"className":29202},[83],[39,29204,15389],{"className":29205},[83],[39,29207,29209],{"className":29208},[192],[39,29210,29212,29232],{"className":29211},[196,649],[39,29213,29215,29229],{"className":29214},[200],[39,29216,29218],{"className":29217,"style":1728},[204],[39,29219,29220,29223],{"style":966},[39,29221],{"className":29222,"style":213},[212],[39,29224,29226],{"className":29225},[217,218,219,220],[39,29227,1696],{"className":29228},[83,220],[39,29230,673],{"className":29231},[672],[39,29233,29235],{"className":29234},[200],[39,29236,29238],{"className":29237,"style":680},[204],[39,29239],{},[39,29241],{"className":29242,"style":231},[230],[39,29244,1824],{"className":29245},[235],[39,29247],{"className":29248,"style":231},[230],[39,29250,29252,29255,29258,29298,29301,29304],{"className":29251},[74],[39,29253],{"className":29254,"style":245},[78],[39,29256,15389],{"className":29257},[83],[39,29259,29261,29264],{"className":29260},[83],[39,29262,4230],{"className":29263,"style":461},[83,84],[39,29265,29267],{"className":29266},[192],[39,29268,29270,29290],{"className":29269},[196,649],[39,29271,29273,29287],{"className":29272},[200],[39,29274,29276],{"className":29275,"style":963},[204],[39,29277,29278,29281],{"style":13607},[39,29279],{"className":29280,"style":213},[212],[39,29282,29284],{"className":29283},[217,218,219,220],[39,29285,546],{"className":29286},[83,220],[39,29288,673],{"className":29289},[672],[39,29291,29293],{"className":29292},[200],[39,29294,29296],{"className":29295,"style":680},[204],[39,29297],{},[39,29299],{"className":29300,"style":461},[230],[39,29302,5378],{"className":29303},[465],[39,29305],{"className":29306,"style":461},[230],[39,29308,29310,29313,29353],{"className":29309},[74],[39,29311],{"className":29312,"style":245},[78],[39,29314,29316,29319],{"className":29315},[83],[39,29317,4230],{"className":29318,"style":461},[83,84],[39,29320,29322],{"className":29321},[192],[39,29323,29325,29345],{"className":29324},[196,649],[39,29326,29328,29342],{"className":29327},[200],[39,29329,29331],{"className":29330,"style":963},[204],[39,29332,29333,29336],{"style":13607},[39,29334],{"className":29335,"style":213},[212],[39,29337,29339],{"className":29338},[217,218,219,220],[39,29340,17359],{"className":29341},[83,220],[39,29343,673],{"className":29344},[672],[39,29346,29348],{"className":29347},[200],[39,29349,29351],{"className":29350,"style":680},[204],[39,29352],{},[39,29354,29356,29359],{"className":29355},[83],[39,29357,15389],{"className":29358},[83],[39,29360,29362],{"className":29361},[192],[39,29363,29365,29385],{"className":29364},[196,649],[39,29366,29368,29382],{"className":29367},[200],[39,29369,29371],{"className":29370,"style":1728},[204],[39,29372,29373,29376],{"style":966},[39,29374],{"className":29375,"style":213},[212],[39,29377,29379],{"className":29378},[217,218,219,220],[39,29380,1696],{"className":29381},[83,220],[39,29383,673],{"className":29384},[672],[39,29386,29388],{"className":29387},[200],[39,29389,29391],{"className":29390,"style":680},[204],[39,29392],{},"，不能保证距离严格缩小。第 5.4 节会给出反例。",[26,29395,29397],{"id":29396},"_34-banach-不动点定理","3.4 Banach 不动点定理",[31,29399,29402,29672,29802,30170],{"id":29400,"title":29401},"theorem-banach","定理 3.3（Banach 不动点定理）",[11,29403,37,29404,29458,29459,29532,29533,29606,29607,742],{},[39,29405,29407,29431],{"className":29406,"translate":43},[42],[39,29408,29410],{"className":29409},[47],[49,29411,29412],{"xmlns":51},[53,29413,29414,29428],{},[56,29415,29416,29418,29421,29423,29426],{},[133,29417,136],{"stretchy":135},[59,29419,29420],{},"X",[133,29422,155],{"separator":70},[59,29424,29425],{},"d",[133,29427,161],{"stretchy":135},[63,29429,29430],{"encoding":65},"(X,d)",[39,29432,29434],{"className":29433,"ariaHidden":70},[69],[39,29435,29437,29440,29443,29446,29449,29452,29455],{"className":29436},[74],[39,29438],{"className":29439,"style":245},[78],[39,29441,136],{"className":29442},[182],[39,29444,29420],{"className":29445,"style":14420},[83,84],[39,29447,155],{"className":29448},[252],[39,29450],{"className":29451,"style":256},[230],[39,29453,29425],{"className":29454},[83,84],[39,29456,161],{"className":29457},[263]," 是完备度量空间，",[39,29460,29462,29484],{"className":29461,"translate":43},[42],[39,29463,29465],{"className":29464},[47],[49,29466,29467],{"xmlns":51},[53,29468,29469,29481],{},[56,29470,29471,29473,29475,29477,29479],{},[59,29472,11772],{},[133,29474,404],{},[59,29476,29420],{},[133,29478,418],{},[59,29480,29420],{},[63,29482,29483],{"encoding":65},"T:X\\to X",[39,29485,29487,29505,29523],{"className":29486,"ariaHidden":70},[69],[39,29488,29490,29493,29496,29499,29502],{"className":29489},[74],[39,29491],{"className":29492,"style":79},[78],[39,29494,11772],{"className":29495,"style":178},[83,84],[39,29497],{"className":29498,"style":231},[230],[39,29500,404],{"className":29501},[235],[39,29503],{"className":29504,"style":231},[230],[39,29506,29508,29511,29514,29517,29520],{"className":29507},[74],[39,29509],{"className":29510,"style":79},[78],[39,29512,29420],{"className":29513,"style":14420},[83,84],[39,29515],{"className":29516,"style":231},[230],[39,29518,418],{"className":29519},[235],[39,29521],{"className":29522,"style":231},[230],[39,29524,29526,29529],{"className":29525},[74],[39,29527],{"className":29528,"style":79},[78],[39,29530,29420],{"className":29531,"style":14420},[83,84]," 满足：存在 ",[39,29534,29536,29561],{"className":29535,"translate":43},[42],[39,29537,29539],{"className":29538},[47],[49,29540,29541],{"xmlns":51},[53,29542,29543,29559],{},[56,29544,29545,29547,29549,29551,29553,29555,29557],{},[59,29546,531],{},[133,29548,534],{},[133,29550,537],{"stretchy":135},[539,29552,541],{},[133,29554,155],{"separator":70},[539,29556,546],{},[133,29558,161],{"stretchy":135},[63,29560,551],{"encoding":65},[39,29562,29564,29582],{"className":29563,"ariaHidden":70},[69],[39,29565,29567,29570,29573,29576,29579],{"className":29566},[74],[39,29568],{"className":29569,"style":561},[78],[39,29571,531],{"className":29572,"style":565},[83,84],[39,29574],{"className":29575,"style":231},[230],[39,29577,534],{"className":29578},[235],[39,29580],{"className":29581,"style":231},[230],[39,29583,29585,29588,29591,29594,29597,29600,29603],{"className":29584},[74],[39,29586],{"className":29587,"style":245},[78],[39,29589,537],{"className":29590},[182],[39,29592,541],{"className":29593},[83],[39,29595,155],{"className":29596},[252],[39,29598],{"className":29599,"style":256},[230],[39,29601,546],{"className":29602},[83],[39,29604,161],{"className":29605},[263],"，使得对任意 ",[39,29608,29610,29633],{"className":29609,"translate":43},[42],[39,29611,29613],{"className":29612},[47],[49,29614,29615],{"xmlns":51},[53,29616,29617,29630],{},[56,29618,29619,29621,29623,29626,29628],{},[59,29620,1755],{},[133,29622,155],{"separator":70},[59,29624,29625],{},"y",[133,29627,534],{},[59,29629,29420],{},[63,29631,29632],{"encoding":65},"x,y\\in X",[39,29634,29636,29663],{"className":29635,"ariaHidden":70},[69],[39,29637,29639,29642,29645,29648,29651,29654,29657,29660],{"className":29638},[74],[39,29640],{"className":29641,"style":561},[78],[39,29643,1755],{"className":29644},[83,84],[39,29646,155],{"className":29647},[252],[39,29649],{"className":29650,"style":256},[230],[39,29652,29625],{"className":29653,"style":2114},[83,84],[39,29655],{"className":29656,"style":231},[230],[39,29658,534],{"className":29659},[235],[39,29661],{"className":29662,"style":231},[230],[39,29664,29666,29669],{"className":29665},[74],[39,29667],{"className":29668,"style":79},[78],[39,29670,29420],{"className":29671,"style":14420},[83,84],[39,29673,29675],{"className":29674,"translate":43},[746],[39,29676,29678,29724],{"className":29677,"translate":43},[42],[39,29679,29681],{"className":29680},[47],[49,29682,29683],{"xmlns":51,"display":755},[53,29684,29685,29721],{},[56,29686,29687,29689,29691,29693,29695,29697,29699,29701,29703,29705,29707,29709,29711,29713,29715,29717,29719],{},[59,29688,29425],{},[133,29690,136],{"stretchy":135},[59,29692,11772],{},[59,29694,1755],{},[133,29696,155],{"separator":70},[59,29698,11772],{},[59,29700,29625],{},[133,29702,161],{"stretchy":135},[133,29704,1824],{},[59,29706,531],{},[59,29708,29425],{},[133,29710,136],{"stretchy":135},[59,29712,1755],{},[133,29714,155],{"separator":70},[59,29716,29625],{},[133,29718,161],{"stretchy":135},[59,29720,2903],{"mathvariant":145},[63,29722,29723],{"encoding":65},"d(Tx,Ty)\\le\\gamma d(x,y).",[39,29725,29727,29769],{"className":29726,"ariaHidden":70},[69],[39,29728,29730,29733,29736,29739,29742,29745,29748,29751,29754,29757,29760,29763,29766],{"className":29729},[74],[39,29731],{"className":29732,"style":245},[78],[39,29734,29425],{"className":29735},[83,84],[39,29737,136],{"className":29738},[182],[39,29740,11772],{"className":29741,"style":178},[83,84],[39,29743,1755],{"className":29744},[83,84],[39,29746,155],{"className":29747},[252],[39,29749],{"className":29750,"style":256},[230],[39,29752,11772],{"className":29753,"style":178},[83,84],[39,29755,29625],{"className":29756,"style":2114},[83,84],[39,29758,161],{"className":29759},[263],[39,29761],{"className":29762,"style":231},[230],[39,29764,1824],{"className":29765},[235],[39,29767],{"className":29768,"style":231},[230],[39,29770,29772,29775,29778,29781,29784,29787,29790,29793,29796,29799],{"className":29771},[74],[39,29773],{"className":29774,"style":245},[78],[39,29776,531],{"className":29777,"style":565},[83,84],[39,29779,29425],{"className":29780},[83,84],[39,29782,136],{"className":29783},[182],[39,29785,1755],{"className":29786},[83,84],[39,29788,155],{"className":29789},[252],[39,29791],{"className":29792,"style":256},[230],[39,29794,29625],{"className":29795,"style":2114},[83,84],[39,29797,161],{"className":29798},[263],[39,29800,2903],{"className":29801},[83],[11,29803,14173,29804,29832,29833,29861,29862,29921,29922,30015,30016,30169],{},[39,29805,29807,29820],{"className":29806,"translate":43},[42],[39,29808,29810],{"className":29809},[47],[49,29811,29812],{"xmlns":51},[53,29813,29814,29818],{},[56,29815,29816],{},[59,29817,11772],{},[63,29819,11772],{"encoding":65},[39,29821,29823],{"className":29822,"ariaHidden":70},[69],[39,29824,29826,29829],{"className":29825},[74],[39,29827],{"className":29828,"style":79},[78],[39,29830,11772],{"className":29831,"style":178},[83,84]," 在 ",[39,29834,29836,29849],{"className":29835,"translate":43},[42],[39,29837,29839],{"className":29838},[47],[49,29840,29841],{"xmlns":51},[53,29842,29843,29847],{},[56,29844,29845],{},[59,29846,29420],{},[63,29848,29420],{"encoding":65},[39,29850,29852],{"className":29851,"ariaHidden":70},[69],[39,29853,29855,29858],{"className":29854},[74],[39,29856],{"className":29857,"style":79},[78],[39,29859,29420],{"className":29860,"style":14420},[83,84]," 中存在唯一不动点 ",[39,29863,29865,29883],{"className":29864,"translate":43},[42],[39,29866,29868],{"className":29867},[47],[49,29869,29870],{"xmlns":51},[53,29871,29872,29880],{},[56,29873,29874],{},[138,29875,29876,29878],{},[59,29877,1755],{},[133,29879,21884],{},[63,29881,29882],{"encoding":65},"x^*",[39,29884,29886],{"className":29885,"ariaHidden":70},[69],[39,29887,29889,29892],{"className":29888},[74],[39,29890],{"className":29891,"style":21897},[78],[39,29893,29895,29898],{"className":29894},[83],[39,29896,1755],{"className":29897},[83,84],[39,29899,29901],{"className":29900},[192],[39,29902,29904],{"className":29903},[196],[39,29905,29907],{"className":29906},[200],[39,29908,29910],{"className":29909,"style":21897},[204],[39,29911,29912,29915],{"style":208},[39,29913],{"className":29914,"style":213},[212],[39,29916,29918],{"className":29917},[217,218,219,220],[39,29919,21884],{"className":29920},[465,220],"。此外，对任意初始点 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X",[39,29948,29950,30006],{"className":29949,"ariaHidden":70},[69],[39,29951,29953,29957,29997,30000,30003],{"className":29952},[74],[39,29954],{"className":29955,"style":29956},[78],"height:0.6891em;vertical-align:-0.15em;",[39,29958,29960,29963],{"className":29959},[83],[39,29961,1755],{"className":29962},[83,84],[39,29964,29966],{"className":29965},[192],[39,29967,29969,29989],{"className":29968},[196,649],[39,29970,29972,29986],{"className":29971},[200],[39,29973,29975],{"className":29974,"style":963},[204],[39,29976,29977,29980],{"style":966},[39,29978],{"className":29979,"style":213},[212],[39,29981,29983],{"className":29982},[217,218,219,220],[39,29984,541],{"className":29985},[83,220],[39,29987,673],{"className":29988},[672],[39,29990,29992],{"className":29991},[200],[39,29993,29995],{"className":29994,"style":680},[204],[39,29996],{},[39,29998],{"className":29999,"style":231},[230],[39,30001,534],{"className":30002},[235],[39,30004],{"className":30005,"style":231},[230],[39,30007,30009,30012],{"className":30008},[74],[39,30010],{"className":30011,"style":79},[78],[39,30013,29420],{"className":30014,"style":14420},[83,84],"，迭代 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满足",[39,30171,30173],{"className":30172,"translate":43},[746],[39,30174,30176,30238],{"className":30175,"translate":43},[42],[39,30177,30179],{"className":30178},[47],[49,30180,30181],{"xmlns":51,"display":755},[53,30182,30183,30235],{},[56,30184,30185,30187,30189,30195,30197,30203,30205,30207,30213,30215,30217,30223,30225,30231,30233],{},[59,30186,29425],{},[133,30188,136],{"stretchy":135},[613,30190,30191,30193],{},[59,30192,1755],{},[59,30194,3196],{},[133,30196,155],{"separator":70},[138,30198,30199,30201],{},[59,30200,1755],{},[133,30202,21884],{},[133,30204,161],{"stretchy":135},[133,30206,1824],{},[138,30208,30209,30211],{},[59,30210,531],{},[59,30212,3196],{},[59,30214,29425],{},[133,30216,136],{"stretchy":135},[613,30218,30219,30221],{},[59,30220,1755],{},[539,30222,541],{},[133,30224,155],{"separator":70},[138,30226,30227,30229],{},[59,30228,1755],{},[133,30230,21884],{},[133,30232,161],{"stretchy":135},[59,30234,2903],{"mathvariant":145},[63,30236,30237],{"encoding":65},"d(x_k,x^*)\\le\\gamma^k 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先证唯一性。若 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都是不动点，则",[39,30588,30590],{"className":30589,"translate":43},[746],[39,30591,30593,30677],{"className":30592,"translate":43},[42],[39,30594,30596],{"className":30595},[47],[49,30597,30598],{"xmlns":51,"display":755},[53,30599,30600,30674],{},[56,30601,30602,30604,30606,30612,30614,30620,30622,30624,30626,30628,30630,30636,30638,30640,30646,30648,30650,30652,30654,30656,30662,30664,30670,30672],{},[59,30603,29425],{},[133,30605,136],{"stretchy":135},[138,30607,30608,30610],{},[59,30609,1755],{},[133,30611,21884],{},[133,30613,155],{"separator":70},[138,30615,30616,30618],{},[59,30617,29625],{},[133,30619,21884],{},[133,30621,161],{"stretchy":135},[133,30623,793],{},[59,30625,29425],{},[133,30627,136],{"stretchy":135},[59,30629,11772],{},[138,30631,30632,30634],{},[59,30633,1755],{},[133,30635,21884],{},[133,30637,155],{"separator":70},[59,30639,11772],{},[138,30641,30642,30644],{},[59,30643,29625],{},[133,30645,21884],{},[133,30647,161],{"stretchy":135},[133,30649,1824],{},[59,30651,531],{},[59,30653,29425],{},[133,30655,136],{"stretchy":135},[138,30657,30658,30660],{},[59,30659,1755],{},[133,30661,21884],{},[133,30663,155],{"separator":70},[138,30665,30666,30668],{},[59,30667,29625],{},[133,30669,21884],{},[133,30671,161],{"stretchy":135},[59,30673,2903],{"mathvariant":145},[63,30675,30676],{"encoding":65},"d(x^*,y^*)=d(Tx^*,Ty^*)\\le\\gamma d(x^*,y^*).",[39,30678,30680,30768,30862],{"className":30679,"ariaHidden":70},[69],[39,30681,30683,30686,30689,30692,30721,30724,30727,30756,30759,30762,30765],{"className":30682},[74],[39,30684],{"className":30685,"style":245},[78],[39,30687,29425],{"className":30688},[83,84],[39,30690,136],{"className":30691},[182],[39,30693,30695,30698],{"className":30694},[83],[39,30696,1755],{"className":30697},[83,84],[39,30699,30701],{"className":30700},[192],[39,30702,30704],{"className":30703},[196],[39,30705,30707],{"className":30706},[200],[39,30708,30710],{"className":30709,"style":22242},[204],[39,30711,30712,30715],{"style":1027},[39,30713],{"className":30714,"style":213},[212],[39,30716,30718],{"className":30717},[217,218,219,220],[39,30719,21884],{"className":30720},[465,220],[39,30722,155],{"className":30723},[252],[39,30725],{"className":30726,"style":256},[230],[39,30728,30730,30733],{"className":30729},[83],[39,30731,29625],{"className":30732,"style":2114},[83,84],[39,30734,30736],{"className":30735},[192],[39,30737,30739],{"className":30738},[196],[39,30740,30742],{"className":30741},[200],[39,30743,30745],{"className":30744,"style":22242},[204],[39,30746,30747,30750],{"style":1027},[39,30748],{"className":30749,"style":213},[212],[39,30751,30753],{"className":30752},[217,218,219,220],[39,30754,21884],{"className":30755},[465,220],[39,30757,161],{"className":30758},[263],[39,30760],{"className":30761,"style":231},[230],[39,30763,793],{"className":30764},[235],[39,30766],{"className":30767,"style":231},[230],[39,30769,30771,30774,30777,30780,30783,30812,30815,30818,30821,30850,30853,30856,30859],{"className":30770},[74],[39,30772],{"className":30773,"style":245},[78],[39,30775,29425],{"className":30776},[83,84],[39,30778,136],{"className":30779},[182],[39,30781,11772],{"className":30782,"style":178},[83,84],[39,30784,30786,30789],{"className":30785},[83],[39,30787,1755],{"className":30788},[83,84],[39,30790,30792],{"className":30791},[192],[39,30793,30795],{"className":30794},[196],[39,30796,30798],{"className":30797},[200],[39,30799,30801],{"className":30800,"style":22242},[204],[39,30802,30803,30806],{"style":1027},[39,30804],{"className":30805,"style":213},[212],[39,30807,30809],{"className":30808},[217,218,219,220],[39,30810,21884],{"className":30811},[465,220],[39,30813,155],{"className":30814},[252],[39,30816],{"className":30817,"style":256},[230],[39,30819,11772],{"className":30820,"style":178},[83,84],[39,30822,30824,30827],{"className":30823},[83],[39,30825,29625],{"className":30826,"style":2114},[83,84],[39,30828,30830],{"className":30829},[192],[39,30831,30833],{"className":30832},[196],[39,30834,30836],{"className":30835},[200],[39,30837,30839],{"className":30838,"style":22242},[204],[39,30840,30841,30844],{"style":1027},[39,30842],{"className":30843,"style":213},[212],[39,30845,30847],{"className":30846},[217,218,219,220],[39,30848,21884],{"className":30849},[465,220],[39,30851,161],{"className":30852},[263],[39,30854],{"className":30855,"style":231},[230],[39,30857,1824],{"className":30858},[235],[39,30860],{"className":30861,"style":231},[230],[39,30863,30865,30868,30871,30874,30877,30906,30909,30912,30941,30944],{"className":30864},[74],[39,30866],{"className":30867,"style":245},[78],[39,30869,531],{"className":30870,"style":565},[83,84],[39,30872,29425],{"className":30873},[83,84],[39,30875,136],{"className":30876},[182],[39,30878,30880,30883],{"className":30879},[83],[39,30881,1755],{"className":30882},[83,84],[39,30884,30886],{"className":30885},[192],[39,30887,30889],{"className":30888},[196],[39,30890,30892],{"className":30891},[200],[39,30893,30895],{"className":30894,"style":22242},[204],[39,30896,30897,30900],{"style":1027},[39,30898],{"className":30899,"style":213},[212],[39,30901,30903],{"className":30902},[217,218,219,220],[39,30904,21884],{"className":30905},[465,220],[39,30907,155],{"className":30908},[252],[39,30910],{"className":30911,"style":256},[230],[39,30913,30915,30918],{"className":30914},[83],[39,30916,29625],{"className":30917,"style":2114},[83,84],[39,30919,30921],{"className":30920},[192],[39,30922,30924],{"className":30923},[196],[39,30925,30927],{"className":30926},[200],[39,30928,30930],{"className":30929,"style":22242},[204],[39,30931,30932,30935],{"style":1027},[39,30933],{"className":30934,"style":213},[212],[39,30936,30938],{"className":30937},[217,218,219,220],[39,30939,21884],{"className":30940},[465,220],[39,30942,161],{"className":30943},[263],[39,30945,2903],{"className":30946},[83],[11,30948,20571,30949,30999,31000,31139,31140,1607],{},[39,30950,30952,30969],{"className":30951,"translate":43},[42],[39,30953,30955],{"className":30954},[47],[49,30956,30957],{"xmlns":51},[53,30958,30959,30967],{},[56,30960,30961,30963,30965],{},[59,30962,531],{},[133,30964,1693],{},[539,30966,546],{},[63,30968,17296],{"encoding":65},[39,30970,30972,30990],{"className":30971,"ariaHidden":70},[69],[39,30973,30975,30978,30981,30984,30987],{"className":30974},[74],[39,30976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",[39,31001,31003,31039],{"className":31002,"translate":43},[42],[39,31004,31006],{"className":31005},[47],[49,31007,31008],{"xmlns":51},[53,31009,31010,31036],{},[56,31011,31012,31014,31016,31022,31024,31030,31032,31034],{},[59,31013,29425],{},[133,31015,136],{"stretchy":135},[138,31017,31018,31020],{},[59,31019,1755],{},[133,31021,21884],{},[133,31023,155],{"separator":70},[138,31025,31026,31028],{},[59,31027,29625],{},[133,31029,21884],{},[133,31031,161],{"stretchy":135},[133,31033,793],{},[539,31035,541],{},[63,31037,31038],{"encoding":65},"d(x^*,y^*)=0",[39,31040,31042,31130],{"className":31041,"ariaHidden":70},[69],[39,31043,31045,31048,31051,31054,31083,31086,31089,31118,31121,31124,31127],{"className":31044},[74],[39,31046],{"className":31047,"style":245},[78],[39,31049,29425],{"className":31050},[83,84],[39,31052,136],{"className":31053},[182],[39,31055,31057,31060],{"className":31056},[83],[39,31058,1755],{"className":31059},[83,84],[39,31061,31063],{"className":31062},[192],[39,31064,31066],{"className":31065},[196],[39,31067,31069],{"className":31068},[200],[39,31070,31072],{"className":31071,"style":21897},[204],[39,31073,31074,31077],{"style":208},[39,31075],{"className":31076,"style":213},[212],[39,31078,31080],{"className":31079},[217,218,219,220],[39,31081,21884],{"className":31082},[465,220],[39,31084,155],{"className":31085},[252],[39,31087],{"className":31088,"style":256},[230],[39,31090,31092,31095],{"className":31091},[83],[39,31093,29625],{"className":31094,"style":2114},[83,84],[39,31096,31098],{"className":31097},[192],[39,31099,31101],{"className":31100},[196],[39,31102,31104],{"className":31103},[200],[39,31105,31107],{"className":31106,"style":21897},[204],[39,31108,31109,31112],{"style":208},[39,31110],{"className":31111,"style":213},[212],[39,31113,31115],{"className":31114},[217,218,219,220],[39,31116,21884],{"className":31117},[465,220],[39,31119,161],{"className":31120},[263],[39,31122],{"className":31123,"style":231},[230],[39,31125,793],{"className":31126},[235],[39,31128],{"className":31129,"style":231},[230],[39,31131,31133,31136],{"className":31132},[74],[39,31134],{"className":31135,"style":2473},[78],[39,31137,541],{"className":31138},[83],"，即 ",[39,31141,31143,31169],{"className":31142,"translate":43},[42],[39,31144,31146],{"className":31145},[47],[49,31147,31148],{"xmlns":51},[53,31149,31150,31166],{},[56,31151,31152,31158,31160],{},[138,31153,31154,31156],{},[59,31155,1755],{},[133,31157,21884],{},[133,31159,793],{},[138,31161,31162,31164],{},[59,31163,29625],{},[133,31165,21884],{},[63,31167,31168],{"encoding":65},"x^*=y^*",[39,31170,31172,31216],{"className":31171,"ariaHidden":70},[69],[39,31173,31175,31178,31207,31210,31213],{"className":31174},[74],[39,31176],{"className":31177,"style":21897},[78],[39,31179,31181,31184],{"className":31180},[83],[39,31182,1755],{"className":31183},[83,84],[39,31185,31187],{"className":31186},[192],[39,31188,31190],{"className":31189},[196],[39,31191,31193],{"className":31192},[200],[39,31194,31196],{"className":31195,"style":21897},[204],[39,31197,31198,31201],{"style":208},[39,31199],{"className":31200,"style":213},[212],[39,31202,31204],{"className":31203},[217,218,219,220],[39,31205,21884],{"className":31206},[465,220],[39,31208],{"className":31209,"style":231},[230],[39,31211,793],{"className":31212},[235],[39,31214],{"className":31215,"style":231},[230],[39,31217,31219,31222],{"className":31218},[74],[39,31220],{"className":31221,"style":30556},[78],[39,31223,31225,31228],{"className":31224},[83],[39,31226,29625],{"className":31227,"style":2114},[83,84],[39,31229,31231],{"className":31230},[192],[39,31232,31234],{"className":31233},[196],[39,31235,31237],{"className":31236},[200],[39,31238,31240],{"className":31239,"style":21897},[204],[39,31241,31242,31245],{"style":208},[39,31243],{"className":31244,"style":213},[212],[39,31246,31248],{"className":31247},[217,218,219,220],[39,31249,21884],{"className":31250},[465,220],[11,31252,31253,31254,31345,31346,23615,31498,31550],{},"再证存在性。任取 ",[39,31255,31257,31278],{"className":31256,"translate":43},[42],[39,31258,31260],{"className":31259},[47],[49,31261,31262],{"xmlns":51},[53,31263,31264,31276],{},[56,31265,31266,31272,31274],{},[613,31267,31268,31270],{},[59,31269,1755],{},[539,31271,541],{},[133,31273,534],{},[59,31275,29420],{},[63,31277,29946],{"encoding":65},[39,31279,31281,31336],{"className":31280,"ariaHidden":70},[69],[39,31282,31284,31287,31327,31330,31333],{"className":31283},[74],[39,31285],{"className":31286,"style":29956},[78],[39,31288,31290,31293],{"className":31289},[83],[39,31291,1755],{"className":31292},[83,84],[39,31294,31296],{"className":31295},[192],[39,31297,31299,31319],{"className":31298},[196,649],[39,31300,31302,31316],{"className":31301},[200],[39,31303,31305],{"className":31304,"style":963},[204],[39,31306,31307,31310],{"style":966},[39,31308],{"className":31309,"style":213},[212],[39,31311,31313],{"className":31312},[217,218,219,220],[39,31314,541],{"className":31315},[83,220],[39,31317,673],{"className":31318},[672],[39,31320,31322],{"className":31321},[200],[39,31323,31325],{"className":31324,"style":680},[204],[39,31326],{},[39,31328],{"className":31329,"style":231},[230],[39,31331,534],{"className":31332},[235],[39,31334],{"className":31335,"style":231},[230],[39,31337,31339,31342],{"className":31338},[74],[39,31340],{"className":31341,"style":79},[78],[39,31343,29420],{"className":31344,"style":14420},[83,84],"，令 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是 Cauchy 列。由 ",[39,32646,32648,32661],{"className":32647,"translate":43},[42],[39,32649,32651],{"className":32650},[47],[49,32652,32653],{"xmlns":51},[53,32654,32655,32659],{},[56,32656,32657],{},[59,32658,29420],{},[63,32660,29420],{"encoding":65},[39,32662,32664],{"className":32663,"ariaHidden":70},[69],[39,32665,32667,32670],{"className":32666},[74],[39,32668],{"className":32669,"style":79},[78],[39,32671,29420],{"className":32672,"style":14420},[83,84]," 完备，存在 ",[39,32675,32677,32699],{"className":32676,"translate":43},[42],[39,32678,32680],{"className":32679},[47],[49,32681,32682],{"xmlns":51},[53,32683,32684,32696],{},[56,32685,32686,32692,32694],{},[138,32687,32688,32690],{},[59,32689,1755],{},[133,32691,21884],{},[133,32693,534],{},[59,32695,29420],{},[63,32697,32698],{"encoding":65},"x^*\\in 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x^*",[39,32786,32788,32843],{"className":32787,"ariaHidden":70},[69],[39,32789,32791,32794,32834,32837,32840],{"className":32790},[74],[39,32792],{"className":32793,"style":2916},[78],[39,32795,32797,32800],{"className":32796},[83],[39,32798,1755],{"className":32799},[83,84],[39,32801,32803],{"className":32802},[192],[39,32804,32806,32826],{"className":32805},[196,649],[39,32807,32809,32823],{"className":32808},[200],[39,32810,32812],{"className":32811,"style":1728},[204],[39,32813,32814,32817],{"style":966},[39,32815],{"className":32816,"style":213},[212],[39,32818,32820],{"className":32819},[217,218,219,220],[39,32821,12962],{"className":32822},[83,84,220],[39,32824,673],{"className":32825},[672],[39,32827,32829],{"className":32828},[200],[39,32830,32832],{"className":32831,"style":680},[204],[39,32833],{},[39,32835],{"className":32836,"style":231},[230],[39,32838,418],{"className":32839},[235],[39,32841],{"className":32842,"style":231},[230],[39,32844,32846,32849],{"className":32845},[74],[39,32847],{"className":32848,"style":21897},[78],[39,32850,32852,32855],{"className":32851},[83],[39,32853,1755],{"className":32854},[83,84],[39,32856,32858],{"className":32857},[192],[39,32859,32861],{"className":32860},[196],[39,32862,32864],{"className":32863},[200],[39,32865,32867],{"className":32866,"style":21897},[204],[39,32868,32869,32872],{"style":208},[39,32870],{"className":32871,"style":213},[212],[39,32873,32875],{"className":32874},[217,218,219,220],[39,32876,21884],{"className":32877},[465,220],"。算子 ",[39,32880,32882,32895],{"className":32881,"translate":43},[42],[39,32883,32885],{"className":32884},[47],[49,32886,32887],{"xmlns":51},[53,32888,32889,32893],{},[56,32890,32891],{},[59,32892,11772],{},[63,32894,11772],{"encoding":65},[39,32896,32898],{"className":32897,"ariaHidden":70},[69],[39,32899,32901,32904],{"className":32900},[74],[39,32902],{"className":32903,"style":79},[78],[39,32905,11772],{"className":32906,"style":178},[83,84]," 是 Lipschitz 连续的，故",[39,32909,32911],{"className":32910,"translate":43},[746],[39,32912,32914,33065],{"className":32913,"translate":43},[42],[39,32915,32917],{"className":32916},[47],[49,32918,32919],{"xmlns":51,"display":755},[53,32920,32921,33062],{},[759,32922,32923,33006],{"rowspacing":761,"columnalign":762,"columnspacing":146},[764,32924,32925,32939],{},[767,32926,32927],{},[770,32928,32929],{"scriptlevel":541,"displaystyle":70},[56,32930,32931,32933],{},[59,32932,11772],{},[138,32934,32935,32937],{},[59,32936,1755],{},[133,32938,21884],{},[767,32940,32941],{},[770,32942,32943],{"scriptlevel":541,"displaystyle":70},[56,32944,32945,32947,32949,32951,32980,32982,32998,33000],{},[56,32946],{},[133,32948,793],{},[59,32950,11772],{},[56,32952,32953,32955,32972,32978],{},[133,32954,136],{"fence":70},[6556,32956,32957,32964],{},[56,32958,32959,32962],{},[59,32960,32961],{},"lim",[133,32963,1690],{},[56,32965,32966,32968,32970],{},[59,32967,12962],{},[133,32969,418],{},[59,32971,1696],{"mathvariant":145},[613,32973,32974,32976],{},[59,32975,1755],{},[59,32977,12962],{},[133,32979,161],{"fence":70},[133,32981,793],{},[6556,32983,32984,32990],{},[56,32985,32986,32988],{},[59,32987,32961],{},[133,32989,1690],{},[56,32991,32992,32994,32996],{},[59,32993,12962],{},[133,32995,418],{},[59,32997,1696],{"mathvariant":145},[59,32999,11772],{},[613,33001,33002,33004],{},[59,33003,1755],{},[59,33005,12962],{},[764,33007,33008,33014],{},[767,33009,33010],{},[770,33011,33012],{"scriptlevel":541,"displaystyle":70},[56,33013],{},[767,33015,33016],{},[770,33017,33018],{"scriptlevel":541,"displaystyle":70},[56,33019,33020,33022,33024,33040,33052,33054,33060],{},[56,33021],{},[133,33023,793],{},[6556,33025,33026,33032],{},[56,33027,33028,33030],{},[59,33029,32961],{},[133,33031,1690],{},[56,33033,33034,33036,33038],{},[59,33035,12962],{},[133,33037,418],{},[59,33039,1696],{"mathvariant":145},[613,33041,33042,33044],{},[59,33043,1755],{},[56,33045,33046,33048,33050],{},[59,33047,12962],{},[133,33049,788],{},[539,33051,546],{},[133,33053,793],{},[138,33055,33056,33058],{},[59,33057,1755],{},[133,33059,21884],{},[59,33061,2903],{"mathvariant":145},[63,33063,33064],{"encoding":65},"\\begin{aligned}\nTx^*&=T\\left(\\lim_{n\\to\\infty}x_n\\right)\n=\\lim_{n\\to\\infty}Tx_n\\\\\n&=\\lim_{n\\to\\infty}x_{n+1}=x^*.\n\\end{aligned}",[39,33066,33068],{"className":33067,"ariaHidden":70},[69],[39,33069,33071,33075],{"className":33070},[74],[39,33072],{"className":33073,"style":33074},[78],"height:3.69em;vertical-align:-1.595em;",[39,33076,33078],{"className":33077},[83],[39,33079,33081,33157],{"className":33080},[759],[39,33082,33084],{"className":33083},[918],[39,33085,33087,33148],{"className":33086},[196,649],[39,33088,33090,33145],{"className":33089},[200],[39,33091,33094,33136],{"className":33092,"style":33093},[204],"height:2.095em;",[39,33095,33097,33101],{"style":33096},"top:-4.095em;",[39,33098],{"className":33099,"style":33100},[212],"height:3.15em;",[39,33102,33104,33107],{"className":33103},[83],[39,33105,11772],{"className":33106,"style":178},[83,84],[39,33108,33110,33113],{"className":33109},[83],[39,33111,1755],{"className":33112},[83,84],[39,33114,33116],{"className":33115},[192],[39,33117,33119],{"className":33118},[196],[39,33120,33122],{"className":33121},[200],[39,33123,33125],{"className":33124,"style":22242},[204],[39,33126,33127,33130],{"style":1027},[39,33128],{"className":33129,"style":213},[212],[39,33131,33133],{"className":33132},[217,218,219,220],[39,33134,21884],{"className":33135},[465,220],[39,33137,33139,33142],{"style":33138},"top:-2.255em;",[39,33140],{"className":33141,"style":33100},[212],[39,33143],{"className":33144},[83],[39,33146,673],{"className":33147},[672],[39,33149,33151],{"className":33150},[200],[39,33152,33155],{"className":33153,"style":33154},[204],"height:1.595em;",[39,33156],{},[39,33158,33160],{"className":33159},[1223],[39,33161,33163,33584],{"className":33162},[196,649],[39,33164,33166,33581],{"className":33165},[200],[39,33167,33169,33415],{"className":33168,"style":33093},[204],[39,33170,33171,33174],{"style":33096},[39,33172],{"className":33173,"style":33100},[212],[39,33175,33177,33180,33183,33186,33189,33192,33195,33307,33310,33313,33316,33369,33372,33375],{"className":33176},[83],[39,33178],{"className":33179},[83],[39,33181],{"className":33182,"style":231},[230],[39,33184,793],{"className":33185},[235],[39,33187],{"className":33188,"style":231},[230],[39,33190,11772],{"className":33191,"style":178},[83,84],[39,33193],{"className":33194,"style":256},[230],[39,33196,33198,33205,33258,33261,33301],{"className":33197},[2842],[39,33199,33201],{"className":33200,"style":4592},[182,4591],[39,33202,136],{"className":33203},[18520,33204],"size2",[39,33206,33208],{"className":33207},[1744,3294],[39,33209,33211,33250],{"className":33210},[196,649],[39,33212,33214,33247],{"className":33213},[200],[39,33215,33217,33237],{"className":33216,"style":28824},[204],[39,33218,33219,33222],{"style":22299},[39,33220],{"className":33221,"style":935},[212],[39,33223,33225],{"className":33224},[217,218,219,220],[39,33226,33228,33231,33234],{"className":33227},[83,220],[39,33229,12962],{"className":33230},[83,84,220],[39,33232,418],{"className":33233},[235,220],[39,33235,1696],{"className":33236},[83,220],[39,33238,33239,33242],{"style":21599},[39,33240],{"className":33241,"style":935},[212],[39,33243,33244],{},[39,33245,32961],{"className":33246},[1744],[39,33248,673],{"className":33249},[672],[39,33251,33253],{"className":33252},[200],[39,33254,33256],{"className":33255,"style":22331},[204],[39,33257],{},[39,33259],{"className":33260,"style":256},[230],[39,33262,33264,33267],{"className":33263},[83],[39,33265,1755],{"className":33266},[83,84],[39,33268,33270],{"className":33269},[192],[39,33271,33273,33293],{"className":33272},[196,649],[39,33274,33276,33290],{"className":33275},[200],[39,33277,33279],{"className":33278,"style":1728},[204],[39,33280,33281,33284],{"style":966},[39,33282],{"className":33283,"style":213},[212],[39,33285,33287],{"className":33286},[217,218,219,220],[39,33288,12962],{"className":33289},[83,84,220],[39,33291,673],{"className":33292},[672],[39,33294,33296],{"className":33295},[200],[39,33297,33299],{"className":33298,"style":680},[204],[39,33300],{},[39,33302,33304],{"className":33303,"style":4592},[263,4591],[39,33305,161],{"className":33306},[18520,33204],[39,33308],{"className":33309,"style":231},[230],[39,33311,793],{"className":33312},[235],[39,33314],{"className":33315,"style":231},[230],[39,33317,33319],{"className":33318},[1744,3294],[39,33320,33322,33361],{"className":33321},[196,649],[39,33323,33325,33358],{"className":33324},[200],[39,33326,33328,33348],{"className":33327,"style":28824},[204],[39,33329,33330,33333],{"style":22299},[39,33331],{"className":33332,"style":935},[212],[39,33334,33336],{"className":33335},[217,218,219,220],[39,33337,33339,33342,33345],{"className":33338},[83,220],[39,33340,12962],{"className":33341},[83,84,220],[39,33343,418],{"className":33344},[235,220],[39,33346,1696],{"className":33347},[83,220],[39,33349,33350,33353],{"style":21599},[39,33351],{"className":33352,"style":935},[212],[39,33354,33355],{},[39,33356,32961],{"className":33357},[1744],[39,33359,673],{"className":33360},[672],[39,33362,33364],{"className":33363},[200],[39,33365,33367],{"className":33366,"style":22331},[204],[39,33368],{},[39,33370],{"className":33371,"style":256},[230],[39,33373,11772],{"className":33374,"style":178},[83,84],[39,33376,33378,33381],{"className":33377},[83],[39,33379,1755],{"className":33380},[83,84],[39,33382,33384],{"className":33383},[192],[39,33385,33387,33407],{"className":33386},[196,649],[39,33388,33390,33404],{"className":33389},[200],[39,33391,33393],{"className":33392,"style":1728},[204],[39,33394,33395,33398],{"style":966},[39,33396],{"className":33397,"style":213},[212],[39,33399,33401],{"className":33400},[217,218,219,220],[39,33402,12962],{"className":33403},[83,84,220],[39,33405,673],{"className":33406},[672],[39,33408,33410],{"className":33409},[200],[39,33411,33413],{"className":33412,"style":680},[204],[39,33414],{},[39,33416,33417,33420],{"style":33138},[39,33418],{"className":33419,"style":33100},[212],[39,33421,33423,33426,33429,33432,33435,33488,33491,33540,33543,33546,33549,33578],{"className":33422},[83],[39,33424],{"className":33425},[83],[39,33427],{"className":33428,"style":231},[230],[39,33430,793],{"className":33431},[235],[39,33433],{"className":33434,"style":231},[230],[39,33436,33438],{"className":33437},[1744,3294],[39,33439,33441,33480],{"className":33440},[196,649],[39,33442,33444,33477],{"className":33443},[200],[39,33445,33447,33467],{"className":33446,"style":28824},[204],[39,33448,33449,33452],{"style":22299},[39,33450],{"className":33451,"style":935},[212],[39,33453,33455],{"className":33454},[217,218,219,220],[39,33456,33458,33461,33464],{"className":33457},[83,220],[39,33459,12962],{"className":33460},[83,84,220],[39,33462,418],{"className":33463},[235,220],[39,33465,1696],{"className":33466},[83,220],[39,33468,33469,33472],{"style":21599},[39,33470],{"className":33471,"style":935},[212],[39,33473,33474],{},[39,33475,32961],{"className":33476},[1744],[39,33478,673],{"className":33479},[672],[39,33481,33483],{"className":33482},[200],[39,33484,33486],{"className":33485,"style":22331},[204],[39,33487],{},[39,33489],{"className":33490,"style":256},[230],[39,33492,33494,33497],{"className":33493},[83],[39,33495,1755],{"className":33496},[83,84],[39,33498,33500],{"className":33499},[192],[39,33501,33503,33532],{"className":33502},[196,649],[39,33504,33506,33529],{"className":33505},[200],[39,33507,33509],{"className":33508,"style":963},[204],[39,33510,33511,33514],{"style":966},[39,33512],{"className":33513,"style":213},[212],[39,33515,33517],{"className":33516},[217,218,219,220],[39,33518,33520,33523,33526],{"className":33519},[83,220],[39,33521,12962],{"className":33522},[83,84,220],[39,33524,788],{"className":33525},[465,220],[39,33527,546],{"className":33528},[83,220],[39,33530,673],{"className":33531},[672],[39,33533,33535],{"className":33534},[200],[39,33536,33538],{"className":33537,"style":994},[204],[39,33539],{},[39,33541],{"className":33542,"style":231},[230],[39,33544,793],{"className":33545},[235],[39,33547],{"className":33548,"style":231},[230],[39,33550,33552,33555],{"className":33551},[83],[39,33553,1755],{"className":33554},[83,84],[39,33556,33558],{"className":33557},[192],[39,33559,33561],{"className":33560},[196],[39,33562,33564],{"className":33563},[200],[39,33565,33567],{"className":33566,"style":22242},[204],[39,33568,33569,33572],{"style":1027},[39,33570],{"className":33571,"style":213},[212],[39,33573,33575],{"className":33574},[217,218,219,220],[39,33576,21884],{"className":33577},[465,220],[39,33579,2903],{"className":33580},[83],[39,33582,673],{"className":33583},[672],[39,33585,33587],{"className":33586},[200],[39,33588,33590],{"className":33589,"style":33154},[204],[39,33591],{},[11,33593,33594],{},"最后，由压缩性直接得到",[39,33596,33598],{"className":33597,"translate":43},[746],[39,33599,33601,33697],{"className":33600,"translate":43},[42],[39,33602,33604],{"className":33603},[47],[49,33605,33606],{"xmlns":51,"display":755},[53,33607,33608,33694],{},[56,33609,33610,33612,33614,33620,33622,33628,33630,33632,33634,33636,33642,33648,33650,33656,33662,33664,33666,33672,33674,33676,33682,33684,33690,33692],{},[59,33611,29425],{},[133,33613,136],{"stretchy":135},[613,33615,33616,33618],{},[59,33617,1755],{},[59,33619,3196],{},[133,33621,155],{"separator":70},[138,33623,33624,33626],{},[59,33625,1755],{},[133,33627,21884],{},[133,33629,161],{"stretchy":135},[133,33631,793],{},[59,33633,29425],{},[133,33635,136],{"stretchy":135},[138,33637,33638,33640],{},[59,33639,11772],{},[59,33641,3196],{},[613,33643,33644,33646],{},[59,33645,1755],{},[539,33647,541],{},[133,33649,155],{"separator":70},[138,33651,33652,33654],{},[59,33653,11772],{},[59,33655,3196],{},[138,33657,33658,33660],{},[59,33659,1755],{},[133,33661,21884],{},[133,33663,161],{"stretchy":135},[133,33665,1824],{},[138,33667,33668,33670],{},[59,33669,531],{},[59,33671,3196],{},[59,33673,29425],{},[133,33675,136],{"stretchy":135},[613,33677,33678,33680],{},[59,33679,1755],{},[539,33681,541],{},[133,33683,155],{"separator":70},[138,33685,33686,33688],{},[59,33687,1755],{},[133,33689,21884],{},[133,33691,161],{"stretchy":135},[59,33693,2903],{"mathvariant":145},[63,33695,33696],{"encoding":65},"d(x_k,x^*)=d(T^kx_0,T^kx^*)\\le\\gamma^k d(x_0,x^*).",[39,33698,33700,33799,33956],{"className":33699,"ariaHidden":70},[69],[39,33701,33703,33706,33709,33712,33752,33755,33758,33787,33790,33793,33796],{"className":33702},[74],[39,33704],{"className":33705,"style":245},[78],[39,33707,29425],{"className":33708},[83,84],[39,33710,136],{"className":33711},[182],[39,33713,33715,33718],{"className":33714},[83],[39,33716,1755],{"className":33717},[83,84],[39,33719,33721],{"className":33720},[192],[39,33722,33724,33744],{"className":33723},[196,649],[39,33725,33727,33741],{"className":33726},[200],[39,33728,33730],{"className":33729,"style":3419},[204],[39,33731,33732,33735],{"style":966},[39,33733],{"className":33734,"style":213},[212],[39,33736,33738],{"className":33737},[217,218,219,220],[39,33739,3196],{"className":33740,"style":3321},[83,84,220],[39,33742,673],{"className":33743},[672],[39,33745,33747],{"className":33746},[200],[39,33748,33750],{"className":33749,"style":680},[204],[39,33751],{},[39,33753,155],{"className":33754},[252],[39,33756],{"className":33757,"style":256},[230],[39,33759,33761,33764],{"className":33760},[83],[39,33762,1755],{"className":33763},[83,84],[39,33765,33767],{"className":33766},[192],[39,33768,33770],{"className":33769},[196],[39,33771,33773],{"className":33772},[200],[39,33774,33776],{"className":33775,"style":22242},[204],[39,33777,33778,33781],{"style":1027},[39,33779],{"className":33780,"style":213},[212],[39,33782,33784],{"className":33783},[217,218,219,220],[39,33785,21884],{"className":33786},[465,220],[39,33788,161],{"className":33789},[263],[39,33791],{"className":33792,"style":231},[230],[39,33794,793],{"className":33795},[235],[39,33797],{"className":33798,"style":231},[230],[39,33800,33802,33805,33808,33811,33840,33880,33883,33886,33915,33944,33947,33950,33953],{"className":33801},[74],[39,33803],{"className":33804,"style":30346},[78],[39,33806,29425],{"className":33807},[83,84],[39,33809,136],{"className":33810},[182],[39,33812,33814,33817],{"className":33813},[83],[39,33815,11772],{"className":33816,"style":178},[83,84],[39,33818,33820],{"className":33819},[192],[39,33821,33823],{"className":33822},[196],[39,33824,33826],{"className":33825},[200],[39,33827,33829],{"className":33828,"style":3389},[204],[39,33830,33831,33834],{"style":1027},[39,33832],{"className":33833,"style":213},[212],[39,33835,33837],{"className":33836},[217,218,219,220],[39,33838,3196],{"className":33839,"style":3321},[83,84,220],[39,33841,33843,33846],{"className":33842},[83],[39,33844,1755],{"className":33845},[83,84],[39,33847,33849],{"className":33848},[192],[39,33850,33852,33872],{"className":33851},[196,649],[39,33853,33855,33869],{"className":33854},[200],[39,33856,33858],{"className":33857,"style":963},[204],[39,33859,33860,33863],{"style":966},[39,33861],{"className":33862,"style":213},[212],[39,33864,33866],{"className":33865},[217,218,219,220],[39,33867,541],{"className":33868},[83,220],[39,33870,673],{"className":33871},[672],[39,33873,33875],{"className":33874},[200],[39,33876,33878],{"className":33877,"style":680},[204],[39,33879],{},[39,33881,155],{"className":33882},[252],[39,33884],{"className":33885,"style":256},[230],[39,33887,33889,33892],{"className":33888},[83],[39,33890,11772],{"className":33891,"style":178},[83,84],[39,33893,33895],{"className":33894},[192],[39,33896,33898],{"className":33897},[196],[39,33899,33901],{"className":33900},[200],[39,33902,33904],{"className":33903,"style":3389},[204],[39,33905,33906,33909],{"style":1027},[39,33907],{"className":33908,"style":213},[212],[39,33910,33912],{"className":33911},[217,218,219,220],[39,33913,3196],{"className":33914,"style":3321},[83,84,220],[39,33916,33918,33921],{"className":33917},[83],[39,33919,1755],{"className":33920},[83,84],[39,33922,33924],{"className":33923},[192],[39,33925,33927],{"className":33926},[196],[39,33928,33930],{"className":33929},[200],[39,33931,33933],{"className":33932,"style":22242},[204],[39,33934,33935,33938],{"style":1027},[39,33936],{"className":33937,"style":213},[212],[39,33939,33941],{"className":33940},[217,218,219,220],[39,33942,21884],{"className":33943},[465,220],[39,33945,161],{"className":33946},[263],[39,33948],{"className":33949,"style":231},[230],[39,33951,1824],{"className":33952},[235],[39,33954],{"className":33955,"style":231},[230],[39,33957,33959,33962,33991,33994,33997,34037,34040,34043,34072,34075],{"className":33958},[74],[39,33960],{"className":33961,"style":30346},[78],[39,33963,33965,33968],{"className":33964},[83],[39,33966,531],{"className":33967,"style":565},[83,84],[39,33969,33971],{"className":33970},[192],[39,33972,33974],{"className":33973},[196],[39,33975,33977],{"className":33976},[200],[39,33978,33980],{"className":33979,"style":3389},[204],[39,33981,33982,33985],{"style":1027},[39,33983],{"className":33984,"style":213},[212],[39,33986,33988],{"className":33987},[217,218,219,220],[39,33989,3196],{"className":33990,"style":3321},[83,84,220],[39,33992,29425],{"className":33993},[83,84],[39,33995,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唯一性与上下界",[31,34085,34088],{"id":34086,"title":34087},"corollary-unique-fixed-point","推论 3.4（不动点的存在唯一性）",[11,34089,37,34090,14627,34162,1639,34220,29832,34278,34321],{},[39,34091,34093,34114],{"className":34092,"translate":43},[42],[39,34094,34096],{"className":34095},[47],[49,34097,34098],{"xmlns":51},[53,34099,34100,34112],{},[56,34101,34102,34104,34106,34108,34110],{},[539,34103,541],{},[133,34105,1824],{},[59,34107,531],{},[133,34109,1693],{},[539,34111,546],{},[63,34113,14577],{"encoding":65},[39,34115,34117,34135,34153],{"className":34116,"ariaHidden":70},[69],[39,34118,34120,34123,34126,34129,34132],{"className":34119},[74],[39,34121],{"className":34122,"style":14587},[78],[39,34124,541],{"className":34125},[83],[39,34127],{"className":34128,"style":231},[230],[39,34130,1824],{"className":34131},[235],[39,34133],{"className":34134,"style":231},[230],[39,34136,34138,34141,34144,34147,34150],{"className":34137},[74],[39,34139],{"className":34140,"style":561},[78],[39,34142,531],{"className":34143,"style":565},[83,84],[39,34145],{"className":34146,"style":231},[230],[39,34148,1693],{"className":34149},[235],[39,34151],{"className":34152,"style":231},[230],[39,34154,34156,34159],{"className":34155},[74],[39,34157],{"className":34158,"style":2473},[78],[39,34160,546],{"className":34161},[83],[39,34163,34165,34182],{"className":34164,"translate":43},[42],[39,34166,34168],{"className":34167},[47],[49,34169,34170],{"xmlns":51},[53,34171,34172,34180],{},[56,34173,34174],{},[138,34175,34176,34178],{},[59,34177,11772],{"mathvariant":617},[59,34179,2098],{},[63,34181,11777],{"encoding":65},[39,34183,34185],{"className":34184,"ariaHidden":70},[69],[39,34186,34188,34191],{"className":34187},[74],[39,34189],{"className":34190,"style":79},[78],[39,34192,34194,34197],{"className":34193},[83],[39,34195,11772],{"className":34196,"style":11793},[83,641],[39,34198,34200],{"className":34199},[192],[39,34201,34203],{"className":34202},[196],[39,34204,34206],{"className":34205},[200],[39,34207,34209],{"className":34208,"style":6410},[204],[39,34210,34211,34214],{"style":208},[39,34212],{"className":34213,"style":213},[212],[39,34215,34217],{"className":34216},[217,218,219,220],[39,34218,2098],{"className":34219,"style":2114},[83,84,220],[39,34221,34223,34240],{"className":34222,"translate":43},[42],[39,34224,34226],{"className":34225},[47],[49,34227,34228],{"xmlns":51},[53,34229,34230,34238],{},[56,34231,34232],{},[138,34233,34234,34236],{},[59,34235,11772],{"mathvariant":617},[133,34237,21884],{},[63,34239,21887],{"encoding":65},[39,34241,34243],{"className":34242,"ariaHidden":70},[69],[39,34244,34246,34249],{"className":34245},[74],[39,34247],{"className":34248,"style":21897},[78],[39,34250,34252,34255],{"className":34251},[83],[39,34253,11772],{"className":34254,"style":11793},[83,641],[39,34256,34258],{"className":34257},[192],[39,34259,34261],{"className":34260},[196],[39,34262,34264],{"className":34263},[200],[39,34265,34267],{"className":34266,"style":21897},[204],[39,34268,34269,34272],{"style":208},[39,34270],{"className":34271,"style":213},[212],[39,34273,34275],{"className":34274},[217,218,219,220],[39,34276,21884],{"className":34277},[465,220],[39,34279,34281,34300],{"className":34280,"translate":43},[42],[39,34282,34284],{"className":34283},[47],[49,34285,34286],{"xmlns":51},[53,34287,34288,34298],{},[56,34289,34290,34292,34294,34296],{},[59,34291,19772],{},[133,34293,136],{"stretchy":135},[59,34295,61],{},[133,34297,161],{"stretchy":135},[63,34299,21380],{"encoding":65},[39,34301,34303],{"className":34302,"ariaHidden":70},[69],[39,34304,34306,34309,34312,34315,34318],{"className":34305},[74],[39,34307],{"className":34308,"style":245},[78],[39,34310,19772],{"className":34311,"style":19812},[83,84],[39,34313,136],{"className":34314},[182],[39,34316,61],{"className":34317,"style":85},[83,84],[39,34319,161],{"className":34320},[263]," 中各自存在唯一不动点。",[11,34323,34324,2083,34326,34369,34370,1639,34428,26827,34486,34514,34515,34519],{},[1489,34325,5595],{},[39,34327,34329,34348],{"className":34328,"translate":43},[42],[39,34330,34332],{"className":34331},[47],[49,34333,34334],{"xmlns":51},[53,34335,34336,34346],{},[56,34337,34338,34340,34342,34344],{},[59,34339,19772],{},[133,34341,136],{"stretchy":135},[59,34343,61],{},[133,34345,161],{"stretchy":135},[63,34347,21380],{"encoding":65},[39,34349,34351],{"className":34350,"ariaHidden":70},[69],[39,34352,34354,34357,34360,34363,34366],{"className":34353},[74],[39,34355],{"className":34356,"style":245},[78],[39,34358,19772],{"className":34359,"style":19812},[83,84],[39,34361,136],{"className":34362},[182],[39,34364,61],{"className":34365,"style":85},[83,84],[39,34367,161],{"className":34368},[263]," 在 sup 范数下完备，",[39,34371,34373,34390],{"className":34372,"translate":43},[42],[39,34374,34376],{"className":34375},[47],[49,34377,34378],{"xmlns":51},[53,34379,34380,34388],{},[56,34381,34382],{},[138,34383,34384,34386],{},[59,34385,11772],{"mathvariant":617},[59,34387,2098],{},[63,34389,11777],{"encoding":65},[39,34391,34393],{"className":34392,"ariaHidden":70},[69],[39,34394,34396,34399],{"className":34395},[74],[39,34397],{"className":34398,"style":79},[78],[39,34400,34402,34405],{"className":34401},[83],[39,34403,11772],{"className":34404,"style":11793},[83,641],[39,34406,34408],{"className":34407},[192],[39,34409,34411],{"className":34410},[196],[39,34412,34414],{"className":34413},[200],[39,34415,34417],{"className":34416,"style":6410},[204],[39,34418,34419,34422],{"style":208},[39,34420],{"className":34421,"style":213},[212],[39,34423,34425],{"className":34424},[217,218,219,220],[39,34426,2098],{"className":34427,"style":2114},[83,84,220],[39,34429,34431,34448],{"className":34430,"translate":43},[42],[39,34432,34434],{"className":34433},[47],[49,34435,34436],{"xmlns":51},[53,34437,34438,34446],{},[56,34439,34440],{},[138,34441,34442,34444],{},[59,34443,11772],{"mathvariant":617},[133,34445,21884],{},[63,34447,21887],{"encoding":65},[39,34449,34451],{"className":34450,"ariaHidden":70},[69],[39,34452,34454,34457],{"className":34453},[74],[39,34455],{"className":34456,"style":21897},[78],[39,34458,34460,34463],{"className":34459},[83],[39,34461,11772],{"className":34462,"style":11793},[83,641],[39,34464,34466],{"className":34465},[192],[39,34467,34469],{"className":34468},[196],[39,34470,34472],{"className":34471},[200],[39,34473,34475],{"className":34474,"style":21897},[204],[39,34476,34477,34480],{"style":208},[39,34478],{"className":34479,"style":213},[212],[39,34481,34483],{"className":34482},[217,218,219,220],[39,34484,21884],{"className":34485},[465,220],[39,34487,34489,34502],{"className":34488,"translate":43},[42],[39,34490,34492],{"className":34491},[47],[49,34493,34494],{"xmlns":51},[53,34495,34496,34500],{},[56,34497,34498],{},[59,34499,531],{},[63,34501,26586],{"encoding":65},[39,34503,34505],{"className":34504,"ariaHidden":70},[69],[39,34506,34508,34511],{"className":34507},[74],[39,34509],{"className":34510,"style":2654},[78],[39,34512,531],{"className":34513,"style":565},[83,84],"-压缩映射。由 ",[158,34516,34518],{"href":34517},"#theorem-banach","定理 3.3"," 即得。证毕。",[11,34521,26443,34522,742,34580,34582,34583,23615,34641,34699,34700,34759],{},[39,34523,34525,34542],{"className":34524,"translate":43},[42],[39,34526,34528],{"className":34527},[47],[49,34529,34530],{"xmlns":51},[53,34531,34532,34540],{},[56,34533,34534],{},[138,34535,34536,34538],{},[59,34537,11772],{"mathvariant":617},[59,34539,2098],{},[63,34541,11777],{"encoding":65},[39,34543,34545],{"className":34544,"ariaHidden":70},[69],[39,34546,34548,34551],{"className":34547},[74],[39,34549],{"className":34550,"style":79},[78],[39,34552,34554,34557],{"className":34553},[83],[39,34555,11772],{"className":34556,"style":11793},[83,641],[39,34558,34560],{"className":34559},[192],[39,34561,34563],{"className":34562},[196],[39,34564,34566],{"className":34565},[200],[39,34567,34569],{"className":34568,"style":6410},[204],[39,34570,34571,34574],{"style":208},[39,34572],{"className":34573,"style":213},[212],[39,34575,34577],{"className":34576},[217,218,219,220],[39,34578,2098],{"className":34579,"style":2114},[83,84,220],[158,34581,21294],{"href":21293}," 已经指出这个唯一不动点就是 ",[39,34584,34586,34603],{"className":34585,"translate":43},[42],[39,34587,34589],{"className":34588},[47],[49,34590,34591],{"xmlns":51},[53,34592,34593,34601],{},[56,34594,34595],{},[138,34596,34597,34599],{},[59,34598,4230],{},[59,34600,2098],{},[63,34602,6382],{"encoding":65},[39,34604,34606],{"className":34605,"ariaHidden":70},[69],[39,34607,34609,34612],{"className":34608},[74],[39,34610],{"className":34611,"style":79},[78],[39,34613,34615,34618],{"className":34614},[83],[39,34616,4230],{"className":34617,"style":461},[83,84],[39,34619,34621],{"className":34620},[192],[39,34622,34624],{"className":34623},[196],[39,34625,34627],{"className":34626},[200],[39,34628,34630],{"className":34629,"style":6410},[204],[39,34631,34632,34635],{"style":208},[39,34633],{"className":34634,"style":213},[212],[39,34636,34638],{"className":34637},[217,218,219,220],[39,34639,2098],{"className":34640,"style":2114},[83,84,220],[39,34642,34644,34661],{"className":34643,"translate":43},[42],[39,34645,34647],{"className":34646},[47],[49,34648,34649],{"xmlns":51},[53,34650,34651,34659],{},[56,34652,34653],{},[138,34654,34655,34657],{},[59,34656,11772],{"mathvariant":617},[133,34658,21884],{},[63,34660,21887],{"encoding":65},[39,34662,34664],{"className":34663,"ariaHidden":70},[69],[39,34665,34667,34670],{"className":34666},[74],[39,34668],{"className":34669,"style":21897},[78],[39,34671,34673,34676],{"className":34672},[83],[39,34674,11772],{"className":34675,"style":11793},[83,641],[39,34677,34679],{"className":34678},[192],[39,34680,34682],{"className":34681},[196],[39,34683,34685],{"className":34684},[200],[39,34686,34688],{"className":34687,"style":21897},[204],[39,34689,34690,34693],{"style":208},[39,34691],{"className":34692,"style":213},[212],[39,34694,34696],{"className":34695},[217,218,219,220],[39,34697,21884],{"className":34698},[465,220],"，第 4 节将证明它的唯一不动点就是最优值函数 ",[39,34701,34703,34721],{"className":34702,"translate":43},[42],[39,34704,34706],{"className":34705},[47],[49,34707,34708],{"xmlns":51},[53,34709,34710,34718],{},[56,34711,34712],{},[138,34713,34714,34716],{},[59,34715,4230],{},[133,34717,21884],{},[63,34719,34720],{"encoding":65},"V^*",[39,34722,34724],{"className":34723,"ariaHidden":70},[69],[39,34725,34727,34730],{"className":34726},[74],[39,34728],{"className":34729,"style":21897},[78],[39,34731,34733,34736],{"className":34732},[83],[39,34734,4230],{"className":34735,"style":461},[83,84],[39,34737,34739],{"className":34738},[192],[39,34740,34742],{"className":34741},[196],[39,34743,34745],{"className":34744},[200],[39,34746,34748],{"className":34747,"style":21897},[204],[39,34749,34750,34753],{"style":208},[39,34751],{"className":34752,"style":213},[212],[39,34754,34756],{"className":34755},[217,218,219,220],[39,34757,21884],{"className":34758},[465,220],"。因此，「贝尔曼方程有唯一解」并不是一个额外的假设，而是压缩性的直接推论。",[11,34761,34762],{},"单调性还给出了一个在理论上很有用的上下界判据。",[31,34764,34767],{"id":34765,"title":34766},"corollary-order-bounds","推论 3.5（上下界判据）",[11,34768,37,34769,21402,34798,34856,34857,34915,34916,34986,34987,35155,35156,34986,35301,35155,35463,1607],{},[39,34770,34772,34786],{"className":34771,"translate":43},[42],[39,34773,34775],{"className":34774},[47],[49,34776,34777],{"xmlns":51},[53,34778,34779,34783],{},[56,34780,34781],{},[59,34782,11772],{"mathvariant":617},[63,34784,34785],{"encoding":65},"\\mathcal{T}",[39,34787,34789],{"className":34788,"ariaHidden":70},[69],[39,34790,34792,34795],{"className":34791},[74],[39,34793],{"className":34794,"style":79},[78],[39,34796,11772],{"className":34797,"style":11793},[83,641],[39,34799,34801,34818],{"className":34800,"translate":43},[42],[39,34802,34804],{"className":34803},[47],[49,34805,34806],{"xmlns":51},[53,34807,34808,34816],{},[56,34809,34810],{},[138,34811,34812,34814],{},[59,34813,11772],{"mathvariant":617},[59,34815,2098],{},[63,34817,11777],{"encoding":65},[39,34819,34821],{"className":34820,"ariaHidden":70},[69],[39,34822,34824,34827],{"className":34823},[74],[39,34825],{"className":34826,"style":79},[78],[39,34828,34830,34833],{"className":34829},[83],[39,34831,11772],{"className":34832,"style":11793},[83,641],[39,34834,34836],{"className":34835},[192],[39,34837,34839],{"className":34838},[196],[39,34840,34842],{"className":34841},[200],[39,34843,34845],{"className":34844,"style":6410},[204],[39,34846,34847,34850],{"style":208},[39,34848],{"className":34849,"style":213},[212],[39,34851,34853],{"className":34852},[217,218,219,220],[39,34854,2098],{"className":34855,"style":2114},[83,84,220]," 或 ",[39,34858,34860,34877],{"className":34859,"translate":43},[42],[39,34861,34863],{"className":34862},[47],[49,34864,34865],{"xmlns":51},[53,34866,34867,34875],{},[56,34868,34869],{},[138,34870,34871,34873],{},[59,34872,11772],{"mathvariant":617},[133,34874,21884],{},[63,34876,21887],{"encoding":65},[39,34878,34880],{"className":34879,"ariaHidden":70},[69],[39,34881,34883,34886],{"className":34882},[74],[39,34884],{"className":34885,"style":21897},[78],[39,34887,34889,34892],{"className":34888},[83],[39,34890,11772],{"className":34891,"style":11793},[83,641],[39,34893,34895],{"className":34894},[192],[39,34896,34898],{"className":34897},[196],[39,34899,34901],{"className":34900},[200],[39,34902,34904],{"className":34903,"style":21897},[204],[39,34905,34906,34909],{"style":208},[39,34907],{"className":34908,"style":213},[212],[39,34910,34912],{"className":34911},[217,218,219,220],[39,34913,21884],{"className":34914},[465,220],"，其唯一不动点为 ",[39,34917,34919,34937],{"className":34918,"translate":43},[42],[39,34920,34922],{"className":34921},[47],[49,34923,34924],{"xmlns":51},[53,34925,34926,34934],{},[56,34927,34928],{},[613,34929,34930,34932],{},[59,34931,4230],{},[59,34933,1696],{"mathvariant":145},[63,34935,34936],{"encoding":65},"V_\\infty",[39,34938,34940],{"className":34939,"ariaHidden":70},[69],[39,34941,34943,34946],{"className":34942},[74],[39,34944],{"className":34945,"style":634},[78],[39,34947,34949,34952],{"className":34948},[83],[39,34950,4230],{"className":34951,"style":461},[83,84],[39,34953,34955],{"className":34954},[192],[39,34956,34958,34978],{"className":34957},[196,649],[39,34959,34961,34975],{"className":34960},[200],[39,34962,34964],{"className":34963,"style":1728},[204],[39,34965,34966,34969],{"style":13607},[39,34967],{"className":34968,"style":213},[212],[39,34970,34972],{"className":34971},[217,218,219,220],[39,34973,1696],{"className":34974},[83,220],[39,34976,673],{"className":34977},[672],[39,34979,34981],{"className":34980},[200],[39,34982,34984],{"className":34983,"style":680},[204],[39,34985],{},"。若 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V_\\infty",[39,36132,36134,36224],{"className":36133,"ariaHidden":70},[69],[39,36135,36137,36140,36169,36215,36218,36221],{"className":36136},[74],[39,36138],{"className":36139,"style":634},[78],[39,36141,36143,36146],{"className":36142},[83],[39,36144,11772],{"className":36145,"style":11793},[83,641],[39,36147,36149],{"className":36148},[192],[39,36150,36152],{"className":36151},[196],[39,36153,36155],{"className":36154},[200],[39,36156,36158],{"className":36157,"style":6410},[204],[39,36159,36160,36163],{"style":208},[39,36161],{"className":36162,"style":213},[212],[39,36164,36166],{"className":36165},[217,218,219,220],[39,36167,12962],{"className":36168},[83,84,220],[39,36170,36172,36175],{"className":36171},[83],[39,36173,4230],{"className":36174,"style":461},[83,84],[39,36176,36178],{"className":36177},[192],[39,36179,36181,36207],{"className":36180},[196,649],[39,36182,36184,36204],{"className":36183},[200],[39,36185,36187],{"className":36186,"style":3419},[204],[39,36188,36189,36192],{"style":13607},[39,36190],{"className":36191,"style":213},[212],[39,36193,36195],{"className":36194},[217,218,219,220],[39,36196,36198],{"className":36197},[83,220],[39,36199,36201],{"className":36200},[83,220],[39,36202,35079],{"className":36203,"style":35078},[83,35077,220],[39,36205,673],{"className":36206},[672],[39,36208,36210],{"className":36209},[200],[39,36211,36213],{"className":36212,"style":680},[204],[39,36214],{},[39,36216],{"className":36217,"style":231},[230],[39,36219,418],{"className":36220},[235],[39,36222],{"className":36223,"style":231},[230],[39,36225,36227,36230],{"className":36226},[74],[39,36228],{"className":36229,"style":634},[78],[39,36231,36233,36236],{"className":36232},[83],[39,36234,4230],{"className":36235,"style":461},[83,84],[39,36237,36239],{"className":36238},[192],[39,36240,36242,36262],{"className":36241},[196,649],[39,36243,36245,36259],{"className":36244},[200],[39,36246,36248],{"className":36247,"style":1728},[204],[39,36249,36250,36253],{"style":13607},[39,36251],{"className":36252,"style":213},[212],[39,36254,36256],{"className":36255},[217,218,219,220],[39,36257,1696],{"className":36258},[83,220],[39,36260,673],{"className":36261},[672],[39,36263,36265],{"className":36264},[200],[39,36266,36268],{"className":36267,"style":680},[204],[39,36269],{},"，故 ",[39,36272,36274,36305],{"className":36273,"translate":43},[42],[39,36275,36277],{"className":36276},[47],[49,36278,36279],{"xmlns":51},[53,36280,36281,36303],{},[56,36282,36283,36295,36297],{},[613,36284,36285,36287],{},[59,36286,4230],{},[56,36288,36289,36291,36293],{},[59,36290,35007],{"mathvariant":145},[59,36292,35010],{"mathvariant":145},[59,36294,35013],{"mathvariant":145},[133,36296,1824],{},[613,36298,36299,36301],{},[59,36300,4230],{},[59,36302,1696],{"mathvariant":145},[63,36304,35190],{"encoding":65},[39,36306,36308,36369],{"className":36307,"ariaHidden":70},[69],[39,36309,36311,36314,36360,36363,36366],{"className":36310},[74],[39,36312],{"className":36313,"style":634},[78],[39,36315,36317,36320],{"className":36316},[83],[39,36318,4230],{"className":36319,"style":461},[83,84],[39,36321,36323],{"className":36322},[192],[39,36324,36326,36352],{"className":36325},[196,649],[39,36327,36329,36349],{"className":36328},[200],[39,36330,36332],{"className":36331,"style":3419},[204],[39,36333,36334,36337],{"style":13607},[39,36335],{"className":36336,"style":213},[212],[39,36338,36340],{"className":36339},[217,218,219,220],[39,36341,36343],{"className":36342},[83,220],[39,36344,36346],{"className":36345},[83,220],[39,36347,35079],{"className":36348,"style":35078},[83,35077,220],[39,36350,673],{"className":36351},[672],[39,36353,36355],{"className":36354},[200],[39,36356,36358],{"className":36357,"style":680},[204],[39,36359],{},[39,36361],{"className":36362,"style":231},[230],[39,36364,1824],{"className":36365},[235],[39,36367],{"className":36368,"style":231},[230],[39,36370,36372,36375],{"className":36371},[74],[39,36373],{"className":36374,"style":634},[78],[39,36376,36378,36381],{"className":36377},[83],[39,36379,4230],{"className":36380,"style":461},[83,84],[39,36382,36384],{"className":36383},[192],[39,36385,36387,36407],{"className":36386},[196,649],[39,36388,36390,36404],{"className":36389},[200],[39,36391,36393],{"className":36392,"style":1728},[204],[39,36394,36395,36398],{"style":13607},[39,36396],{"className":36397,"style":213},[212],[39,36399,36401],{"className":36400},[217,218,219,220],[39,36402,1696],{"className":36403},[83,220],[39,36405,673],{"className":36406},[672],[39,36408,36410],{"className":36409},[200],[39,36411,36413],{"className":36412,"style":680},[204],[39,36414],{},"。上界情形同理，只是迭代序列单调下降。证毕。",[11,36417,36418,36419,36475,36476,36532],{},"这个判据说明，只要能够构造一个满足 ",[39,36420,36422,36442],{"className":36421,"translate":43},[42],[39,36423,36425],{"className":36424},[47],[49,36426,36427],{"xmlns":51},[53,36428,36429,36439],{},[56,36430,36431,36433,36435,36437],{},[59,36432,4230],{},[133,36434,1824],{},[59,36436,11772],{"mathvariant":617},[59,36438,4230],{},[63,36440,36441],{"encoding":65},"V\\le\\mathcal{T}V",[39,36443,36445,36463],{"className":36444,"ariaHidden":70},[69],[39,36446,36448,36451,36454,36457,36460],{"className":36447},[74],[39,36449],{"className":36450,"style":23796},[78],[39,36452,4230],{"className":36453,"style":461},[83,84],[39,36455],{"className":36456,"style":231},[230],[39,36458,1824],{"className":36459},[235],[39,36461],{"className":36462,"style":231},[230],[39,36464,36466,36469,36472],{"className":36465},[74],[39,36467],{"className":36468,"style":79},[78],[39,36470,11772],{"className":36471,"style":11793},[83,641],[39,36473,4230],{"className":36474,"style":461},[83,84]," 的下界函数，它就一直位于真实值函数之下。满足 ",[39,36477,36479,36499],{"className":36478,"translate":43},[42],[39,36480,36482],{"className":36481},[47],[49,36483,36484],{"xmlns":51},[53,36485,36486,36496],{},[56,36487,36488,36490,36492,36494],{},[59,36489,4230],{},[133,36491,25173],{},[59,36493,11772],{"mathvariant":617},[59,36495,4230],{},[63,36497,36498],{"encoding":65},"V\\ge\\mathcal{T}V",[39,36500,36502,36520],{"className":36501,"ariaHidden":70},[69],[39,36503,36505,36508,36511,36514,36517],{"className":36504},[74],[39,36506],{"className":36507,"style":23796},[78],[39,36509,4230],{"className":36510,"style":461},[83,84],[39,36512],{"className":36513,"style":231},[230],[39,36515,25173],{"className":36516},[235],[39,36518],{"className":36519,"style":231},[230],[39,36521,36523,36526,36529],{"className":36522},[74],[39,36524],{"className":36525,"style":79},[78],[39,36527,11772],{"className":36528,"style":11793},[83,641],[39,36530,4230],{"className":36531,"style":461},[83,84]," 的上界函数则一直位于真实值函数之上。这为值函数的区间估计提供了严格的数学基础。",[21,36534,36536],{"id":36535},"四贝尔曼最优方程","四、贝尔曼最优方程",[26,36538,36540],{"id":36539},"_41-最优值函数","4.1 最优值函数",[31,36542,36545,36548,36976],{"id":36543,"title":36544},"definition-optimal-value","定义 4.1（最优值函数）",[11,36546,36547],{},"最优状态值函数与最优动作值函数分别定义为",[39,36549,36551],{"className":36550,"translate":43},[746],[39,36552,36554,36652],{"className":36553,"translate":43},[42],[39,36555,36557],{"className":36556},[47],[49,36558,36559],{"xmlns":51,"display":755},[53,36560,36561,36649],{},[56,36562,36563,36569,36571,36573,36575,36577,36587,36593,36595,36597,36599,36601,36603,36609,36611,36613,36615,36617,36619,36621,36631,36637,36639,36641,36643,36645,36647],{},[138,36564,36565,36567],{},[59,36566,4230],{},[133,36568,21884],{},[133,36570,136],{"stretchy":135},[59,36572,142],{},[133,36574,161],{"stretchy":135},[133,36576,793],{},[6556,36578,36579,36585],{},[56,36580,36581,36583],{},[59,36582,21465],{},[133,36584,1690],{},[59,36586,2098],{},[138,36588,36589,36591],{},[59,36590,4230],{},[59,36592,2098],{},[133,36594,136],{"stretchy":135},[59,36596,142],{},[133,36598,161],{"stretchy":135},[133,36600,155],{"separator":70},[230,36602],{"width":10251},[138,36604,36605,36607],{},[59,36606,4295],{},[133,36608,21884],{},[133,36610,136],{"stretchy":135},[59,36612,142],{},[133,36614,155],{"separator":70},[59,36616,158],{},[133,36618,161],{"stretchy":135},[133,36620,793],{},[6556,36622,36623,36629],{},[56,36624,36625,36627],{},[59,36626,21465],{},[133,36628,1690],{},[59,36630,2098],{},[138,36632,36633,36635],{},[59,36634,4295],{},[59,36636,2098],{},[133,36638,136],{"stretchy":135},[59,36640,142],{},[133,36642,155],{"separator":70},[59,36644,158],{},[133,36646,161],{"stretchy":135},[59,36648,2903],{"mathvariant":145},[63,36650,36651],{"encoding":65},"V^*(s)=\\sup_{\\pi}V^\\pi(s),\n\\qquad\nQ^*(s,a)=\\sup_{\\pi}Q^\\pi(s,a).",[39,36653,36655,36708,36870],{"className":36654,"ariaHidden":70},[69],[39,36656,36658,36661,36690,36693,36696,36699,36702,36705],{"className":36657},[74],[39,36659],{"className":36660,"style":245},[78],[39,36662,36664,36667],{"className":36663},[83],[39,36665,4230],{"className":36666,"style":461},[83,84],[39,36668,36670],{"className":36669},[192],[39,36671,36673],{"className":36672},[196],[39,36674,36676],{"className":36675},[200],[39,36677,36679],{"className":36678,"style":22242},[204],[39,36680,36681,36684],{"style":1027},[39,36682],{"className":36683,"style":213},[212],[39,36685,36687],{"className":36686},[217,218,219,220],[39,36688,21884],{"className":36689},[465,220],[39,36691,136],{"className":36692},[182],[39,36694,142],{"className":36695},[83,84],[39,36697,161],{"className":36698},[263],[39,36700],{"className":36701,"style":231},[230],[39,36703,793],{"className":36704},[235],[39,36706],{"className":36707,"style":231},[230],[39,36709,36711,36715,36764,36767,36796,36799,36802,36805,36808,36811,36814,36843,36846,36849,36852,36855,36858,36861,36864,36867],{"className":36710},[74],[39,36712],{"className":36713,"style":36714},[78],"height:1.6444em;vertical-align:-0.8944em;",[39,36716,36718],{"className":36717},[1744,3294],[39,36719,36721,36755],{"className":36720},[196,649],[39,36722,36724,36752],{"className":36723},[200],[39,36725,36727,36742],{"className":36726,"style":290},[204],[39,36728,36730,36733],{"style":36729},"top:-2.2056em;margin-left:0em;",[39,36731],{"className":36732,"style":935},[212],[39,36734,36736],{"className":36735},[217,218,219,220],[39,36737,36739],{"className":36738},[83,220],[39,36740,2098],{"className":36741,"style":2114},[83,84,220],[39,36743,36744,36747],{"style":21599},[39,36745],{"className":36746,"style":935},[212],[39,36748,36749],{},[39,36750,21465],{"className":36751},[1744],[39,36753,673],{"className":36754},[672],[39,36756,36758],{"className":36757},[200],[39,36759,36762],{"className":36760,"style":36761},[204],"height:0.8944em;",[39,36763],{},[39,36765],{"className":36766,"style":256},[230],[39,36768,36770,36773],{"className":36769},[83],[39,36771,4230],{"className":36772,"style":461},[83,84],[39,36774,36776],{"className":36775},[192],[39,36777,36779],{"className":36778},[196],[39,36780,36782],{"className":36781},[200],[39,36783,36785],{"className":36784,"style":4421},[204],[39,36786,36787,36790],{"style":1027},[39,36788],{"className":36789,"style":213},[212],[39,36791,36793],{"className":36792},[217,218,219,220],[39,36794,2098],{"className":36795,"style":2114},[83,84,220],[39,36797,136],{"className":36798},[182],[39,36800,142],{"className":36801},[83,84],[39,36803,161],{"className":36804},[263],[39,36806,155],{"className":36807},[252],[39,36809],{"className":36810,"style":10787},[230],[39,36812],{"className":36813,"style":256},[230],[39,36815,36817,36820],{"className":36816},[83],[39,36818,4295],{"className":36819},[83,84],[39,36821,36823],{"className":36822},[192],[39,36824,36826],{"className":36825},[196],[39,36827,36829],{"className":36828},[200],[39,36830,36832],{"className":36831,"style":22242},[204],[39,36833,36834,36837],{"style":1027},[39,36835],{"className":36836,"style":213},[212],[39,36838,36840],{"className":36839},[217,218,219,220],[39,36841,21884],{"className":36842},[465,220],[39,36844,136],{"className":36845},[182],[39,36847,142],{"className":36848},[83,84],[39,36850,155],{"className":36851},[252],[39,36853],{"className":36854,"style":256},[230],[39,36856,158],{"className":36857},[83,84],[39,36859,161],{"className":36860},[263],[39,36862],{"className":36863,"style":231},[230],[39,36865,793],{"className":36866},[235],[39,36868],{"className":36869,"style":231},[230],[39,36871,36873,36876,36923,36926,36955,36958,36961,36964,36967,36970,36973],{"className":36872},[74],[39,36874],{"className":36875,"style":36714},[78],[39,36877,36879],{"className":36878},[1744,3294],[39,36880,36882,36915],{"className":36881},[196,649],[39,36883,36885,36912],{"className":36884},[200],[39,36886,36888,36902],{"className":36887,"style":290},[204],[39,36889,36890,36893],{"style":36729},[39,36891],{"className":36892,"style":935},[212],[39,36894,36896],{"className":36895},[217,218,219,220],[39,36897,36899],{"className":36898},[83,220],[39,36900,2098],{"className":36901,"style":2114},[83,84,220],[39,36903,36904,36907],{"style":21599},[39,36905],{"className":36906,"style":935},[212],[39,36908,36909],{},[39,36910,21465],{"className":36911},[1744],[39,36913,673],{"className":36914},[672],[39,36916,36918],{"className":36917},[200],[39,36919,36921],{"className":36920,"style":36761},[204],[39,36922],{},[39,36924],{"className":36925,"style":256},[230],[39,36927,36929,36932],{"className":36928},[83],[39,36930,4295],{"className":36931},[83,84],[39,36933,36935],{"className":36934},[192],[39,36936,36938],{"className":36937},[196],[39,36939,36941],{"className":36940},[200],[39,36942,36944],{"className":36943,"style":4421},[204],[39,36945,36946,36949],{"style":1027},[39,36947],{"className":36948,"style":213},[212],[39,36950,36952],{"className":36951},[217,218,219,220],[39,36953,2098],{"className":36954,"style":2114},[83,84,220],[39,36956,136],{"className":36957},[182],[39,36959,142],{"className":36960},[83,84],[39,36962,155],{"className":36963},[252],[39,36965],{"className":36966,"style":256},[230],[39,36968,158],{"className":36969},[83,84],[39,36971,161],{"className":36972},[263],[39,36974,2903],{"className":36975},[83],[11,36977,36978,36979,16486,37038,2477,37185,2506,37243,1607],{},"若策略 ",[39,36980,36982,37000],{"className":36981,"translate":43},[42],[39,36983,36985],{"className":36984},[47],[49,36986,36987],{"xmlns":51},[53,36988,36989,36997],{},[56,36990,36991],{},[138,36992,36993,36995],{},[59,36994,2098],{},[133,36996,21884],{},[63,36998,36999],{"encoding":65},"\\pi^*",[39,37001,37003],{"className":37002,"ariaHidden":70},[69],[39,37004,37006,37009],{"className":37005},[74],[39,37007],{"className":37008,"style":21897},[78],[39,37010,37012,37015],{"className":37011},[83],[39,37013,2098],{"className":37014,"style":2114},[83,84],[39,37016,37018],{"className":37017},[192],[39,37019,37021],{"className":37020},[196],[39,37022,37024],{"className":37023},[200],[39,37025,37027],{"className":37026,"style":21897},[204],[39,37028,37029,37032],{"style":208},[39,37030],{"className":37031,"style":213},[212],[39,37033,37035],{"className":37034},[217,218,219,220],[39,37036,21884],{"className":37037},[465,220],[39,37039,37041,37071],{"className":37040,"translate":43},[42],[39,37042,37044],{"className":37043},[47],[49,37045,37046],{"xmlns":51},[53,37047,37048,37068],{},[56,37049,37050,37060,37062],{},[138,37051,37052,37054],{},[59,37053,4230],{},[138,37055,37056,37058],{},[59,37057,2098],{},[133,37059,21884],{},[133,37061,793],{},[138,37063,37064,37066],{},[59,37065,4230],{},[133,37067,21884],{},[63,37069,37070],{"encoding":65},"V^{\\pi^*}=V^*",[39,37072,37074,37150],{"className":37073,"ariaHidden":70},[69],[39,37075,37077,37081,37141,37144,37147],{"className":37076},[74],[39,37078],{"className":37079,"style":37080},[78],"height:0.8973em;",[39,37082,37084,37087],{"className":37083},[83],[39,37085,4230],{"className":37086,"style":461},[83,84],[39,37088,37090],{"className":37089},[192],[39,37091,37093],{"className":37092},[196],[39,37094,37096],{"className":37095},[200],[39,37097,37099],{"className":37098,"style":37080},[204],[39,37100,37101,37104],{"style":208},[39,37102],{"className":37103,"style":213},[212],[39,37105,37107],{"className":37106},[217,218,219,220],[39,37108,37110],{"className":37109},[83,220],[39,37111,37113,37116],{"className":37112},[83,220],[39,37114,2098],{"className":37115,"style":2114},[83,84,220],[39,37117,37119],{"className":37118},[192],[39,37120,37122],{"className":37121},[196],[39,37123,37125],{"className":37124},[200],[39,37126,37129],{"className":37127,"style":37128},[204],"height:0.7633em;",[39,37130,37132,37135],{"style":37131},"top:-2.931em;margin-right:0.0714em;",[39,37133],{"className":37134,"style":4004},[212],[39,37136,37138],{"className":37137},[217,4008,4009,220],[39,37139,21884],{"className":37140},[465,220],[39,37142],{"className":37143,"style":231},[230],[39,37145,793],{"className":37146},[235],[39,37148],{"className":37149,"style":231},[230],[39,37151,37153,37156],{"className":37152},[74],[39,37154],{"className":37155,"style":21897},[78],[39,37157,37159,37162],{"className":37158},[83],[39,37160,4230],{"className":37161,"style":461},[83,84],[39,37163,37165],{"className":37164},[192],[39,37166,37168],{"className":37167},[196],[39,37169,37171],{"className":37170},[200],[39,37172,37174],{"className":37173,"style":21897},[204],[39,37175,37176,37179],{"style":208},[39,37177],{"className":37178,"style":213},[212],[39,37180,37182],{"className":37181},[217,218,219,220],[39,37183,21884],{"className":37184},[465,220],[39,37186,37188,37205],{"className":37187,"translate":43},[42],[39,37189,37191],{"className":37190},[47],[49,37192,37193],{"xmlns":51},[53,37194,37195,37203],{},[56,37196,37197],{},[138,37198,37199,37201],{},[59,37200,2098],{},[133,37202,21884],{},[63,37204,36999],{"encoding":65},[39,37206,37208],{"className":37207,"ariaHidden":70},[69],[39,37209,37211,37214],{"className":37210},[74],[39,37212],{"className":37213,"style":21897},[78],[39,37215,37217,37220],{"className":37216},[83],[39,37218,2098],{"className":37219,"style":2114},[83,84],[39,37221,37223],{"className":37222},[192],[39,37224,37226],{"className":37225},[196],[39,37227,37229],{"className":37228},[200],[39,37230,37232],{"className":37231,"style":21897},[204],[39,37233,37234,37237],{"style":208},[39,37235],{"className":37236,"style":213},[212],[39,37238,37240],{"className":37239},[217,218,219,220],[39,37241,21884],{"className":37242},[465,220],[1489,37244,37245],{},"最优策略",[11,37247,37248,37249,37253,37254,37285,37286,37317,37318,37346],{},"在有限状态与有限动作的折扣 MDP 中，上确界实际上可以取到，而且存在一个平稳确定性最优策略。这一点将由",[158,37250,37252],{"href":37251},"#theorem-bellman-optimality","定理 4.2"," 的证明一并给出。因此，本节的 ",[39,37255,37257,37273],{"className":37256,"translate":43},[42],[39,37258,37260],{"className":37259},[47],[49,37261,37262],{"xmlns":51},[53,37263,37264,37270],{},[56,37265,37266,37268],{},[59,37267,21465],{},[133,37269,1690],{},[63,37271,37272],{"encoding":65},"\\sup",[39,37274,37276],{"className":37275,"ariaHidden":70},[69],[39,37277,37279,37282],{"className":37278},[74],[39,37280],{"className":37281,"style":2654},[78],[39,37283,21465],{"className":37284},[1744]," 可以理解为 ",[39,37287,37289,37305],{"className":37288,"translate":43},[42],[39,37290,37292],{"className":37291},[47],[49,37293,37294],{"xmlns":51},[53,37295,37296,37302],{},[56,37297,37298,37300],{},[59,37299,1687],{},[133,37301,1690],{},[63,37303,37304],{"encoding":65},"\\max",[39,37306,37308],{"className":37307,"ariaHidden":70},[69],[39,37309,37311,37314],{"className":37310},[74],[39,37312],{"className":37313,"style":290},[78],[39,37315,1687],{"className":37316},[1744],"。本文的 ",[39,37319,37321,37334],{"className":37320,"translate":43},[42],[39,37322,37324],{"className":37323},[47],[49,37325,37326],{"xmlns":51},[53,37327,37328,37332],{},[56,37329,37330],{},[59,37331,2098],{},[63,37333,2101],{"encoding":65},[39,37335,37337],{"className":37336,"ariaHidden":70},[69],[39,37338,37340,37343],{"className":37339},[74],[39,37341],{"className":37342,"style":290},[78],[39,37344,2098],{"className":37345,"style":2114},[83,84]," 均指平稳策略。对于折扣有限 MDP，平稳策略类已经包含最优策略，因此这一限制不失一般性。",[26,37348,37350],{"id":37349},"_42-最优方程的证明","4.2 最优方程的证明",[11,37352,37353,37354,2083,37357,13349],{},"第 3.1 节已经定义了",[1489,37355,37356],{},"贝尔曼最优算子",[39,37358,37360,37377],{"className":37359,"translate":43},[42],[39,37361,37363],{"className":37362},[47],[49,37364,37365],{"xmlns":51},[53,37366,37367,37375],{},[56,37368,37369],{},[138,37370,37371,37373],{},[59,37372,11772],{"mathvariant":617},[133,37374,21884],{},[63,37376,21887],{"encoding":65},[39,37378,37380],{"className":37379,"ariaHidden":70},[69],[39,37381,37383,37386],{"className":37382},[74],[39,37384],{"className":37385,"style":21897},[78],[39,37387,37389,37392],{"className":37388},[83],[39,37390,11772],{"className":37391,"style":11793},[83,641],[39,37393,37395],{"className":37394},[192],[39,37396,37398],{"className":37397},[196],[39,37399,37401],{"className":37400},[200],[39,37402,37404],{"className":37403,"style":21897},[204],[39,37405,37406,37409],{"style":208},[39,37407],{"className":37408,"style":213},[212],[39,37410,37412],{"className":37411},[217,218,219,220],[39,37413,21884],{"className":37414},[465,220],[39,37416,37418],{"className":37417,"translate":43},[746],[39,37419,37421,37536],{"className":37420,"translate":43},[42],[39,37422,37424],{"className":37423},[47],[49,37425,37426],{"xmlns":51,"display":755},[53,37427,37428,37534],{},[56,37429,37430,37432,37438,37440,37442,37444,37446,37448,37450,37460,37470,37472,37474,37480,37482,37484,37486,37488,37490,37532],{},[133,37431,136],{"stretchy":135},[138,37433,37434,37436],{},[59,37435,11772],{"mathvariant":617},[133,37437,21884],{},[59,37439,4230],{},[133,37441,161],{"stretchy":135},[133,37443,136],{"stretchy":135},[59,37445,142],{},[133,37447,161],{"stretchy":135},[133,37449,793],{},[6556,37451,37452,37458],{},[56,37453,37454,37456],{},[59,37455,1687],{},[133,37457,1690],{},[59,37459,158],{},[6556,37461,37462,37464],{},[133,37463,3191],{},[138,37465,37466,37468],{},[59,37467,142],{},[133,37469,147],{"mathvariant":145,"lspace":146,"rspace":146},[59,37471,131],{},[133,37473,136],{"stretchy":135},[138,37475,37476,37478],{},[59,37477,142],{},[133,37479,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,37481,150],{},[59,37483,142],{},[133,37485,155],{"separator":70},[59,37487,158],{},[133,37489,161],{"stretchy":135},[56,37491,37492,37494,37496,37498,37500,37502,37504,37506,37512,37514,37516,37518,37520,37522,37528,37530],{},[133,37493,537],{"fence":70},[59,37495,401],{},[133,37497,136],{"stretchy":135},[59,37499,142],{},[133,37501,155],{"separator":70},[59,37503,158],{},[133,37505,155],{"separator":70},[138,37507,37508,37510],{},[59,37509,142],{},[133,37511,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,37513,161],{"stretchy":135},[133,37515,788],{},[59,37517,531],{},[59,37519,4230],{},[133,37521,136],{"stretchy":135},[138,37523,37524,37526],{},[59,37525,142],{},[133,37527,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,37529,161],{"stretchy":135},[133,37531,4280],{"fence":70},[59,37533,2903],{"mathvariant":145},[63,37535,22211],{"encoding":65},[39,37537,37539,37601,37783],{"className":37538,"ariaHidden":70},[69],[39,37540,37542,37545,37548,37577,37580,37583,37586,37589,37592,37595,37598],{"className":37541},[74],[39,37543],{"className":37544,"style":245},[78],[39,37546,136],{"className":37547},[182],[39,37549,37551,37554],{"className":37550},[83],[39,37552,11772],{"className":37553,"style":11793},[83,641],[39,37555,37557],{"className":37556},[192],[39,37558,37560],{"className":37559},[196],[39,37561,37563],{"className":37562},[200],[39,37564,37566],{"className":37565,"style":22242},[204],[39,37567,37568,37571],{"style":1027},[39,37569],{"className":37570,"style":213},[212],[39,37572,37574],{"className":37573},[217,218,219,220],[39,37575,21884],{"className":37576},[465,220],[39,37578,4230],{"className":37579,"style":461},[83,84],[39,37581,161],{"className":37582},[263],[39,37584,136],{"className":37585},[182],[39,37587,142],{"className":37588},[83,84],[39,37590,161],{"className":37591},[263],[39,37593],{"className":37594,"style":231},[230],[39,37596,793],{"className":37597},[235],[39,37599],{"className":37600,"style":231},[230],[39,37602,37604,37607,37654,37657,37733,37736,37739,37742,37774,37777,37780],{"className":37603},[74],[39,37605],{"className":37606,"style":22284},[78],[39,37608,37610],{"className":37609},[1744,3294],[39,37611,37613,37646],{"className":37612},[196,649],[39,37614,37616,37643],{"className":37615},[200],[39,37617,37619,37633],{"className":37618,"style":290},[204],[39,37620,37621,37624],{"style":22299},[39,37622],{"className":37623,"style":935},[212],[39,37625,37627],{"className":37626},[217,218,219,220],[39,37628,37630],{"className":37629},[83,220],[39,37631,158],{"className":37632},[83,84,220],[39,37634,37635,37638],{"style":21599},[39,37636],{"className":37637,"style":935},[212],[39,37639,37640],{},[39,37641,1687],{"className":37642},[1744],[39,37644,673],{"className":37645},[672],[39,37647,37649],{"className":37648},[200],[39,37650,37652],{"className":37651,"style":22331},[204],[39,37653],{},[39,37655],{"className":37656,"style":256},[230],[39,37658,37660],{"className":37659},[1744,3294],[39,37661,37663,37725],{"className":37662},[196,649],[39,37664,37666,37722],{"className":37665},[200],[39,37667,37669,37712],{"className":37668,"style":6670},[204],[39,37670,37671,37674],{"style":7097},[39,37672],{"className":37673,"style":3311},[212],[39,37675,37677],{"className":37676},[217,218,219,220],[39,37678,37680],{"className":37679},[83,220],[39,37681,37683,37686],{"className":37682},[83,220],[39,37684,142],{"className":37685},[83,84,220],[39,37687,37689],{"className":37688},[192],[39,37690,37692],{"className":37691},[196],[39,37693,37695],{"className":37694},[200],[39,37696,37698],{"className":37697,"style":3997},[204],[39,37699,37700,37703],{"style":4000},[39,37701],{"className":37702,"style":4004},[212],[39,37704,37706],{"className":37705},[217,4008,4009,220],[39,37707,37709],{"className":37708},[83,220],[39,37710,147],{"className":37711},[83,220],[39,37713,37714,37717],{"style":3330},[39,37715],{"className":37716,"style":3311},[212],[39,37718,37719],{},[39,37720,3191],{"className":37721},[1744,3339,3340],[39,37723,673],{"className":37724},[672],[39,37726,37728],{"className":37727},[200],[39,37729,37731],{"className":37730,"style":7158},[204],[39,37732],{},[39,37734],{"className":37735,"style":256},[230],[39,37737,131],{"className":37738,"style":178},[83,84],[39,37740,136],{"className":37741},[182],[39,37743,37745,37748],{"className":37744},[83],[39,37746,142],{"className":37747},[83,84],[39,37749,37751],{"className":37750},[192],[39,37752,37754],{"className":37753},[196],[39,37755,37757],{"className":37756},[200],[39,37758,37760],{"className":37759,"style":1024},[204],[39,37761,37762,37765],{"style":1027},[39,37763],{"className":37764,"style":213},[212],[39,37766,37768],{"className":37767},[217,218,219,220],[39,37769,37771],{"className":37770},[83,220],[39,37772,147],{"className":37773},[83,220],[39,37775],{"className":37776,"style":231},[230],[39,37778,150],{"className":37779},[235],[39,37781],{"className":37782,"style":231},[230],[39,37784,37786,37789,37792,37795,37798,37801,37804,37807,37928,37931],{"className":37785},[74],[39,37787],{"className":37788,"style":22469},[78],[39,37790,142],{"className":37791},[83,84],[39,37793,155],{"className":37794},[252],[39,37796],{"className":37797,"style":256},[230],[39,37799,158],{"className":37800},[83,84],[39,37802,161],{"className":37803},[263],[39,37805],{"className":37806,"style":256},[230],[39,37808,37810,37813,37816,37819,37822,37825,37828,37831,37834,37837,37869,37872,37875,37878,37881,37884,37887,37890,37922,37925],{"className":37809},[2842],[39,37811,537],{"className":37812,"style":4592},[182,4591],[39,37814,401],{"className":37815,"style":438},[83,84],[39,37817,136],{"className":37818},[182],[39,37820,142],{"className":37821},[83,84],[39,37823,155],{"className":37824},[252],[39,37826],{"className":37827,"style":256},[230],[39,37829,158],{"className":37830},[83,84],[39,37832,155],{"className":37833},[252],[39,37835],{"className":37836,"style":256},[230],[39,37838,37840,37843],{"className":37839},[83],[39,37841,142],{"className":37842},[83,84],[39,37844,37846],{"className":37845},[192],[39,37847,37849],{"className":37848},[196],[39,37850,37852],{"className":37851},[200],[39,37853,37855],{"className":37854,"style":1024},[204],[39,37856,37857,37860],{"style":1027},[39,37858],{"className":37859,"style":213},[212],[39,37861,37863],{"className":37862},[217,218,219,220],[39,37864,37866],{"className":37865},[83,220],[39,37867,147],{"className":37868},[83,220],[39,37870,161],{"className":37871},[263],[39,37873],{"className":37874,"style":461},[230],[39,37876,788],{"className":37877},[465],[39,37879],{"className":37880,"style":461},[230],[39,37882,531],{"className":37883,"style":565},[83,84],[39,37885,4230],{"className":37886,"style":461},[83,84],[39,37888,136],{"className":37889},[182],[39,37891,37893,37896],{"className":37892},[83],[39,37894,142],{"className":37895},[83,84],[39,37897,37899],{"className":37898},[192],[39,37900,37902],{"className":37901},[196],[39,37903,37905],{"className":37904},[200],[39,37906,37908],{"className":37907,"style":1024},[204],[39,37909,37910,37913],{"style":1027},[39,37911],{"className":37912,"style":213},[212],[39,37914,37916],{"className":37915},[217,218,219,220],[39,37917,37919],{"className":37918},[83,220],[39,37920,147],{"className":37921},[83,220],[39,37923,161],{"className":37924},[263],[39,37926,4280],{"className":37927,"style":4592},[263,4591],[39,37929],{"className":37930,"style":256},[230],[39,37932,2903],{"className":37933},[83],[11,37935,37936,37937,37995,37996,38054],{},"与 ",[39,37938,37940,37957],{"className":37939,"translate":43},[42],[39,37941,37943],{"className":37942},[47],[49,37944,37945],{"xmlns":51},[53,37946,37947,37955],{},[56,37948,37949],{},[138,37950,37951,37953],{},[59,37952,11772],{"mathvariant":617},[59,37954,2098],{},[63,37956,11777],{"encoding":65},[39,37958,37960],{"className":37959,"ariaHidden":70},[69],[39,37961,37963,37966],{"className":37962},[74],[39,37964],{"className":37965,"style":79},[78],[39,37967,37969,37972],{"className":37968},[83],[39,37970,11772],{"className":37971,"style":11793},[83,641],[39,37973,37975],{"className":37974},[192],[39,37976,37978],{"className":37977},[196],[39,37979,37981],{"className":37980},[200],[39,37982,37984],{"className":37983,"style":6410},[204],[39,37985,37986,37989],{"style":208},[39,37987],{"className":37988,"style":213},[212],[39,37990,37992],{"className":37991},[217,218,219,220],[39,37993,2098],{"className":37994,"style":2114},[83,84,220]," 不同，",[39,37997,37999,38016],{"className":37998,"translate":43},[42],[39,38000,38002],{"className":38001},[47],[49,38003,38004],{"xmlns":51},[53,38005,38006,38014],{},[56,38007,38008],{},[138,38009,38010,38012],{},[59,38011,11772],{"mathvariant":617},[133,38013,21884],{},[63,38015,21887],{"encoding":65},[39,38017,38019],{"className":38018,"ariaHidden":70},[69],[39,38020,38022,38025],{"className":38021},[74],[39,38023],{"className":38024,"style":21897},[78],[39,38026,38028,38031],{"className":38027},[83],[39,38029,11772],{"className":38030,"style":11793},[83,641],[39,38032,38034],{"className":38033},[192],[39,38035,38037],{"className":38036},[196],[39,38038,38040],{"className":38039},[200],[39,38041,38043],{"className":38042,"style":21897},[204],[39,38044,38045,38048],{"style":208},[39,38046],{"className":38047,"style":213},[212],[39,38049,38051],{"className":38050},[217,218,219,220],[39,38052,21884],{"className":38053},[465,220]," 先对每个动作计算期望，再在动作上取最大值。这个顺序不能交换，原因见 4.3 节。",[31,38056,38059,38294,38926,39016],{"id":38057,"title":38058},"theorem-bellman-optimality","定理 4.2（贝尔曼最优方程）",[11,38060,37,38061,742,38134,1639,38162,38190,38191,38249,38250,38293],{},[39,38062,38064,38089],{"className":38063,"translate":43},[42],[39,38065,38067],{"className":38066},[47],[49,38068,38069],{"xmlns":51},[53,38070,38071,38087],{},[56,38072,38073,38075,38077,38079,38081,38083,38085],{},[59,38074,531],{},[133,38076,534],{},[133,38078,537],{"stretchy":135},[539,38080,541],{},[133,38082,155],{"separator":70},[539,38084,546],{},[133,38086,161],{"stretchy":135},[63,38088,551],{"encoding":65},[39,38090,38092,38110],{"className":38091,"ariaHidden":70},[69],[39,38093,38095,38098,38101,38104,38107],{"className":38094},[74],[39,38096],{"className":38097,"style":561},[78],[39,38099,531],{"className":38100,"style":565},[83,84],[39,38102],{"className":38103,"style":231},[230],[39,38105,534],{"className":38106},[235],[39,38108],{"className":38109,"style":231},[230],[39,38111,38113,38116,38119,38122,38125,38128,38131],{"className":38112},[74],[39,38114],{"className":38115,"style":245},[78],[39,38117,537],{"className":38118},[182],[39,38120,541],{"className":38121},[83],[39,38123,155],{"className":38124},[252],[39,38126],{"className":38127,"style":256},[230],[39,38129,546],{"className":38130},[83],[39,38132,161],{"className":38133},[263],[39,38135,38137,38150],{"className":38136,"translate":43},[42],[39,38138,38140],{"className":38139},[47],[49,38141,38142],{"xmlns":51},[53,38143,38144,38148],{},[56,38145,38146],{},[59,38147,61],{},[63,38149,61],{"encoding":65},[39,38151,38153],{"className":38152,"ariaHidden":70},[69],[39,38154,38156,38159],{"className":38155},[74],[39,38157],{"className":38158,"style":79},[78],[39,38160,61],{"className":38161,"style":85},[83,84],[39,38163,38165,38178],{"className":38164,"translate":43},[42],[39,38166,38168],{"className":38167},[47],[49,38169,38170],{"xmlns":51},[53,38171,38172,38176],{},[56,38173,38174],{},[59,38175,101],{},[63,38177,101],{"encoding":65},[39,38179,38181],{"className":38180,"ariaHidden":70},[69],[39,38182,38184,38187],{"className":38183},[74],[39,38185],{"className":38186,"style":79},[78],[39,38188,101],{"className":38189},[83,84]," 有限。则 ",[39,38192,38194,38211],{"className":38193,"translate":43},[42],[39,38195,38197],{"className":38196},[47],[49,38198,38199],{"xmlns":51},[53,38200,38201,38209],{},[56,38202,38203],{},[138,38204,38205,38207],{},[59,38206,4230],{},[133,38208,21884],{},[63,38210,34720],{"encoding":65},[39,38212,38214],{"className":38213,"ariaHidden":70},[69],[39,38215,38217,38220],{"className":38216},[74],[39,38218],{"className":38219,"style":21897},[78],[39,38221,38223,38226],{"className":38222},[83],[39,38224,4230],{"className":38225,"style":461},[83,84],[39,38227,38229],{"className":38228},[192],[39,38230,38232],{"className":38231},[196],[39,38233,38235],{"className":38234},[200],[39,38236,38238],{"className":38237,"style":21897},[204],[39,38239,38240,38243],{"style":208},[39,38241],{"className":38242,"style":213},[212],[39,38244,38246],{"className":38245},[217,218,219,220],[39,38247,21884],{"className":38248},[465,220]," 是 ",[39,38251,38253,38272],{"className":38252,"translate":43},[42],[39,38254,38256],{"className":38255},[47],[49,38257,38258],{"xmlns":51},[53,38259,38260,38270],{},[56,38261,38262,38264,38266,38268],{},[59,38263,19772],{},[133,38265,136],{"stretchy":135},[59,38267,61],{},[133,38269,161],{"stretchy":135},[63,38271,21380],{"encoding":65},[39,38273,38275],{"className":38274,"ariaHidden":70},[69],[39,38276,38278,38281,38284,38287,38290],{"className":38277},[74],[39,38279],{"className":38280,"style":245},[78],[39,38282,19772],{"className":38283,"style":19812},[83,84],[39,38285,136],{"className":38286},[182],[39,38288,61],{"className":38289,"style":85},[83,84],[39,38291,161],{"className":38292},[263]," 中满足",[39,38295,38297],{"className":38296,"translate":43},[746],[39,38298,38300,38452],{"className":38299,"translate":43},[42],[39,38301,38303],{"className":38302},[47],[49,38304,38305],{"xmlns":51,"display":755},[53,38306,38307,38449],{},[759,38308,38309,38375],{"rowspacing":761,"columnalign":762,"columnspacing":146},[764,38310,38311,38325],{},[767,38312,38313],{},[770,38314,38315],{"scriptlevel":541,"displaystyle":70},[56,38316,38317,38319,38321,38323],{},[59,38318,4230],{},[133,38320,136],{"stretchy":135},[59,38322,142],{},[133,38324,161],{"stretchy":135},[767,38326,38327],{},[770,38328,38329],{"scriptlevel":541,"displaystyle":70},[56,38330,38331,38333,38335,38345,38355,38357,38359,38365,38367,38369,38371,38373],{},[56,38332],{},[133,38334,793],{},[6556,38336,38337,38343],{},[56,38338,38339,38341],{},[59,38340,1687],{},[133,38342,1690],{},[59,38344,158],{},[6556,38346,38347,38349],{},[133,38348,3191],{},[138,38350,38351,38353],{},[59,38352,142],{},[133,38354,147],{"mathvariant":145,"lspace":146,"rspace":146},[59,38356,131],{},[133,38358,136],{"stretchy":135},[138,38360,38361,38363],{},[59,38362,142],{},[133,38364,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,38366,150],{},[59,38368,142],{},[133,38370,155],{"separator":70},[59,38372,158],{},[133,38374,161],{"stretchy":135},[764,38376,38377,38383],{},[767,38378,38379],{},[770,38380,38381],{"scriptlevel":541,"displaystyle":70},[56,38382],{},[767,38384,38385],{},[770,38386,38387],{"scriptlevel":541,"displaystyle":70},[56,38388,38389,38391,38393,38395,38437,38439,38441,38443,38445,38447],{},[56,38390],{},[230,38392],{"width":6891},[133,38394,409],{},[56,38396,38397,38399,38401,38403,38405,38407,38409,38411,38417,38419,38421,38423,38425,38427,38433,38435],{},[133,38398,537],{"fence":70},[59,38400,401],{},[133,38402,136],{"stretchy":135},[59,38404,142],{},[133,38406,155],{"separator":70},[59,38408,158],{},[133,38410,155],{"separator":70},[138,38412,38413,38415],{},[59,38414,142],{},[133,38416,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,38418,161],{"stretchy":135},[133,38420,788],{},[59,38422,531],{},[59,38424,4230],{},[133,38426,136],{"stretchy":135},[138,38428,38429,38431],{},[59,38430,142],{},[133,38432,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,38434,161],{"stretchy":135},[133,38436,4280],{"fence":70},[133,38438,155],{"separator":70},[230,38440],{"width":10251},[59,38442,10254],{"mathvariant":145},[59,38444,142],{},[133,38446,534],{},[59,38448,61],{},[63,38450,38451],{"encoding":65},"\\begin{aligned}\nV(s)&=\\max_{a}\\sum_{s'}P(s'\\mid s,a)\\\\\n&\\quad\\times\\left[r(s,a,s')+\\gamma V(s')\\right],\n\\qquad \\forall s\\in 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中存在唯一不动点，记作 ",[39,39677,39679,39699],{"className":39678,"translate":43},[42],[39,39680,39682],{"className":39681},[47],[49,39683,39684],{"xmlns":51},[53,39685,39686,39696],{},[56,39687,39688],{},[39689,39690,39691,39693],"mover",{"accent":70},[59,39692,4230],{},[133,39694,39695],{},"ˉ",[63,39697,39698],{"encoding":65},"\\bar 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取上确界得 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V=\\mathcal{T}^*\\bar V=\\bar V",[39,42050,42052,42170,42245],{"className":42051,"ariaHidden":70},[69],[39,42053,42055,42058,42130,42161,42164,42167],{"className":42054},[74],[39,42056],{"className":42057,"style":39708},[78],[39,42059,42061,42064],{"className":42060},[83],[39,42062,11772],{"className":42063,"style":11793},[83,641],[39,42065,42067],{"className":42066},[192],[39,42068,42070],{"className":42069},[196],[39,42071,42073],{"className":42072},[200],[39,42074,42076],{"className":42075,"style":6410},[204],[39,42077,42078,42081],{"style":208},[39,42079],{"className":42080,"style":213},[212],[39,42082,42084],{"className":42083},[217,218,219,220],[39,42085,42087],{"className":42086},[83,220],[39,42088,42090,42093],{"className":42089},[83,220],[39,42091,2098],{"className":42092,"style":2114},[83,84,220],[39,42094,42096],{"className":42095},[192],[39,42097,42099,42121],{"className":42098},[196,649],[39,42100,42102,42118],{"className":42101},[200],[39,42103,42106],{"className":42104,"style":42105},[204],"height:0.1645em;",[39,42107,42109,42112],{"style":42108},"top:-2.357em;margin-left:-0.0359em;margin-right:0.0714em;",[39,42110],{"className":42111,"style":4004},[212],[39,42113,42115],{"className":42114},[217,4008,4009,220],[39,42116,39209],{"className":42117,"style":2114},[83,84,220],[39,42119,673],{"className":42120},[672],[39,42122,42124],{"className":42123},[200],[39,42125,42128],{"className":42126,"style":42127},[204],"height:0.2819em;",[39,42129],{},[39,42131,42133],{"className":42132},[83,39712],[39,42134,42136],{"className":42135},[196],[39,42137,42139],{"className":42138},[200],[39,42140,42142,42150],{"className":42141,"style":39708},[204],[39,42143,42144,42147],{"style":21599},[39,42145],{"className":42146,"style":935},[212],[39,42148,4230],{"className":42149,"style":461},[83,84],[39,42151,42152,42155],{"style":39732},[39,42153],{"className":42154,"style":935},[212],[39,42156,42158],{"className":42157,"style":39740},[39739],[39,42159,39695],{"className":42160},[83],[39,42162],{"className":42163,"style":231},[230],[39,42165,793],{"className":42166},[235],[39,42168],{"className":42169,"style":231},[230],[39,42171,42173,42176,42205,42236,42239,42242],{"className":42172},[74],[39,42174],{"className":42175,"style":39708},[78],[39,42177,42179,42182],{"className":42178},[83],[39,42180,11772],{"className":42181,"style":11793},[83,641],[39,42183,42185],{"className":42184},[192],[39,42186,42188],{"className":42187},[196],[39,42189,42191],{"className":42190},[200],[39,42192,42194],{"className":42193,"style":21897},[204],[39,42195,42196,42199],{"style":208},[39,42197],{"className":42198,"style":213},[212],[39,42200,42202],{"className":42201},[217,218,219,220],[39,42203,21884],{"className":42204},[465,220],[39,42206,42208],{"className":42207},[83,39712],[39,42209,42211],{"className":42210},[196],[39,42212,42214],{"className":42213},[200],[39,42215,42217,42225],{"className":42216,"style":39708},[204],[39,42218,42219,42222],{"style":21599},[39,42220],{"className":42221,"style":935},[212],[39,42223,4230],{"className":42224,"style":461},[83,84],[39,42226,42227,42230],{"style":39732},[39,42228],{"className":42229,"style":935},[212],[39,42231,42233],{"className":42232,"style":39740},[39739],[39,42234,39695],{"className":42235},[83],[39,42237],{"className":42238,"style":231},[230],[39,42240,793],{"className":42241},[235],[39,42243],{"className":42244,"style":231},[230],[39,42246,42248,42251],{"className":42247},[74],[39,42249],{"className":42250,"style":39708},[78],[39,42252,42254],{"className":42253},[83,39712],[39,42255,42257],{"className":42256},[196],[39,42258,42260],{"className":42259},[200],[39,42261,42263,42271],{"className":42262,"style":39708},[204],[39,42264,42265,42268],{"style":21599},[39,42266],{"className":42267,"style":935},[212],[39,42269,4230],{"className":42270,"style":461},[83,84],[39,42272,42273,42276],{"style":39732},[39,42274],{"className":42275,"style":935},[212],[39,42277,42279],{"className":42278,"style":39740},[39739],[39,42280,39695],{"className":42281},[83],"。由",[158,42284,21294],{"href":21293},[39,42286,42288,42318],{"className":42287,"translate":43},[42],[39,42289,42291],{"className":42290},[47],[49,42292,42293],{"xmlns":51},[53,42294,42295,42315],{},[56,42296,42297,42303,42305],{},[39689,42298,42299,42301],{"accent":70},[59,42300,4230],{},[133,42302,39695],{},[133,42304,793],{},[138,42306,42307,42309],{},[59,42308,4230],{},[613,42310,42311,42313],{},[59,42312,2098],{},[59,42314,39209],{},[63,42316,42317],{"encoding":65},"\\bar V=V^{\\pi_g}",[39,42319,42321,42367],{"className":42320,"ariaHidden":70},[69],[39,42322,42324,42327,42358,42361,42364],{"className":42323},[74],[39,42325],{"className":42326,"style":39708},[78],[39,42328,42330],{"className":42329},[83,39712],[39,42331,42333],{"className":42332},[196],[39,42334,42336],{"className":42335},[200],[39,42337,42339,42347],{"className":42338,"style":39708},[204],[39,42340,42341,42344],{"style":21599},[39,42342],{"className":42343,"style":935},[212],[39,42345,4230],{"className":42346,"style":461},[83,84],[39,42348,42349,42352],{"style":39732},[39,42350],{"className":42351,"style":935},[212],[39,42353,42355],{"className":42354,"style":39740},[39739],[39,42356,39695],{"className":42357},[83],[39,42359],{"className":42360,"style":231},[230],[39,42362,793],{"className":42363},[235],[39,42365],{"className":42366,"style":231},[230],[39,42368,42370,42373],{"className":42369},[74],[39,42371],{"className":42372,"style":79},[78],[39,42374,42376,42379],{"className":42375},[83],[39,42377,4230],{"className":42378,"style":461},[83,84],[39,42380,42382],{"className":42381},[192],[39,42383,42385],{"className":42384},[196],[39,42386,42388],{"className":42387},[200],[39,42389,42391],{"className":42390,"style":6410},[204],[39,42392,42393,42396],{"style":208},[39,42394],{"className":42395,"style":213},[212],[39,42397,42399],{"className":42398},[217,218,219,220],[39,42400,42402],{"className":42401},[83,220],[39,42403,42405,42408],{"className":42404},[83,220],[39,42406,2098],{"className":42407,"style":2114},[83,84,220],[39,42409,42411],{"className":42410},[192],[39,42412,42414,42434],{"className":42413},[196,649],[39,42415,42417,42431],{"className":42416},[200],[39,42418,42420],{"className":42419,"style":42105},[204],[39,42421,42422,42425],{"style":42108},[39,42423],{"className":42424,"style":4004},[212],[39,42426,42428],{"className":42427},[217,4008,4009,220],[39,42429,39209],{"className":42430,"style":2114},[83,84,220],[39,42432,673],{"className":42433},[672],[39,42435,42437],{"className":42436},[200],[39,42438,42440],{"className":42439,"style":42127},[204],[39,42441],{},[39,42443,42445,42470],{"className":42444,"translate":43},[42],[39,42446,42448],{"className":42447},[47],[49,42449,42450],{"xmlns":51},[53,42451,42452,42468],{},[56,42453,42454,42460,42462],{},[39689,42455,42456,42458],{"accent":70},[59,42457,4230],{},[133,42459,39695],{},[133,42461,1824],{},[138,42463,42464,42466],{},[59,42465,4230],{},[133,42467,21884],{},[63,42469,41180],{"encoding":65},[39,42471,42473,42519],{"className":42472,"ariaHidden":70},[69],[39,42474,42476,42479,42510,42513,42516],{"className":42475},[74],[39,42477],{"className":42478,"style":40185},[78],[39,42480,42482],{"className":42481},[83,39712],[39,42483,42485],{"className":42484},[196],[39,42486,42488],{"className":42487},[200],[39,42489,42491,42499],{"className":42490,"style":39708},[204],[39,42492,42493,42496],{"style":21599},[39,42494],{"className":42495,"style":935},[212],[39,42497,4230],{"className":42498,"style":461},[83,84],[39,42500,42501,42504],{"style":39732},[39,42502],{"className":42503,"style":935},[212],[39,42505,42507],{"className":42506,"style":39740},[39739],[39,42508,39695],{"className":42509},[83],[39,42511],{"className":42512,"style":231},[230],[39,42514,1824],{"className":42515},[235],[39,42517],{"className":42518,"style":231},[230],[39,42520,42522,42525],{"className":42521},[74],[39,42523],{"className":42524,"style":21897},[78],[39,42526,42528,42531],{"className":42527},[83],[39,42529,4230],{"className":42530,"style":461},[83,84],[39,42532,42534],{"className":42533},[192],[39,42535,42537],{"className":42536},[196],[39,42538,42540],{"className":42539},[200],[39,42541,42543],{"className":42542,"style":21897},[204],[39,42544,42545,42548],{"style":208},[39,42546],{"className":42547,"style":213},[212],[39,42549,42551],{"className":42550},[217,218,219,220],[39,42552,21884],{"className":42553},[465,220],"。结合两个不等式得 ",[39,42556,42558,42583],{"className":42557,"translate":43},[42],[39,42559,42561],{"className":42560},[47],[49,42562,42563],{"xmlns":51},[53,42564,42565,42581],{},[56,42566,42567,42573,42575],{},[39689,42568,42569,42571],{"accent":70},[59,42570,4230],{},[133,42572,39695],{},[133,42574,793],{},[138,42576,42577,42579],{},[59,42578,4230],{},[133,42580,21884],{},[63,42582,39773],{"encoding":65},[39,42584,42586,42632],{"className":42585,"ariaHidden":70},[69],[39,42587,42589,42592,42623,42626,42629],{"className":42588},[74],[39,42590],{"className":42591,"style":39708},[78],[39,42593,42595],{"className":42594},[83,39712],[39,42596,42598],{"className":42597},[196],[39,42599,42601],{"className":42600},[200],[39,42602,42604,42612],{"className":42603,"style":39708},[204],[39,42605,42606,42609],{"style":21599},[39,42607],{"className":42608,"style":935},[212],[39,42610,4230],{"className":42611,"style":461},[83,84],[39,42613,42614,42617],{"style":39732},[39,42615],{"className":42616,"style":935},[212],[39,42618,42620],{"className":42619,"style":39740},[39739],[39,42621,39695],{"className":42622},[83],[39,42624],{"className":42625,"style":231},[230],[39,42627,793],{"className":42628},[235],[39,42630],{"className":42631,"style":231},[230],[39,42633,42635,42638],{"className":42634},[74],[39,42636],{"className":42637,"style":21897},[78],[39,42639,42641,42644],{"className":42640},[83],[39,42642,4230],{"className":42643,"style":461},[83,84],[39,42645,42647],{"className":42646},[192],[39,42648,42650],{"className":42649},[196],[39,42651,42653],{"className":42652},[200],[39,42654,42656],{"className":42655,"style":21897},[204],[39,42657,42658,42661],{"style":208},[39,42659],{"className":42660,"style":213},[212],[39,42662,42664],{"className":42663},[217,218,219,220],[39,42665,21884],{"className":42666},[465,220],[11,42668,42669,42670,42698,42699,42932,42933,42935,42936,43047,43048,43158],{},"最后证明策略的最优性判据。若确定性策略 ",[39,42671,42673,42686],{"className":42672,"translate":43},[42],[39,42674,42676],{"className":42675},[47],[49,42677,42678],{"xmlns":51},[53,42679,42680,42684],{},[56,42681,42682],{},[59,42683,2098],{},[63,42685,2101],{"encoding":65},[39,42687,42689],{"className":42688,"ariaHidden":70},[69],[39,42690,42692,42695],{"className":42691},[74],[39,42693],{"className":42694,"style":290},[78],[39,42696,2098],{"className":42697,"style":2114},[83,84]," 在每个状态都取到最大值，则 ",[39,42700,42702,42748],{"className":42701,"translate":43},[42],[39,42703,42705],{"className":42704},[47],[49,42706,42707],{"xmlns":51},[53,42708,42709,42745],{},[56,42710,42711,42717,42723,42725,42731,42737,42739],{},[138,42712,42713,42715],{},[59,42714,11772],{"mathvariant":617},[59,42716,2098],{},[138,42718,42719,42721],{},[59,42720,4230],{},[133,42722,21884],{},[133,42724,793],{},[138,42726,42727,42729],{},[59,42728,11772],{"mathvariant":617},[133,42730,21884],{},[138,42732,42733,42735],{},[59,42734,4230],{},[133,42736,21884],{},[133,42738,793],{},[138,42740,42741,42743],{},[59,42742,4230],{},[133,42744,21884],{},[63,42746,42747],{"encoding":65},"\\mathcal{T}^\\pi V^*=\\mathcal{T}^*V^*=V^*",[39,42749,42751,42824,42897],{"className":42750,"ariaHidden":70},[69],[39,42752,42754,42757,42786,42815,42818,42821],{"className":42753},[74],[39,42755],{"className":42756,"style":21897},[78],[39,42758,42760,42763],{"className":42759},[83],[39,42761,11772],{"className":42762,"style":11793},[83,641],[39,42764,42766],{"className":42765},[192],[39,42767,42769],{"className":42768},[196],[39,42770,42772],{"className":42771},[200],[39,42773,42775],{"className":42774,"style":6410},[204],[39,42776,42777,42780],{"style":208},[39,42778],{"className":42779,"style":213},[212],[39,42781,42783],{"className":42782},[217,218,219,220],[39,42784,2098],{"className":42785,"style":2114},[83,84,220],[39,42787,42789,42792],{"className":42788},[83],[39,42790,4230],{"className":42791,"style":461},[83,84],[39,42793,42795],{"className":42794},[192],[39,42796,42798],{"className":42797},[196],[39,42799,42801],{"className":42800},[200],[39,42802,42804],{"className":42803,"style":21897},[204],[39,42805,42806,42809],{"style":208},[39,42807],{"className":42808,"style":213},[212],[39,42810,42812],{"className":42811},[217,218,219,220],[39,42813,21884],{"className":42814},[465,220],[39,42816],{"className":42817,"style":231},[230],[39,42819,793],{"className":42820},[235],[39,42822],{"className":42823,"style":231},[230],[39,42825,42827,42830,42859,42888,42891,42894],{"className":42826},[74],[39,42828],{"className":42829,"style":21897},[78],[39,42831,42833,42836],{"className":42832},[83],[39,42834,11772],{"className":42835,"style":11793},[83,641],[39,42837,42839],{"className":42838},[192],[39,42840,42842],{"className":42841},[196],[39,42843,42845],{"className":42844},[200],[39,42846,42848],{"className":42847,"style":21897},[204],[39,42849,42850,42853],{"style":208},[39,42851],{"className":42852,"style":213},[212],[39,42854,42856],{"className":42855},[217,218,219,220],[39,42857,21884],{"className":42858},[465,220],[39,42860,42862,42865],{"className":42861},[83],[39,42863,4230],{"className":42864,"style":461},[83,84],[39,42866,42868],{"className":42867},[192],[39,42869,42871],{"className":42870},[196],[39,42872,42874],{"className":42873},[200],[39,42875,42877],{"className":42876,"style":21897},[204],[39,42878,42879,42882],{"style":208},[39,42880],{"className":42881,"style":213},[212],[39,42883,42885],{"className":42884},[217,218,219,220],[39,42886,21884],{"className":42887},[465,220],[39,42889],{"className":42890,"style":231},[230],[39,42892,793],{"className":42893},[235],[39,42895],{"className":42896,"style":231},[230],[39,42898,42900,42903],{"className":42899},[74],[39,42901],{"className":42902,"style":21897},[78],[39,42904,42906,42909],{"className":42905},[83],[39,42907,4230],{"className":42908,"style":461},[83,84],[39,42910,42912],{"className":42911},[192],[39,42913,42915],{"className":42914},[196],[39,42916,42918],{"className":42917},[200],[39,42919,42921],{"className":42920,"style":21897},[204],[39,42922,42923,42926],{"style":208},[39,42924],{"className":42925,"style":213},[212],[39,42927,42929],{"className":42928},[217,218,219,220],[39,42930,21884],{"className":42931},[465,220],"，由",[158,42934,21294],{"href":21293}," 得 ",[39,42937,42939,42965],{"className":42938,"translate":43},[42],[39,42940,42942],{"className":42941},[47],[49,42943,42944],{"xmlns":51},[53,42945,42946,42962],{},[56,42947,42948,42954,42956],{},[138,42949,42950,42952],{},[59,42951,4230],{},[59,42953,2098],{},[133,42955,793],{},[138,42957,42958,42960],{},[59,42959,4230],{},[133,42961,21884],{},[63,42963,42964],{"encoding":65},"V^\\pi=V^*",[39,42966,42968,43012],{"className":42967,"ariaHidden":70},[69],[39,42969,42971,42974,43003,43006,43009],{"className":42970},[74],[39,42972],{"className":42973,"style":79},[78],[39,42975,42977,42980],{"className":42976},[83],[39,42978,4230],{"className":42979,"style":461},[83,84],[39,42981,42983],{"className":42982},[192],[39,42984,42986],{"className":42985},[196],[39,42987,42989],{"className":42988},[200],[39,42990,42992],{"className":42991,"style":6410},[204],[39,42993,42994,42997],{"style":208},[39,42995],{"className":42996,"style":213},[212],[39,42998,43000],{"className":42999},[217,218,219,220],[39,43001,2098],{"className":43002,"style":2114},[83,84,220],[39,43004],{"className":43005,"style":231},[230],[39,43007,793],{"className":43008},[235],[39,43010],{"className":43011,"style":231},[230],[39,43013,43015,43018],{"className":43014},[74],[39,43016],{"className":43017,"style":21897},[78],[39,43019,43021,43024],{"className":43020},[83],[39,43022,4230],{"className":43023,"style":461},[83,84],[39,43025,43027],{"className":43026},[192],[39,43028,43030],{"className":43029},[196],[39,43031,43033],{"className":43032},[200],[39,43034,43036],{"className":43035,"style":21897},[204],[39,43037,43038,43041],{"style":208},[39,43039],{"className":43040,"style":213},[212],[39,43042,43044],{"className":43043},[217,218,219,220],[39,43045,21884],{"className":43046},[465,220],"。反之，若 ",[39,43049,43051,43076],{"className":43050,"translate":43},[42],[39,43052,43054],{"className":43053},[47],[49,43055,43056],{"xmlns":51},[53,43057,43058,43074],{},[56,43059,43060,43066,43068],{},[138,43061,43062,43064],{},[59,43063,4230],{},[59,43065,2098],{},[133,43067,793],{},[138,43069,43070,43072],{},[59,43071,4230],{},[133,43073,21884],{},[63,43075,42964],{"encoding":65},[39,43077,43079,43123],{"className":43078,"ariaHidden":70},[69],[39,43080,43082,43085,43114,43117,43120],{"className":43081},[74],[39,43083],{"className":43084,"style":79},[78],[39,43086,43088,43091],{"className":43087},[83],[39,43089,4230],{"className":43090,"style":461},[83,84],[39,43092,43094],{"className":43093},[192],[39,43095,43097],{"className":43096},[196],[39,43098,43100],{"className":43099},[200],[39,43101,43103],{"className":43102,"style":6410},[204],[39,43104,43105,43108],{"style":208},[39,43106],{"className":43107,"style":213},[212],[39,43109,43111],{"className":43110},[217,218,219,220],[39,43112,2098],{"className":43113,"style":2114},[83,84,220],[39,43115],{"className":43116,"style":231},[230],[39,43118,793],{"className":43119},[235],[39,43121],{"className":43122,"style":231},[230],[39,43124,43126,43129],{"className":43125},[74],[39,43127],{"className":43128,"style":21897},[78],[39,43130,43132,43135],{"className":43131},[83],[39,43133,4230],{"className":43134,"style":461},[83,84],[39,43136,43138],{"className":43137},[192],[39,43139,43141],{"className":43140},[196],[39,43142,43144],{"className":43143},[200],[39,43145,43147],{"className":43146,"style":21897},[204],[39,43148,43149,43152],{"style":208},[39,43150],{"className":43151,"style":213},[212],[39,43153,43155],{"className":43154},[217,218,219,220],[39,43156,21884],{"className":43157},[465,220],"，则",[39,43160,43162],{"className":43161,"translate":43},[746],[39,43163,43165,43267],{"className":43164,"translate":43},[42],[39,43166,43168],{"className":43167},[47],[49,43169,43170],{"xmlns":51,"display":755},[53,43171,43172,43264],{},[759,43173,43174,43224],{"rowspacing":761,"columnalign":762,"columnspacing":146},[764,43175,43176,43194],{},[767,43177,43178],{},[770,43179,43180],{"scriptlevel":541,"displaystyle":70},[56,43181,43182,43188],{},[138,43183,43184,43186],{},[59,43185,11772],{"mathvariant":617},[59,43187,2098],{},[138,43189,43190,43192],{},[59,43191,4230],{},[133,43193,21884],{},[767,43195,43196],{},[770,43197,43198],{"scriptlevel":541,"displaystyle":70},[56,43199,43200,43202,43204,43210,43216,43218],{},[56,43201],{},[133,43203,793],{},[138,43205,43206,43208],{},[59,43207,11772],{"mathvariant":617},[59,43209,2098],{},[138,43211,43212,43214],{},[59,43213,4230],{},[59,43215,2098],{},[133,43217,793],{},[138,43219,43220,43222],{},[59,43221,4230],{},[59,43223,2098],{},[764,43225,43226,43232],{},[767,43227,43228],{},[770,43229,43230],{"scriptlevel":541,"displaystyle":70},[56,43231],{},[767,43233,43234],{},[770,43235,43236],{"scriptlevel":541,"displaystyle":70},[56,43237,43238,43240,43242,43248,43250,43256,43262],{},[56,43239],{},[133,43241,793],{},[138,43243,43244,43246],{},[59,43245,4230],{},[133,43247,21884],{},[133,43249,793],{},[138,43251,43252,43254],{},[59,43253,11772],{"mathvariant":617},[133,43255,21884],{},[138,43257,43258,43260],{},[59,43259,4230],{},[133,43261,21884],{},[59,43263,2903],{"mathvariant":145},[63,43265,43266],{"encoding":65},"\\begin{aligned}\n\\mathcal{T}^\\pi V^*\n&=\\mathcal{T}^\\pi V^\\pi=V^\\pi\\\\\n&=V^*=\\mathcal{T}^*V^*.\n\\end{aligned}",[39,43268,43270],{"className":43269,"ariaHidden":70},[69],[39,43271,43273,43276],{"className":43272},[74],[39,43274],{"className":43275,"style":4374},[78],[39,43277,43279],{"className":43278},[83],[39,43280,43282,43379],{"className":43281},[759],[39,43283,43285],{"className":43284},[918],[39,43286,43288,43371],{"className":43287},[196,649],[39,43289,43291,43368],{"className":43290},[200],[39,43292,43294,43360],{"className":43293,"style":4393},[204],[39,43295,43296,43299],{"style":4396},[39,43297],{"className":43298,"style":935},[212],[39,43300,43302,43331],{"className":43301},[83],[39,43303,43305,43308],{"className":43304},[83],[39,43306,11772],{"className":43307,"style":11793},[83,641],[39,43309,43311],{"className":43310},[192],[39,43312,43314],{"className":43313},[196],[39,43315,43317],{"className":43316},[200],[39,43318,43320],{"className":43319,"style":4421},[204],[39,43321,43322,43325],{"style":1027},[39,43323],{"className":43324,"style":213},[212],[39,43326,43328],{"className":43327},[217,218,219,220],[39,43329,2098],{"className":43330,"style":2114},[83,84,220],[39,43332,43334,43337],{"className":43333},[83],[39,43335,4230],{"className":43336,"style":461},[83,84],[39,43338,43340],{"className":43339},[192],[39,43341,43343],{"className":43342},[196],[39,43344,43346],{"className":43345},[200],[39,43347,43349],{"className":43348,"style":22242},[204],[39,43350,43351,43354],{"style":1027},[39,43352],{"className":43353,"style":213},[212],[39,43355,43357],{"className":43356},[217,218,219,220],[39,43358,21884],{"className":43359},[465,220],[39,43361,43362,43365],{"style":4444},[39,43363],{"className":43364,"style":935},[212],[39,43366],{"className":43367},[83],[39,43369,673],{"className":43370},[672],[39,43372,43374],{"className":43373},[200],[39,43375,43377],{"className":43376,"style":4507},[204],[39,43378],{},[39,43380,43382],{"className":43381},[1223],[39,43383,43385,43629],{"className":43384},[196,649],[39,43386,43388,43626],{"className":43387},[200],[39,43389,43391,43507],{"className":43390,"style":4393},[204],[39,43392,43393,43396],{"style":4396},[39,43394],{"className":43395,"style":935},[212],[39,43397,43399,43402,43405,43408,43411,43440,43469,43472,43475,43478],{"className":43398},[83],[39,43400],{"className":43401},[83],[39,43403],{"className":43404,"style":231},[230],[39,43406,793],{"className":43407},[235],[39,43409],{"className":43410,"style":231},[230],[39,43412,43414,43417],{"className":43413},[83],[39,43415,11772],{"className":43416,"style":11793},[83,641],[39,43418,43420],{"className":43419},[192],[39,43421,43423],{"className":43422},[196],[39,43424,43426],{"className":43425},[200],[39,43427,43429],{"className":43428,"style":4421},[204],[39,43430,43431,43434],{"style":1027},[39,43432],{"className":43433,"style":213},[212],[39,43435,43437],{"className":43436},[217,218,219,220],[39,43438,2098],{"className":43439,"style":2114},[83,84,220],[39,43441,43443,43446],{"className":43442},[83],[39,43444,4230],{"className":43445,"style":461},[83,84],[39,43447,43449],{"className":43448},[192],[39,43450,43452],{"className":43451},[196],[39,43453,43455],{"className":43454},[200],[39,43456,43458],{"className":43457,"style":4421},[204],[39,43459,43460,43463],{"style":1027},[39,43461],{"className":43462,"style":213},[212],[39,43464,43466],{"className":43465},[217,218,219,220],[39,43467,2098],{"className":43468,"style":2114},[83,84,220],[39,43470],{"className":43471,"style":231},[230],[39,43473,793],{"className":43474},[235],[39,43476],{"className":43477,"style":231},[230],[39,43479,43481,43484],{"className":43480},[83],[39,43482,4230],{"className":43483,"style":461},[83,84],[39,43485,43487],{"className":43486},[192],[39,43488,43490],{"className":43489},[196],[39,43491,43493],{"className":43492},[200],[39,43494,43496],{"className":43495,"style":4421},[204],[39,43497,43498,43501],{"style":1027},[39,43499],{"className":43500,"style":213},[212],[39,43502,43504],{"className":43503},[217,218,219,220],[39,43505,2098],{"className":43506,"style":2114},[83,84,220],[39,43508,43509,43512],{"style":4444},[39,43510],{"className":43511,"style":935},[212],[39,43513,43515,43518,43521,43524,43527,43556,43559,43562,43565,43594,43623],{"className":43514},[83],[39,43516],{"className":43517},[83],[39,43519],{"className":43520,"style":231},[230],[39,43522,793],{"className":43523},[235],[39,43525],{"className":43526,"style":231},[230],[39,43528,43530,43533],{"className":43529},[83],[39,43531,4230],{"className":43532,"style":461},[83,84],[39,43534,43536],{"className":43535},[192],[39,43537,43539],{"className":43538},[196],[39,43540,43542],{"className":43541},[200],[39,43543,43545],{"className":43544,"style":22242},[204],[39,43546,43547,43550],{"style":1027},[39,43548],{"className":43549,"style":213},[212],[39,43551,43553],{"className":43552},[217,218,219,220],[39,43554,21884],{"className":43555},[465,220],[39,43557],{"className":43558,"style":231},[230],[39,43560,793],{"className":43561},[235],[39,43563],{"className":43564,"style":231},[230],[39,43566,43568,43571],{"className":43567},[83],[39,43569,11772],{"className":43570,"style":11793},[83,641],[39,43572,43574],{"className":43573},[192],[39,43575,43577],{"className":43576},[196],[39,43578,43580],{"className":43579},[200],[39,43581,43583],{"className":43582,"style":22242},[204],[39,43584,43585,43588],{"style":1027},[39,43586],{"className":43587,"style":213},[212],[39,43589,43591],{"className":43590},[217,218,219,220],[39,43592,21884],{"className":43593},[465,220],[39,43595,43597,43600],{"className":43596},[83],[39,43598,4230],{"className":43599,"style":461},[83,84],[39,43601,43603],{"className":43602},[192],[39,43604,43606],{"className":43605},[196],[39,43607,43609],{"className":43608},[200],[39,43610,43612],{"className":43611,"style":22242},[204],[39,43613,43614,43617],{"style":1027},[39,43615],{"className":43616,"style":213},[212],[39,43618,43620],{"className":43619},[217,218,219,220],[39,43621,21884],{"className":43622},[465,220],[39,43624,2903],{"className":43625},[83],[39,43627,673],{"className":43628},[672],[39,43630,43632],{"className":43631},[200],[39,43633,43635],{"className":43634,"style":4507},[204],[39,43636],{},[11,43638,20571,43639,40062,43748,43857,43858,43886,43887,43915,43916,21228],{},[39,43640,43642,43672],{"className":43641,"translate":43},[42],[39,43643,43645],{"className":43644},[47],[49,43646,43647],{"xmlns":51},[53,43648,43649,43669],{},[56,43650,43651,43657,43663,43665,43667],{},[138,43652,43653,43655],{},[59,43654,11772],{"mathvariant":617},[59,43656,2098],{},[138,43658,43659,43661],{},[59,43660,4230],{},[133,43662,21884],{},[133,43664,136],{"stretchy":135},[59,43666,142],{},[133,43668,161],{"stretchy":135},[63,43670,43671],{"encoding":65},"\\mathcal{T}^\\pi V^*(s)",[39,43673,43675],{"className":43674,"ariaHidden":70},[69],[39,43676,43678,43681,43710,43739,43742,43745],{"className":43677},[74],[39,43679],{"className":43680,"style":245},[78],[39,43682,43684,43687],{"className":43683},[83],[39,43685,11772],{"className":43686,"style":11793},[83,641],[39,43688,43690],{"className":43689},[192],[39,43691,43693],{"className":43692},[196],[39,43694,43696],{"className":43695},[200],[39,43697,43699],{"className":43698,"style":6410},[204],[39,43700,43701,43704],{"style":208},[39,43702],{"className":43703,"style":213},[212],[39,43705,43707],{"className":43706},[217,218,219,220],[39,43708,2098],{"className":43709,"style":2114},[83,84,220],[39,43711,43713,43716],{"className":43712},[83],[39,43714,4230],{"className":43715,"style":461},[83,84],[39,43717,43719],{"className":43718},[192],[39,43720,43722],{"className":43721},[196],[39,43723,43725],{"className":43724},[200],[39,43726,43728],{"className":43727,"style":21897},[204],[39,43729,43730,43733],{"style":208},[39,43731],{"className":43732,"style":213},[212],[39,43734,43736],{"className":43735},[217,218,219,220],[39,43737,21884],{"className":43738},[465,220],[39,43740,136],{"className":43741},[182],[39,43743,142],{"className":43744},[83,84],[39,43746,161],{"className":43747},[263],[39,43749,43751,43781],{"className":43750,"translate":43},[42],[39,43752,43754],{"className":43753},[47],[49,43755,43756],{"xmlns":51},[53,43757,43758,43778],{},[56,43759,43760,43766,43772,43774,43776],{},[138,43761,43762,43764],{},[59,43763,11772],{"mathvariant":617},[133,43765,21884],{},[138,43767,43768,43770],{},[59,43769,4230],{},[133,43771,21884],{},[133,43773,136],{"stretchy":135},[59,43775,142],{},[133,43777,161],{"stretchy":135},[63,43779,43780],{"encoding":65},"\\mathcal{T}^*V^*(s)",[39,43782,43784],{"className":43783,"ariaHidden":70},[69],[39,43785,43787,43790,43819,43848,43851,43854],{"className":43786},[74],[39,43788],{"className":43789,"style":245},[78],[39,43791,43793,43796],{"className":43792},[83],[39,43794,11772],{"className":43795,"style":11793},[83,641],[39,43797,43799],{"className":43798},[192],[39,43800,43802],{"className":43801},[196],[39,43803,43805],{"className":43804},[200],[39,43806,43808],{"className":43807,"style":21897},[204],[39,43809,43810,43813],{"style":208},[39,43811],{"className":43812,"style":213},[212],[39,43814,43816],{"className":43815},[217,218,219,220],[39,43817,21884],{"className":43818},[465,220],[39,43820,43822,43825],{"className":43821},[83],[39,43823,4230],{"className":43824,"style":461},[83,84],[39,43826,43828],{"className":43827},[192],[39,43829,43831],{"className":43830},[196],[39,43832,43834],{"className":43833},[200],[39,43835,43837],{"className":43836,"style":21897},[204],[39,43838,43839,43842],{"style":208},[39,43840],{"className":43841,"style":213},[212],[39,43843,43845],{"className":43844},[217,218,219,220],[39,43846,21884],{"className":43847},[465,220],[39,43849,136],{"className":43850},[182],[39,43852,142],{"className":43853},[83,84],[39,43855,161],{"className":43856},[263]," 是这些动作值的最大值，等号成立当且仅当 ",[39,43859,43861,43874],{"className":43860,"translate":43},[42],[39,43862,43864],{"className":43863},[47],[49,43865,43866],{"xmlns":51},[53,43867,43868,43872],{},[56,43869,43870],{},[59,43871,2098],{},[63,43873,2101],{"encoding":65},[39,43875,43877],{"className":43876,"ariaHidden":70},[69],[39,43878,43880,43883],{"className":43879},[74],[39,43881],{"className":43882,"style":290},[78],[39,43884,2098],{"className":43885,"style":2114},[83,84]," 在该状态以概率 ",[39,43888,43890,43903],{"className":43889,"translate":43},[42],[39,43891,43893],{"className":43892},[47],[49,43894,43895],{"xmlns":51},[53,43896,43897,43901],{},[56,43898,43899],{},[539,43900,546],{},[63,43902,546],{"encoding":65},[39,43904,43906],{"className":43905,"ariaHidden":70},[69],[39,43907,43909,43912],{"className":43908},[74],[39,43910],{"className":43911,"style":2473},[78],[39,43913,546],{"className":43914},[83]," 选择取得最大值的动作。对确定性策略，这正是 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",[39,44052,44054,44071],{"className":44053,"translate":43},[42],[39,44055,44057],{"className":44056},[47],[49,44058,44059],{"xmlns":51},[53,44060,44061,44069],{},[56,44062,44063],{},[138,44064,44065,44067],{},[59,44066,11772],{"mathvariant":617},[133,44068,21884],{},[63,44070,21887],{"encoding":65},[39,44072,44074],{"className":44073,"ariaHidden":70},[69],[39,44075,44077,44080],{"className":44076},[74],[39,44078],{"className":44079,"style":21897},[78],[39,44081,44083,44086],{"className":44082},[83],[39,44084,11772],{"className":44085,"style":11793},[83,641],[39,44087,44089],{"className":44088},[192],[39,44090,44092],{"className":44091},[196],[39,44093,44095],{"className":44094},[200],[39,44096,44098],{"className":44097,"style":21897},[204],[39,44099,44100,44103],{"style":208},[39,44101],{"className":44102,"style":213},[212],[39,44104,44106],{"className":44105},[217,218,219,220],[39,44107,21884],{"className":44108},[465,220]," 的不动点。另一方面，关于该不动点的贪婪策略恰好达到它。两个方向合起来，既证明了最优方程，也证明了最优策略的存在性。",[11,44111,16423,44112,1639,44114,44173,44174,44202,44203,44231],{},[158,44113,37252],{"href":37251},[39,44115,44117,44135],{"className":44116,"translate":43},[42],[39,44118,44120],{"className":44119},[47],[49,44121,44122],{"xmlns":51},[53,44123,44124,44132],{},[56,44125,44126],{},[138,44127,44128,44130],{},[59,44129,4295],{},[133,44131,21884],{},[63,44133,44134],{"encoding":65},"Q^*",[39,44136,44138],{"className":44137,"ariaHidden":70},[69],[39,44139,44141,44144],{"className":44140},[74],[39,44142],{"className":44143,"style":30556},[78],[39,44145,44147,44150],{"className":44146},[83],[39,44148,4295],{"className":44149},[83,84],[39,44151,44153],{"className":44152},[192],[39,44154,44156],{"className":44155},[196],[39,44157,44159],{"className":44158},[200],[39,44160,44162],{"className":44161,"style":21897},[204],[39,44163,44164,44167],{"style":208},[39,44165],{"className":44166,"style":213},[212],[39,44168,44170],{"className":44169},[217,218,219,220],[39,44171,21884],{"className":44172},[465,220]," 的定义，还可以得到最优动作值函数的表达式。若在状态 ",[39,44175,44177,44190],{"className":44176,"translate":43},[42],[39,44178,44180],{"className":44179},[47],[49,44181,44182],{"xmlns":51},[53,44183,44184,44188],{},[56,44185,44186],{},[59,44187,142],{},[63,44189,142],{"encoding":65},[39,44191,44193],{"className":44192,"ariaHidden":70},[69],[39,44194,44196,44199],{"className":44195},[74],[39,44197],{"className":44198,"style":290},[78],[39,44200,142],{"className":44201},[83,84]," 先固定动作 ",[39,44204,44206,44219],{"className":44205,"translate":43},[42],[39,44207,44209],{"className":44208},[47],[49,44210,44211],{"xmlns":51},[53,44212,44213,44217],{},[56,44214,44215],{},[59,44216,158],{},[63,44218,158],{"encoding":65},[39,44220,44222],{"className":44221,"ariaHidden":70},[69],[39,44223,44225,44228],{"className":44224},[74],[39,44226],{"className":44227,"style":290},[78],[39,44229,158],{"className":44230},[83,84],"，此后采用最优策略，则",[39,44233,44235],{"className":44234,"translate":43},[746],[39,44236,44238,44420],{"className":44237,"translate":43},[42],[39,44239,44241],{"className":44240},[47],[49,44242,44243],{"xmlns":51,"display":755},[53,44244,44245,44417],{},[759,44246,44247,44359],{"rowspacing":761,"columnalign":762,"columnspacing":146},[764,44248,44249,44271],{},[767,44250,44251],{},[770,44252,44253],{"scriptlevel":541,"displaystyle":70},[56,44254,44255,44261,44263,44265,44267,44269],{},[138,44256,44257,44259],{},[59,44258,4295],{},[133,44260,21884],{},[133,44262,136],{"stretchy":135},[59,44264,142],{},[133,44266,155],{"separator":70},[59,44268,158],{},[133,44270,161],{"stretchy":135},[767,44272,44273],{},[770,44274,44275],{"scriptlevel":541,"displaystyle":70},[56,44276,44277,44279,44281,44291,44293,44295,44301,44303,44305,44307,44309,44311,44357],{},[56,44278],{},[133,44280,793],{},[6556,44282,44283,44285],{},[133,44284,3191],{},[138,44286,44287,44289],{},[59,44288,142],{},[133,44290,147],{"mathvariant":145,"lspace":146,"rspace":146},[59,44292,131],{},[133,44294,136],{"stretchy":135},[138,44296,44297,44299],{},[59,44298,142],{},[133,44300,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,44302,150],{},[59,44304,142],{},[133,44306,155],{"separator":70},[59,44308,158],{},[133,44310,161],{"stretchy":135},[56,44312,44313,44315,44317,44319,44321,44323,44325,44327,44333,44335,44337,44339,44345,44347,44353,44355],{},[133,44314,537],{"fence":70},[59,44316,401],{},[133,44318,136],{"stretchy":135},[59,44320,142],{},[133,44322,155],{"separator":70},[59,44324,158],{},[133,44326,155],{"separator":70},[138,44328,44329,44331],{},[59,44330,142],{},[133,44332,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,44334,161],{"stretchy":135},[133,44336,788],{},[59,44338,531],{},[138,44340,44341,44343],{},[59,44342,4230],{},[133,44344,21884],{},[133,44346,136],{"stretchy":135},[138,44348,44349,44351],{},[59,44350,142],{},[133,44352,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,44354,161],{"stretchy":135},[133,44356,4280],{"fence":70},[133,44358,155],{"separator":70},[764,44360,44361,44379],{},[767,44362,44363],{},[770,44364,44365],{"scriptlevel":541,"displaystyle":70},[56,44366,44367,44373,44375,44377],{},[138,44368,44369,44371],{},[59,44370,4230],{},[133,44372,21884],{},[133,44374,136],{"stretchy":135},[59,44376,142],{},[133,44378,161],{"stretchy":135},[767,44380,44381],{},[770,44382,44383],{"scriptlevel":541,"displaystyle":70},[56,44384,44385,44387,44389,44399,44405,44407,44409,44411,44413,44415],{},[56,44386],{},[133,44388,793],{},[6556,44390,44391,44397],{},[56,44392,44393,44395],{},[59,44394,1687],{},[133,44396,1690],{},[59,44398,158],{},[138,44400,44401,44403],{},[59,44402,4295],{},[133,44404,21884],{},[133,44406,136],{"stretchy":135},[59,44408,142],{},[133,44410,155],{"separator":70},[59,44412,158],{},[133,44414,161],{"stretchy":135},[59,44416,2903],{"mathvariant":145},[63,44418,44419],{"encoding":65},"\\begin{aligned}\nQ^*(s,a)&=\\sum_{s'}P(s'\\mid s,a)\n\\left[r(s,a,s')+\\gamma V^*(s')\\right],\\\\\nV^*(s)&=\\max_{a}Q^*(s,a).\n\\end{aligned}",[39,44421,44423],{"className":44422,"ariaHidden":70},[69],[39,44424,44426,44430],{"className":44425},[74],[39,44427],{"className":44428,"style":44429},[78],"height:4.184em;vertical-align:-1.842em;",[39,44431,44433],{"className":44432},[83],[39,44434,44436,44564],{"className":44435},[759],[39,44437,44439],{"className":44438},[918],[39,44440,44442,44555],{"className":44441},[196,649],[39,44443,44445,44552],{"className":44444},[200],[39,44446,44449,44505],{"className":44447,"style":44448},[204],"height:2.342em;",[39,44450,44452,44455],{"style":44451},"top:-4.342em;",[39,44453],{"className":44454,"style":3311},[212],[39,44456,44458,44487,44490,44493,44496,44499,44502],{"className":44457},[83],[39,44459,44461,44464],{"className":44460},[83],[39,44462,4295],{"className":44463},[83,84],[39,44465,44467],{"className":44466},[192],[39,44468,44470],{"className":44469},[196],[39,44471,44473],{"className":44472},[200],[39,44474,44476],{"className":44475,"style":22242},[204],[39,44477,44478,44481],{"style":1027},[39,44479],{"className":44480,"style":213},[212],[39,44482,44484],{"className":44483},[217,218,219,220],[39,44485,21884],{"className":44486},[465,220],[39,44488,136],{"className":44489},[182],[39,44491,142],{"className":44492},[83,84],[39,44494,155],{"className":44495},[252],[39,44497],{"className":44498,"style":256},[230],[39,44500,158],{"className":44501},[83,84],[39,44503,161],{"className":44504},[263],[39,44506,44508,44511],{"style":44507},"top:-1.908em;",[39,44509],{"className":44510,"style":3311},[212],[39,44512,44514,44543,44546,44549],{"className":44513},[83],[39,44515,44517,44520],{"className":44516},[83],[39,44518,4230],{"className":44519,"style":461},[83,84],[39,44521,44523],{"className":44522},[192],[39,44524,44526],{"className":44525},[196],[39,44527,44529],{"className":44528},[200],[39,44530,44532],{"className":44531,"style":22242},[204],[39,44533,44534,44537],{"style":1027},[39,44535],{"className":44536,"style":213},[212],[39,44538,44540],{"className":44539},[217,218,219,220],[39,44541,21884],{"className":44542},[465,220],[39,44544,136],{"className":44545},[182],[39,44547,142],{"className":44548},[83,84],[39,44550,161],{"className":44551},[263],[39,44553,673],{"className":44554},[672],[39,44556,44558],{"className":44557},[200],[39,44559,44562],{"className":44560,"style":44561},[204],"height:1.842em;",[39,44563],{},[39,44565,44567],{"className":44566},[1223],[39,44568,44570,45016],{"className":44569},[196,649],[39,44571,44573,45013],{"className":44572},[200],[39,44574,44576,44893],{"className":44575,"style":44448},[204],[39,44577,44578,44581],{"style":44451},[39,44579],{"className":44580,"style":3311},[212],[39,44582,44584,44587,44590,44593,44596,44672,44675,44678,44681,44713,44716,44719,44722,44725,44728,44731,44734,44737,44740,44887,44890],{"className":44583},[83],[39,44585],{"className":44586},[83],[39,44588],{"className":44589,"style":231},[230],[39,44591,793],{"className":44592},[235],[39,44594],{"className":44595,"style":231},[230],[39,44597,44599],{"className":44598},[1744,3294],[39,44600,44602,44664],{"className":44601},[196,649],[39,44603,44605,44661],{"className":44604},[200],[39,44606,44608,44651],{"className":44607,"style":6670},[204],[39,44609,44610,44613],{"style":7097},[39,44611],{"className":44612,"style":3311},[212],[39,44614,44616],{"className":44615},[217,218,219,220],[39,44617,44619],{"className":44618},[83,220],[39,44620,44622,44625],{"className":44621},[83,220],[39,44623,142],{"className":44624},[83,84,220],[39,44626,44628],{"className":44627},[192],[39,44629,44631],{"className":44630},[196],[39,44632,44634],{"className":44633},[200],[39,44635,44637],{"className":44636,"style":3997},[204],[39,44638,44639,44642],{"style":4000},[39,44640],{"className":44641,"style":4004},[212],[39,44643,44645],{"className":44644},[217,4008,4009,220],[39,44646,44648],{"className":44647},[83,220],[39,44649,147],{"className":44650},[83,220],[39,44652,44653,44656],{"style":3330},[39,44654],{"className":44655,"style":3311},[212],[39,44657,44658],{},[39,44659,3191],{"className":44660},[1744,3339,3340],[39,44662,673],{"className":44663},[672],[39,44665,44667],{"className":44666},[200],[39,44668,44670],{"className":44669,"style":7158},[204],[39,44671],{},[39,44673],{"className":44674,"style":256},[230],[39,44676,131],{"className":44677,"style":178},[83,84],[39,44679,136],{"className":44680},[182],[39,44682,44684,44687],{"className":44683},[83],[39,44685,142],{"className":44686},[83,84],[39,44688,44690],{"className":44689},[192],[39,44691,44693],{"className":44692},[196],[39,44694,44696],{"className":44695},[200],[39,44697,44699],{"className":44698,"style":1024},[204],[39,44700,44701,44704],{"style":1027},[39,44702],{"className":44703,"style":213},[212],[39,44705,44707],{"className":44706},[217,218,219,220],[39,44708,44710],{"className":44709},[83,220],[39,44711,147],{"className":44712},[83,220],[39,44714],{"className":44715,"style":231},[230],[39,44717,150],{"className":44718},[235],[39,44720],{"className":44721,"style":231},[230],[39,44723,142],{"className":44724},[83,84],[39,44726,155],{"className":44727},[252],[39,44729],{"className":44730,"style":256},[230],[39,44732,158],{"className":44733},[83,84],[39,44735,161],{"className":44736},[263],[39,44738],{"className":44739,"style":256},[230],[39,44741,44743,44746,44749,44752,44755,44758,44761,44764,44767,44770,44802,44805,44808,44811,44814,44817,44846,44849,44881,44884],{"className":44742},[2842],[39,44744,537],{"className":44745,"style":4592},[182,4591],[39,44747,401],{"className":44748,"style":438},[83,84],[39,44750,136],{"className":44751},[182],[39,44753,142],{"className":44754},[83,84],[39,44756,155],{"className":44757},[252],[39,44759],{"className":44760,"style":256},[230],[39,44762,158],{"className":44763},[83,84],[39,44765,155],{"className":44766},[252],[39,44768],{"className":44769,"style":256},[230],[39,44771,44773,44776],{"className":44772},[83],[39,44774,142],{"className":44775},[83,84],[39,44777,44779],{"className":44778},[192],[39,44780,44782],{"className":44781},[196],[39,44783,44785],{"className":44784},[200],[39,44786,44788],{"className":44787,"style":1024},[204],[39,44789,44790,44793],{"style":1027},[39,44791],{"className":44792,"style":213},[212],[39,44794,44796],{"className":44795},[217,218,219,220],[39,44797,44799],{"className":44798},[83,220],[39,44800,147],{"className":44801},[83,220],[39,44803,161],{"className":44804},[263],[39,44806],{"className":44807,"style":461},[230],[39,44809,788],{"className":44810},[465],[39,44812],{"className":44813,"style":461},[230],[39,44815,531],{"className":44816,"style":565},[83,84],[39,44818,44820,44823],{"className":44819},[83],[39,44821,4230],{"className":44822,"style":461},[83,84],[39,44824,44826],{"className":44825},[192],[39,44827,44829],{"className":44828},[196],[39,44830,44832],{"className":44831},[200],[39,44833,44835],{"className":44834,"style":22242},[204],[39,44836,44837,44840],{"style":1027},[39,44838],{"className":44839,"style":213},[212],[39,44841,44843],{"className":44842},[217,218,219,220],[39,44844,21884],{"className":44845},[465,220],[39,44847,136],{"className":44848},[182],[39,44850,44852,44855],{"className":44851},[83],[39,44853,142],{"className":44854},[83,84],[39,44856,44858],{"className":44857},[192],[39,44859,44861],{"className":44860},[196],[39,44862,44864],{"className":44863},[200],[39,44865,44867],{"className":44866,"style":1024},[204],[39,44868,44869,44872],{"style":1027},[39,44870],{"className":44871,"style":213},[212],[39,44873,44875],{"className":44874},[217,218,219,220],[39,44876,44878],{"className":44877},[83,220],[39,44879,147],{"className":44880},[83,220],[39,44882,161],{"className":44883},[263],[39,44885,4280],{"className":44886,"style":4592},[263,4591],[39,44888],{"className":44889,"style":256},[230],[39,44891,155],{"className":44892},[252],[39,44894,44895,44898],{"style":44507},[39,44896],{"className":44897,"style":3311},[212],[39,44899,44901,44904,44907,44910,44913,44960,44963,44992,44995,44998,45001,45004,45007,45010],{"className":44900},[83],[39,44902],{"className":44903},[83],[39,44905],{"className":44906,"style":231},[230],[39,44908,793],{"className":44909},[235],[39,44911],{"className":44912,"style":231},[230],[39,44914,44916],{"className":44915},[1744,3294],[39,44917,44919,44952],{"className":44918},[196,649],[39,44920,44922,44949],{"className":44921},[200],[39,44923,44925,44939],{"className":44924,"style":290},[204],[39,44926,44927,44930],{"style":22299},[39,44928],{"className":44929,"style":935},[212],[39,44931,44933],{"className":44932},[217,218,219,220],[39,44934,44936],{"className":44935},[83,220],[39,44937,158],{"className":44938},[83,84,220],[39,44940,44941,44944],{"style":21599},[39,44942],{"className":44943,"style":935},[212],[39,44945,44946],{},[39,44947,1687],{"className":44948},[1744],[39,44950,673],{"className":44951},[672],[39,44953,44955],{"className":44954},[200],[39,44956,44958],{"className":44957,"style":22331},[204],[39,44959],{},[39,44961],{"className":44962,"style":256},[230],[39,44964,44966,44969],{"className":44965},[83],[39,44967,4295],{"className":44968},[83,84],[39,44970,44972],{"className":44971},[192],[39,44973,44975],{"className":44974},[196],[39,44976,44978],{"className":44977},[200],[39,44979,44981],{"className":44980,"style":22242},[204],[39,44982,44983,44986],{"style":1027},[39,44984],{"className":44985,"style":213},[212],[39,44987,44989],{"className":44988},[217,218,219,220],[39,44990,21884],{"className":44991},[465,220],[39,44993,136],{"className":44994},[182],[39,44996,142],{"className":44997},[83,84],[39,44999,155],{"className":45000},[252],[39,45002],{"className":45003,"style":256},[230],[39,45005,158],{"className":45006},[83,84],[39,45008,161],{"className":45009},[263],[39,45011,2903],{"className":45012},[83],[39,45014,673],{"className":45015},[672],[39,45017,45019],{"className":45018},[200],[39,45020,45022],{"className":45021,"style":44561},[204],[39,45023],{},[11,45025,45026,45027,45138,45139,45226,45227,1639,45285,45343],{},"第一式的上界由 ",[39,45028,45030,45056],{"className":45029,"translate":43},[42],[39,45031,45033],{"className":45032},[47],[49,45034,45035],{"xmlns":51},[53,45036,45037,45053],{},[56,45038,45039,45045,45047],{},[138,45040,45041,45043],{},[59,45042,4230],{},[59,45044,2098],{},[133,45046,1824],{},[138,45048,45049,45051],{},[59,45050,4230],{},[133,45052,21884],{},[63,45054,45055],{"encoding":65},"V^\\pi\\le V^*",[39,45057,45059,45103],{"className":45058,"ariaHidden":70},[69],[39,45060,45062,45065,45094,45097,45100],{"className":45061},[74],[39,45063],{"className":45064,"style":23796},[78],[39,45066,45068,45071],{"className":45067},[83],[39,45069,4230],{"className":45070,"style":461},[83,84],[39,45072,45074],{"className":45073},[192],[39,45075,45077],{"className":45076},[196],[39,45078,45080],{"className":45079},[200],[39,45081,45083],{"className":45082,"style":6410},[204],[39,45084,45085,45088],{"style":208},[39,45086],{"className":45087,"style":213},[212],[39,45089,45091],{"className":45090},[217,218,219,220],[39,45092,2098],{"className":45093,"style":2114},[83,84,220],[39,45095],{"className":45096,"style":231},[230],[39,45098,1824],{"className":45099},[235],[39,45101],{"className":45102,"style":231},[230],[39,45104,45106,45109],{"className":45105},[74],[39,45107],{"className":45108,"style":21897},[78],[39,45110,45112,45115],{"className":45111},[83],[39,45113,4230],{"className":45114,"style":461},[83,84],[39,45116,45118],{"className":45117},[192],[39,45119,45121],{"className":45120},[196],[39,45122,45124],{"className":45123},[200],[39,45125,45127],{"className":45126,"style":21897},[204],[39,45128,45129,45132],{"style":208},[39,45130],{"className":45131,"style":213},[212],[39,45133,45135],{"className":45134},[217,218,219,220],[39,45136,21884],{"className":45137},[465,220]," 得到，下界由在后续状态选取最优续策略达到。第二式说明，最优动作正是使 ",[39,45140,45142,45170],{"className":45141,"translate":43},[42],[39,45143,45145],{"className":45144},[47],[49,45146,45147],{"xmlns":51},[53,45148,45149,45167],{},[56,45150,45151,45157,45159,45161,45163,45165],{},[138,45152,45153,45155],{},[59,45154,4295],{},[133,45156,21884],{},[133,45158,136],{"stretchy":135},[59,45160,142],{},[133,45162,155],{"separator":70},[59,45164,158],{},[133,45166,161],{"stretchy":135},[63,45168,45169],{"encoding":65},"Q^*(s,a)",[39,45171,45173],{"className":45172,"ariaHidden":70},[69],[39,45174,45176,45179,45208,45211,45214,45217,45220,45223],{"className":45175},[74],[39,45177],{"className":45178,"style":245},[78],[39,45180,45182,45185],{"className":45181},[83],[39,45183,4295],{"className":45184},[83,84],[39,45186,45188],{"className":45187},[192],[39,45189,45191],{"className":45190},[196],[39,45192,45194],{"className":45193},[200],[39,45195,45197],{"className":45196,"style":21897},[204],[39,45198,45199,45202],{"style":208},[39,45200],{"className":45201,"style":213},[212],[39,45203,45205],{"className":45204},[217,218,219,220],[39,45206,21884],{"className":45207},[465,220],[39,45209,136],{"className":45210},[182],[39,45212,142],{"className":45213},[83,84],[39,45215,155],{"className":45216},[252],[39,45218],{"className":45219,"style":256},[230],[39,45221,158],{"className":45222},[83,84],[39,45224,161],{"className":45225},[263]," 最大的动作。因此，",[39,45228,45230,45247],{"className":45229,"translate":43},[42],[39,45231,45233],{"className":45232},[47],[49,45234,45235],{"xmlns":51},[53,45236,45237,45245],{},[56,45238,45239],{},[138,45240,45241,45243],{},[59,45242,4295],{},[133,45244,21884],{},[63,45246,44134],{"encoding":65},[39,45248,45250],{"className":45249,"ariaHidden":70},[69],[39,45251,45253,45256],{"className":45252},[74],[39,45254],{"className":45255,"style":30556},[78],[39,45257,45259,45262],{"className":45258},[83],[39,45260,4295],{"className":45261},[83,84],[39,45263,45265],{"className":45264},[192],[39,45266,45268],{"className":45267},[196],[39,45269,45271],{"className":45270},[200],[39,45272,45274],{"className":45273,"style":21897},[204],[39,45275,45276,45279],{"style":208},[39,45277],{"className":45278,"style":213},[212],[39,45280,45282],{"className":45281},[217,218,219,220],[39,45283,21884],{"className":45284},[465,220],[39,45286,45288,45305],{"className":45287,"translate":43},[42],[39,45289,45291],{"className":45290},[47],[49,45292,45293],{"xmlns":51},[53,45294,45295,45303],{},[56,45296,45297],{},[138,45298,45299,45301],{},[59,45300,4230],{},[133,45302,21884],{},[63,45304,34720],{"encoding":65},[39,45306,45308],{"className":45307,"ariaHidden":70},[69],[39,45309,45311,45314],{"className":45310},[74],[39,45312],{"className":45313,"style":21897},[78],[39,45315,45317,45320],{"className":45316},[83],[39,45318,4230],{"className":45319,"style":461},[83,84],[39,45321,45323],{"className":45322},[192],[39,45324,45326],{"className":45325},[196],[39,45327,45329],{"className":45328},[200],[39,45330,45332],{"className":45331,"style":21897},[204],[39,45333,45334,45337],{"style":208},[39,45335],{"className":45336,"style":213},[212],[39,45338,45340],{"className":45339},[217,218,219,220],[39,45341,21884],{"className":45342},[465,220]," 的差别只在于第一步是否已经固定动作。",[26,45345,45347],{"id":45346},"_43-max-与期望不可交换","4.3 max 与期望不可交换",[11,45349,45350,45351,45434,45435,45517],{},"最优贝尔曼方程中是 ",[39,45352,45354,45378],{"className":45353,"translate":43},[42],[39,45355,45357],{"className":45356},[47],[49,45358,45359],{"xmlns":51},[53,45360,45361,45375],{},[56,45362,45363,45373],{},[613,45364,45365,45371],{},[56,45366,45367,45369],{},[59,45368,1687],{},[133,45370,1690],{},[59,45372,158],{},[59,45374,4253],{"mathvariant":421},[63,45376,45377],{"encoding":65},"\\max_a\\mathbb{E}",[39,45379,45381],{"className":45380,"ariaHidden":70},[69],[39,45382,45384,45388,45428,45431],{"className":45383},[74],[39,45385],{"className":45386,"style":45387},[78],"height:0.8389em;vertical-align:-0.15em;",[39,45389,45391,45394],{"className":45390},[1744],[39,45392,1687],{"className":45393},[1744],[39,45395,45397],{"className":45396},[192],[39,45398,45400,45420],{"className":45399},[196,649],[39,45401,45403,45417],{"className":45402},[200],[39,45404,45406],{"className":45405,"style":1728},[204],[39,45407,45408,45411],{"style":44021},[39,45409],{"className":45410,"style":213},[212],[39,45412,45414],{"className":45413},[217,218,219,220],[39,45415,158],{"className":45416},[83,84,220],[39,45418,673],{"className":45419},[672],[39,45421,45423],{"className":45422},[200],[39,45424,45426],{"className":45425,"style":680},[204],[39,45427],{},[39,45429],{"className":45430,"style":256},[230],[39,45432,4253],{"className":45433},[83,515],"，而不是 ",[39,45436,45438,45462],{"className":45437,"translate":43},[42],[39,45439,45441],{"className":45440},[47],[49,45442,45443],{"xmlns":51},[53,45444,45445,45459],{},[56,45446,45447,45449],{},[59,45448,4253],{"mathvariant":421},[613,45450,45451,45457],{},[56,45452,45453,45455],{},[59,45454,1687],{},[133,45456,1690],{},[59,45458,158],{},[63,45460,45461],{"encoding":65},"\\mathbb{E}\\max_a",[39,45463,45465],{"className":45464,"ariaHidden":70},[69],[39,45466,45468,45471,45474,45477],{"className":45467},[74],[39,45469],{"className":45470,"style":45387},[78],[39,45472,4253],{"className":45473},[83,515],[39,45475],{"className":45476,"style":256},[230],[39,45478,45480,45483],{"className":45479},[1744],[39,45481,1687],{"className":45482},[1744],[39,45484,45486],{"className":45485},[192],[39,45487,45489,45509],{"className":45488},[196,649],[39,45490,45492,45506],{"className":45491},[200],[39,45493,45495],{"className":45494,"style":1728},[204],[39,45496,45497,45500],{"style":44021},[39,45498],{"className":45499,"style":213},[212],[39,45501,45503],{"className":45502},[217,218,219,220],[39,45504,158],{"className":45505},[83,84,220],[39,45507,673],{"className":45508},[672],[39,45510,45512],{"className":45511},[200],[39,45513,45515],{"className":45514,"style":680},[204],[39,45516],{},"。这两个量一般不相等，原因在于动作必须在观察到下一状态之前选定。",[11,45519,45520,45521,45549,45550,45674,45675,45709,45710,45780,45781,45709,45813,45883],{},"考虑一个状态 ",[39,45522,45524,45537],{"className":45523,"translate":43},[42],[39,45525,45527],{"className":45526},[47],[49,45528,45529],{"xmlns":51},[53,45530,45531,45535],{},[56,45532,45533],{},[59,45534,142],{},[63,45536,142],{"encoding":65},[39,45538,45540],{"className":45539,"ariaHidden":70},[69],[39,45541,45543,45546],{"className":45542},[74],[39,45544],{"className":45545,"style":290},[78],[39,45547,142],{"className":45548},[83,84],"，两个动作 ",[39,45551,45553,45579],{"className":45552,"translate":43},[42],[39,45554,45556],{"className":45555},[47],[49,45557,45558],{"xmlns":51},[53,45559,45560,45576],{},[56,45561,45562,45568,45570],{},[613,45563,45564,45566],{},[59,45565,158],{},[539,45567,546],{},[133,45569,155],{"separator":70},[613,45571,45572,45574],{},[59,45573,158],{},[539,45575,17359],{},[63,45577,45578],{"encoding":65},"a_1,a_2",[39,45580,45582],{"className":45581,"ariaHidden":70},[69],[39,45583,45585,45588,45628,45631,45634],{"className":45584},[74],[39,45586],{"className":45587,"style":2654},[78],[39,45589,45591,45594],{"className":45590},[83],[39,45592,158],{"className":45593},[83,84],[39,45595,45597],{"className":45596},[192],[39,45598,45600,45620],{"className":45599},[196,649],[39,45601,45603,45617],{"className":45602},[200],[39,45604,45606],{"className":45605,"style":963},[204],[39,45607,45608,45611],{"style":966},[39,45609],{"className":45610,"style":213},[212],[39,45612,45614],{"className":45613},[217,218,219,220],[39,45615,546],{"className":45616},[83,220],[39,45618,673],{"className":45619},[672],[39,45621,45623],{"className":45622},[200],[39,45624,45626],{"className":45625,"style":680},[204],[39,45627],{},[39,45629,155],{"className":45630},[252],[39,45632],{"className":45633,"style":256},[230],[39,45635,45637,45640],{"className":45636},[83],[39,45638,158],{"className":45639},[83,84],[39,45641,45643],{"className":45642},[192],[39,45644,45646,45666],{"className":45645},[196,649],[39,45647,45649,45663],{"className":45648},[200],[39,45650,45652],{"className":45651,"style":963},[204],[39,45653,45654,45657],{"style":966},[39,45655],{"className":45656,"style":213},[212],[39,45658,45660],{"className":45659},[217,218,219,220],[39,45661,17359],{"className":45662},[83,220],[39,45664,673],{"className":45665},[672],[39,45667,45669],{"className":45668},[200],[39,45670,45672],{"className":45671,"style":680},[204],[39,45673],{},"。下一状态以 ",[39,45676,45678,45697],{"className":45677,"translate":43},[42],[39,45679,45681],{"className":45680},[47],[49,45682,45683],{"xmlns":51},[53,45684,45685,45694],{},[56,45686,45687,45689,45692],{},[539,45688,546],{},[59,45690,45691],{"mathvariant":145},"\u002F",[539,45693,17359],{},[63,45695,45696],{"encoding":65},"1\u002F2",[39,45698,45700],{"className":45699,"ariaHidden":70},[69],[39,45701,45703,45706],{"className":45702},[74],[39,45704],{"className":45705,"style":245},[78],[39,45707,45696],{"className":45708},[83]," 的概率为 ",[39,45711,45713,45731],{"className":45712,"translate":43},[42],[39,45714,45716],{"className":45715},[47],[49,45717,45718],{"xmlns":51},[53,45719,45720,45728],{},[56,45721,45722],{},[613,45723,45724,45726],{},[59,45725,142],{},[539,45727,546],{},[63,45729,45730],{"encoding":65},"s_1",[39,45732,45734],{"className":45733,"ariaHidden":70},[69],[39,45735,45737,45740],{"className":45736},[74],[39,45738],{"className":45739,"style":2916},[78],[39,45741,45743,45746],{"className":45742},[83],[39,45744,142],{"className":45745},[83,84],[39,45747,45749],{"className":45748},[192],[39,45750,45752,45772],{"className":45751},[196,649],[39,45753,45755,45769],{"className":45754},[200],[39,45756,45758],{"className":45757,"style":963},[204],[39,45759,45760,45763],{"style":966},[39,45761],{"className":45762,"style":213},[212],[39,45764,45766],{"className":45765},[217,218,219,220],[39,45767,546],{"className":45768},[83,220],[39,45770,673],{"className":45771},[672],[39,45773,45775],{"className":45774},[200],[39,45776,45778],{"className":45777,"style":680},[204],[39,45779],{},"，以 ",[39,45782,45784,45801],{"className":45783,"translate":43},[42],[39,45785,45787],{"className":45786},[47],[49,45788,45789],{"xmlns":51},[53,45790,45791,45799],{},[56,45792,45793,45795,45797],{},[539,45794,546],{},[59,45796,45691],{"mathvariant":145},[539,45798,17359],{},[63,45800,45696],{"encoding":65},[39,45802,45804],{"className":45803,"ariaHidden":70},[69],[39,45805,45807,45810],{"className":45806},[74],[39,45808],{"className":45809,"style":245},[78],[39,45811,45696],{"className":45812},[83],[39,45814,45816,45834],{"className":45815,"translate":43},[42],[39,45817,45819],{"className":45818},[47],[49,45820,45821],{"xmlns":51},[53,45822,45823,45831],{},[56,45824,45825],{},[613,45826,45827,45829],{},[59,45828,142],{},[539,45830,17359],{},[63,45832,45833],{"encoding":65},"s_2",[39,45835,45837],{"className":45836,"ariaHidden":70},[69],[39,45838,45840,45843],{"className":45839},[74],[39,45841],{"className":45842,"style":2916},[78],[39,45844,45846,45849],{"className":45845},[83],[39,45847,142],{"className":45848},[83,84],[39,45850,45852],{"className":45851},[192],[39,45853,45855,45875],{"className":45854},[196,649],[39,45856,45858,45872],{"className":45857},[200],[39,45859,45861],{"className":45860,"style":963},[204],[39,45862,45863,45866],{"style":966},[39,45864],{"className":45865,"style":213},[212],[39,45867,45869],{"className":45868},[217,218,219,220],[39,45870,17359],{"className":45871},[83,220],[39,45873,673],{"className":45874},[672],[39,45876,45878],{"className":45877},[200],[39,45879,45881],{"className":45880,"style":680},[204],[39,45882],{},"，并设未来价值为零。奖励为",[39,45885,45887],{"className":45886,"translate":43},[746],[39,45888,45890,46026],{"className":45889,"translate":43},[42],[39,45891,45893],{"className":45892},[47],[49,45894,45895],{"xmlns":51,"display":755},[53,45896,45897,46023],{},[56,45898,45899,45901,45903,45905,45907,45913,45915,45921,45923,45925,45927,45929,45931,45933,45935,45937,45939,45945,45947,45953,45955,45957,45959,45961,45963,45965,45967,45969,45971,45977,45979,45985,45987,45989,45991,45993,45995,45997,45999,46001,46003,46009,46011,46017,46019,46021],{},[59,45900,401],{},[133,45902,136],{"stretchy":135},[59,45904,142],{},[133,45906,155],{"separator":70},[613,45908,45909,45911],{},[59,45910,158],{},[539,45912,546],{},[133,45914,155],{"separator":70},[613,45916,45917,45919],{},[59,45918,142],{},[539,45920,546],{},[133,45922,161],{"stretchy":135},[133,45924,793],{},[539,45926,546],{},[133,45928,155],{"separator":70},[230,45930],{"width":6891},[59,45932,401],{},[133,45934,136],{"stretchy":135},[59,45936,142],{},[133,45938,155],{"separator":70},[613,45940,45941,45943],{},[59,45942,158],{},[539,45944,546],{},[133,45946,155],{"separator":70},[613,45948,45949,45951],{},[59,45950,142],{},[539,45952,17359],{},[133,45954,161],{"stretchy":135},[133,45956,793],{},[539,45958,541],{},[133,45960,155],{"separator":70},[230,45962],{"width":10251},[59,45964,401],{},[133,45966,136],{"stretchy":135},[59,45968,142],{},[133,45970,155],{"separator":70},[613,45972,45973,45975],{},[59,45974,158],{},[539,45976,17359],{},[133,45978,155],{"separator":70},[613,45980,45981,45983],{},[59,45982,142],{},[539,45984,546],{},[133,45986,161],{"stretchy":135},[133,45988,793],{},[539,45990,541],{},[133,45992,155],{"separator":70},[230,45994],{"width":6891},[59,45996,401],{},[133,45998,136],{"stretchy":135},[59,46000,142],{},[133,46002,155],{"separator":70},[613,46004,46005,46007],{},[59,46006,158],{},[539,46008,17359],{},[133,46010,155],{"separator":70},[613,46012,46013,46015],{},[59,46014,142],{},[539,46016,17359],{},[133,46018,161],{"stretchy":135},[133,46020,793],{},[539,46022,25857],{},[63,46024,46025],{"encoding":65},"r(s,a_1,s_1)=1,\\quad r(s,a_1,s_2)=0,\n\\qquad\nr(s,a_2,s_1)=0,\\quad r(s,a_2,s_2)=1.",[39,46027,46029,46148,46279,46410,46541],{"className":46028,"ariaHidden":70},[69],[39,46030,46032,46035,46038,46041,46044,46047,46050,46090,46093,46096,46136,46139,46142,46145],{"className":46031},[74],[39,46033],{"className":46034,"style":245},[78],[39,46036,401],{"className":46037,"style":438},[83,84],[39,46039,136],{"className":46040},[182],[39,46042,142],{"className":46043},[83,84],[39,46045,155],{"className":46046},[252],[39,46048],{"className":46049,"style":256},[230],[39,46051,46053,46056],{"className":46052},[83],[39,46054,158],{"className":46055},[83,84],[39,46057,46059],{"className":46058},[192],[39,46060,46062,46082],{"className":46061},[196,649],[39,46063,46065,46079],{"className":46064},[200],[39,46066,46068],{"className":46067,"style":963},[204],[39,46069,46070,46073],{"style":966},[39,46071],{"className":46072,"style":213},[212],[39,46074,46076],{"className":46075},[217,218,219,220],[39,46077,546],{"className":46078},[83,220],[39,46080,673],{"className":46081},[672],[39,46083,46085],{"className":46084},[200],[39,46086,46088],{"className":46087,"style":680},[204],[39,46089],{},[39,46091,155],{"className":46092},[252],[39,46094],{"className":46095,"style":256},[230],[39,46097,46099,46102],{"className":46098},[83],[39,46100,142],{"className":46101},[83,84],[39,46103,46105],{"className":46104},[192],[39,46106,46108,46128],{"className":46107},[196,649],[39,46109,46111,46125],{"className":46110},[200],[39,46112,46114],{"className":46113,"style":963},[204],[39,46115,46116,46119],{"style":966},[39,46117],{"className":46118,"style":213},[212],[39,46120,46122],{"className":46121},[217,218,219,220],[39,46123,546],{"className":46124},[83,220],[39,46126,673],{"className":46127},[672],[39,46129,46131],{"className":46130},[200],[39,46132,46134],{"className":46133,"style":680},[204],[39,46135],{},[39,46137,161],{"className":46138},[263],[39,46140],{"className":46141,"style":231},[230],[39,46143,793],{"className":46144},[235],[39,46146],{"className":46147,"style":231},[230],[39,46149,46151,46154,46157,46160,46163,46166,46169,46172,46175,46178,46181,46221,46224,46227,46267,46270,46273,46276],{"className":46150},[74],[39,46152],{"className":46153,"style":245},[78],[39,46155,546],{"className":46156},[83],[39,46158,155],{"className":46159},[252],[39,46161],{"className":46162,"style":7240},[230],[39,46164],{"className":46165,"style":256},[230],[39,46167,401],{"className":46168,"style":438},[83,84],[39,46170,136],{"className":46171},[182],[39,46173,142],{"className":46174},[83,84],[39,46176,155],{"className":46177},[252],[39,46179],{"className":46180,"style":256},[230],[39,46182,46184,46187],{"className":46183},[83],[39,46185,158],{"className":46186},[83,84],[39,46188,46190],{"className":46189},[192],[39,46191,46193,46213],{"className":46192},[196,649],[39,46194,46196,46210],{"className":46195},[200],[39,46197,46199],{"className":46198,"style":963},[204],[39,46200,46201,46204],{"style":966},[39,46202],{"className":46203,"style":213},[212],[39,46205,46207],{"className":46206},[217,218,219,220],[39,46208,546],{"className":46209},[83,220],[39,46211,673],{"className":46212},[672],[39,46214,46216],{"className":46215},[200],[39,46217,46219],{"className":46218,"style":680},[204],[39,46220],{},[39,46222,155],{"className":46223},[252],[39,46225],{"className":46226,"style":256},[230],[39,46228,46230,46233],{"className":46229},[83],[39,46231,142],{"className":46232},[83,84],[39,46234,46236],{"className":46235},[192],[39,46237,46239,46259],{"className":46238},[196,649],[39,46240,46242,46256],{"className":46241},[200],[39,46243,46245],{"className":46244,"style":963},[204],[39,46246,46247,46250],{"style":966},[39,46248],{"className":46249,"style":213},[212],[39,46251,46253],{"className":46252},[217,218,219,220],[39,46254,17359],{"className":46255},[83,220],[39,46257,673],{"className":46258},[672],[39,46260,46262],{"className":46261},[200],[39,46263,46265],{"className":46264,"style":680},[204],[39,46266],{},[39,46268,161],{"className":46269},[263],[39,46271],{"className":46272,"style":231},[230],[39,46274,793],{"className":46275},[235],[39,46277],{"className":46278,"style":231},[230],[39,46280,46282,46285,46288,46291,46294,46297,46300,46303,46306,46309,46312,46352,46355,46358,46398,46401,46404,46407],{"className":46281},[74],[39,46283],{"className":46284,"style":245},[78],[39,46286,541],{"className":46287},[83],[39,46289,155],{"className":46290},[252],[39,46292],{"className":46293,"style":10787},[230],[39,46295],{"className":46296,"style":256},[230],[39,46298,401],{"className":46299,"style":438},[83,84],[39,46301,136],{"className":46302},[182],[39,46304,142],{"className":46305},[83,84],[39,46307,155],{"className":46308},[252],[39,46310],{"className":46311,"style":256},[230],[39,46313,46315,46318],{"className":46314},[83],[39,46316,158],{"className":46317},[83,84],[39,46319,46321],{"className":46320},[192],[39,46322,46324,46344],{"className":46323},[196,649],[39,46325,46327,46341],{"className":46326},[200],[39,463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s,a]&=\\frac12,\\\\\n\\mathbb{E}\\left[\\max_{a}r(s,a,s')\\mid s\\right]&=1.\n\\end{aligned}",[39,46686,46688],{"className":46687,"ariaHidden":70},[69],[39,46689,46691,46695],{"className":46690},[74],[39,46692],{"className":46693,"style":46694},[78],"height:4.1714em;vertical-align:-1.8357em;",[39,46696,46698],{"className":46697},[83],[39,46699,46701,46970],{"className":46700},[759],[39,46702,46704],{"className":46703},[918],[39,46705,46707,46961],{"className":46706},[196,649],[39,46708,46710,46958],{"className":46709},[200],[39,46711,46714,46807],{"className":46712,"style":46713},[204],"height:2.3357em;",[39,46715,46717,46721],{"style":46716},"top:-4.3357em;",[39,46718],{"className":46719,"style":46720},[212],"height:3.3214em;",[39,46722,46724,46771,46774,46777,46780,46783,46786,46789,46792,46795,46798,46801,46804],{"className":46723},[83],[39,46725,46727],{"className":46726},[1744,3294],[39,46728,46730,46763],{"className":46729},[196,649],[39,46731,46733,46760],{"className":46732},[200],[39,46734,46736,46750],{"className":46735,"style":290},[204],[39,46737,46738,46741],{"style":22299},[39,46739],{"className":46740,"style":935},[212],[39,46742,46744],{"className":46743},[217,218,219,220],[39,46745,46747],{"className":46746},[83,220],[39,46748,158],{"className":46749},[83,84,220],[39,46751,46752,46755],{"style":21599},[39,46753],{"className":46754,"style":935},[212],[39,46756,46757],{},[39,46758,1687],{"className":46759},[1744],[39,46761,673],{"className":46762},[672],[39,46764,46766],{"className":46765},[200],[39,46767,46769],{"className":46768,"style":22331},[204],[39,46770],{},[39,46772],{"className":46773,"style":256},[230],[39,46775,4253],{"className":46776},[83,515],[39,46778,537],{"className":46779},[182],[39,46781,401],{"className":46782,"style":438},[83,84],[39,46784],{"className":46785,"style":231},[230],[39,46787,150],{"className":46788},[235],[39,46790],{"className":46791,"style":231},[230],[39,46793,142],{"className":46794},[83,84],[39,46796,155],{"className":46797},[252],[39,46799],{"className":46800,"style":256},[230],[39,46802,158],{"className":46803},[83,84],[39,46805,4280],{"className":46806},[263],[39,46808,46810,46813],{"style":46809},"top:-2.1857em;",[39,46811],{"className":46812,"style":46720},[212],[39,46814,46816,46819,46822],{"className":46815},[83],[39,46817,4253],{"className":46818},[83,515],[39,46820],{"className":46821,"style":256},[230],[39,46823,46825,46831,46878,46881,46884,46887,46890,46893,46896,46899,46902,46905,46937,46940,46943,46946,46949,46952],{"className":46824},[2842],[39,46826,46828],{"className":46827,"style":4592},[182,4591],[39,46829,537],{"className":46830},[18520,33204],[39,46832,46834],{"className":46833},[1744,3294],[39,46835,46837,46870],{"className":46836},[196,649],[39,46838,46840,46867],{"className":46839},[200],[39,46841,46843,46857],{"className":46842,"style":290},[204],[39,46844,46845,46848],{"style":22299},[39,46846],{"className":46847,"style":935},[212],[39,46849,46851],{"className":46850},[217,218,219,220],[39,46852,46854],{"className":46853},[83,220],[39,46855,158],{"className":46856},[83,84,220],[39,46858,46859,46862],{"style":21599},[39,46860],{"className":46861,"style":935},[212],[39,46863,46864],{},[39,46865,1687],{"className":46866},[1744],[39,46868,673],{"className":46869},[672],[39,46871,46873],{"className":46872},[200],[39,46874,46876],{"className":46875,"style":22331},[204],[39,46877],{},[39,46879],{"className":46880,"style":256},[230],[39,46882,401],{"className":46883,"style":438},[83,84],[39,46885,136],{"className":46886},[182],[39,46888,142],{"className":46889},[83,84],[39,46891,155],{"className":46892},[252],[39,46894],{"className":46895,"style":256},[230],[39,46897,158],{"className":46898},[83,84],[39,46900,155],{"className":46901},[252],[39,46903],{"className":46904,"style":256},[230],[39,46906,46908,46911],{"className":46907},[83],[39,46909,142],{"className":46910},[83,84],[39,46912,46914],{"className":46913},[192],[39,46915,46917],{"className":46916},[196],[39,46918,46920],{"className":46919},[200],[39,46921,46923],{"className":46922,"style":1024},[204],[39,46924,46925,46928],{"style":1027},[39,46926],{"className":46927,"style":213},[212],[39,46929,46931],{"className":46930},[217,218,219,220],[39,46932,46934],{"className":46933},[83,220],[39,46935,147],{"className":46936},[83,220],[39,46938,161],{"className":46939},[263],[39,46941],{"className":46942,"style":231},[230],[39,46944,150],{"className":46945},[235],[39,46947],{"className":46948,"style":231},[230],[39,46950,142],{"className":46951},[83,84],[39,46953,46955],{"className":46954,"style":4592},[263,4591],[39,46956,4280],{"className":46957},[18520,33204],[39,46959,673],{"className":46960},[672],[39,46962,46964],{"className":46963},[200],[39,46965,46968],{"className":46966,"style":46967},[204],"height:1.8357em;",[39,46969],{},[39,46971,46973],{"className":46972},[1223],[39,46974,46976,47093],{"className":46975},[196,649],[39,46977,46979,47090],{"className":46978},[200],[39,46980,46982,47067],{"className":46981,"style":46713},[204],[39,46983,46984,46987],{"style":46716},[39,46985],{"className":46986,"style":46720},[212],[39,46988,46990,46993,46996,46999,47002,47064],{"className":46989},[83],[39,46991],{"className":46992},[83],[39,46994],{"className":46995,"style":231},[230],[39,46997,793],{"className":46998},[235],[39,47000],{"className":47001,"style":231},[230],[39,47003,47005,47008,47061],{"className":47004},[83],[39,47006],{"className":47007},[182,5463],[39,47009,47011],{"className":47010},[5362],[39,47012,47014,47053],{"className":47013},[196,649],[39,47015,47017,47050],{"className":47016},[200],[39,47018,47020,47031,47039],{"className":47019,"style":18240},[204],[39,47021,47022,47025],{"style":5479},[39,47023],{"className":47024,"style":935},[212],[39,47026,47028],{"className":47027},[83],[39,47029,17359],{"className":47030},[83],[39,47032,47033,47036],{"style":5503},[39,47034],{"className":47035,"style":935},[212],[39,47037],{"className":47038,"style":5511},[5510],[39,47040,47041,47044],{"style":5514},[39,47042],{"className":47043,"style":935},[212],[39,47045,47047],{"className":47046},[83],[39,47048,546],{"className":47049},[83],[39,47051,673],{"className":47052},[672],[39,47054,47056],{"className":47055},[200],[39,47057,47059],{"className":47058,"style":19487},[204],[39,47060],{},[39,47062],{"className":47063},[263,5463],[39,47065,155],{"className":47066},[252],[39,47068,47069,47072],{"style":46809},[39,47070],{"className":47071,"style":46720},[212],[39,47073,47075,47078,47081,47084,47087],{"className":47074},[83],[39,47076],{"className":47077},[83],[39,47079],{"className":47080,"style":231},[230],[39,47082,793],{"className":47083},[235],[39,47085],{"className":47086,"style":231},[230],[39,47088,25857],{"className":47089},[83],[39,47091,673],{"className":47092},[672],[39,47094,47096],{"className":47095},[200],[39,47097,47099],{"className":47098,"style":46967},[204],[39,47100],{},[11,47102,47103,47104,47165],{},"后者相当于在观察到 ",[39,47105,47107,47124],{"className":47106,"translate":43},[42],[39,47108,47110],{"className":47109},[47],[49,47111,47112],{"xmlns":51},[53,47113,47114,47122],{},[56,47115,47116],{},[138,47117,47118,47120],{},[59,47119,142],{},[133,47121,147],{"mathvariant":145,"lspace":146,"rspace":146},[63,47123,344],{"encoding":65},[39,47125,47127],{"className":47126,"ariaHidden":70},[69],[39,47128,47130,47133],{"className":47129},[74],[39,47131],{"className":47132,"style":205},[78],[39,47134,47136,47139],{"className":47135},[83],[39,47137,142],{"className":47138},[83,84],[39,47140,47142],{"className":47141},[192],[39,47143,47145],{"className":47144},[196],[39,47146,47148],{"className":47147},[200],[39,47149,47151],{"className":47150,"style":205},[204],[39,47152,47153,47156],{"style":208},[39,47154],{"className":47155,"style":213},[212],[39,47157,47159],{"className":47158},[217,218,219,220],[39,47160,47162],{"className":47161},[83,220],[39,47163,147],{"className":47164},[83,220]," 之后才选动作，因此获得了更大的灵活性，但它不是原问题的可行决策顺序。贝尔曼最优方程采用前者，是对决策时序的正确编码。",[26,47167,47169],{"id":47168},"_44-验证定理与最优策略","4.4 验证定理与最优策略",[31,47171,47174,47467,47954],{"id":47172,"title":47173},"corollary-optimal-verification","推论 4.3（最优性的验证定理）",[11,47175,37,47176,16486,47241,14627,47327,47408,47409,47437,47438,47466],{},[39,47177,47179,47202],{"className":47178,"translate":43},[42],[39,47180,47182],{"className":47181},[47],[49,47183,47184],{"xmlns":51},[53,47185,47186,47200],{},[56,47187,47188,47190,47192,47194,47196,47198],{},[59,47189,4230],{},[133,47191,534],{},[59,47193,19772],{},[133,47195,136],{"stretchy":135},[59,47197,61],{},[133,47199,161],{"stretchy":135},[63,47201,19781],{"encoding":65},[39,47203,47205,47223],{"className":47204,"ariaHidden":70},[69],[39,47206,47208,47211,47214,47217,47220],{"className":47207},[74],[39,47209],{"className":47210,"style":13061},[78],[39,47212,4230],{"className":47213,"style":461},[83,84],[39,47215],{"className":47216,"style":231},[230],[39,47218,534],{"className":47219},[235],[39,47221],{"className":47222,"style":231},[230],[39,47224,47226,47229,47232,47235,47238],{"className":47225},[74],[39,47227],{"className":47228,"style":245},[78],[39,47230,19772],{"className":47231,"style":19812},[83,84],[39,47233,136],{"className":47234},[182],[39,47236,61],{"className":47237,"style":85},[83,84],[39,47239,161],{"className":47240},[263],[39,47242,47244,47268],{"className":47243,"translate":43},[42],[39,47245,47247],{"className":47246},[47],[49,47248,47249],{"xmlns":51},[53,47250,47251,47265],{},[56,47252,47253,47255,47257,47263],{},[59,47254,4230],{},[133,47256,793],{},[138,47258,47259,47261],{},[59,47260,11772],{"mathvariant":617},[133,47262,21884],{},[59,47264,4230],{},[63,47266,47267],{"encoding":65},"V=\\mathcal{T}^*V",[39,47269,47271,47289],{"className":47270,"ariaHidden":70},[69],[39,47272,47274,47277,47280,47283,47286],{"className":47273},[74],[39,47275],{"className":47276,"style":79},[78],[39,47278,4230],{"className":47279,"style":461},[83,84],[39,47281],{"className":47282,"style":231},[230],[39,47284,793],{"className":47285},[235],[39,47287],{"className":47288,"style":231},[230],[39,47290,47292,47295,47324],{"className":47291},[74],[39,47293],{"className":47294,"style":21897},[78],[39,47296,47298,47301],{"className":47297},[83],[39,47299,11772],{"className":47300,"style":11793},[83,641],[39,47302,47304],{"className":47303},[192],[39,47305,47307],{"className":47306},[196],[39,47308,47310],{"className":47309},[200],[39,47311,47313],{"className":47312,"style":21897},[204],[39,47314,47315,47318],{"style":208},[39,47316],{"className":47317,"style":213},[212],[39,47319,47321],{"className":47320},[217,218,219,220],[39,47322,21884],{"className":47323},[465,220],[39,47325,4230],{"className":47326,"style":461},[83,84],[39,47328,47330,47352],{"className":47329,"translate":43},[42],[39,47331,47333],{"className":47332},[47],[49,47334,47335],{"xmlns":51},[53,47336,47337,47349],{},[56,47338,47339,47341,47343],{},[59,47340,4230],{},[133,47342,793],{},[138,47344,47345,47347],{},[59,47346,4230],{},[133,47348,21884],{},[63,47350,47351],{"encoding":65},"V=V^*",[39,47353,47355,47373],{"className":47354,"ariaHidden":70},[69],[39,47356,47358,47361,47364,47367,47370],{"className":47357},[74],[39,47359],{"className":47360,"style":79},[78],[39,47362,4230],{"className":47363,"style":461},[83,84],[39,47365],{"className":47366,"style":231},[230],[39,47368,793],{"className":47369},[235],[39,47371],{"className":47372,"style":231},[230],[39,47374,47376,47379],{"className":47375},[74],[39,47377],{"className":47378,"style":21897},[78],[39,47380,47382,47385],{"className":47381},[83],[39,47383,4230],{"className":47384,"style":461},[83,84],[39,47386,47388],{"className":47387},[192],[39,47389,47391],{"className":47390},[196],[39,47392,47394],{"className":47393},[200],[39,47395,47397],{"className":47396,"style":21897},[204],[39,47398,47399,47402],{"style":208},[39,47400],{"className":47401,"style":213},[212],[39,47403,47405],{"className":47404},[217,218,219,220],[39,47406,21884],{"className":47407},[465,220],"。若策略 ",[39,47410,47412,47425],{"className":47411,"translate":43},[42],[39,47413,47415],{"className":47414},[47],[49,47416,47417],{"xmlns":51},[53,47418,47419,47423],{},[56,47420,47421],{},[59,47422,2098],{},[63,47424,2101],{"encoding":65},[39,47426,47428],{"className":47427,"ariaHidden":70},[69],[39,47429,47431,47434],{"className":47430},[74],[39,47432],{"className":47433,"style":290},[78],[39,47435,2098],{"className":47436,"style":2114},[83,84]," 关于 ",[39,47439,47441,47454],{"className":47440,"translate":43},[42],[39,47442,47444],{"className":47443},[47],[49,47445,47446],{"xmlns":51},[53,47447,47448,47452],{},[56,47449,47450],{},[59,47451,4230],{},[63,47453,4230],{"encoding":65},[39,47455,47457],{"className":47456,"ariaHidden":70},[69],[39,47458,47460,47463],{"className":47459},[74],[39,47461],{"className":47462,"style":79},[78],[39,47464,4230],{"className":47465,"style":461},[83,84]," 贪婪，即",[39,47468,47470],{"className":47469,"translate":43},[746],[39,47471,47473,47583],{"className":47472,"translate":43},[42],[39,47474,47476],{"className":47475},[47],[49,47477,47478],{"xmlns":51,"display":755},[53,47479,47480,47580],{},[56,47481,47482,47484,47486,47488,47490,47492,47494,47496,47506,47516,47518,47520,47526,47528,47530,47532,47534,47536,47578],{},[59,47483,2098],{},[133,47485,136],{"stretchy":135},[59,47487,142],{},[133,47489,161],{"stretchy":135},[133,47491,534],{},[59,47493,39047],{},[133,47495,1690],{},[6556,47497,47498,47504],{},[56,47499,47500,47502],{},[59,47501,1687],{},[133,47503,1690],{},[59,47505,158],{},[6556,47507,47508,47510],{},[133,47509,3191],{},[138,47511,47512,47514],{},[59,47513,142],{},[133,47515,147],{"mathvariant":145,"lspace":146,"rspace":146},[59,47517,131],{},[133,47519,136],{"stretchy":135},[138,47521,47522,47524],{},[59,47523,142],{},[133,47525,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,47527,150],{},[59,47529,142],{},[133,47531,155],{"separator":70},[59,47533,158],{},[133,47535,161],{"stretchy":135},[56,47537,47538,47540,47542,47544,47546,47548,47550,47552,47558,47560,47562,47564,47566,47568,47574,47576],{},[133,47539,537],{"fence":70},[59,47541,401],{},[133,47543,136],{"stretchy":135},[59,47545,142],{},[133,47547,155],{"separator":70},[59,47549,158],{},[133,47551,155],{"separator":70},[138,47553,47554,47556],{},[59,47555,142],{},[133,47557,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,47559,161],{"stretchy":135},[133,47561,788],{},[59,47563,531],{},[59,47565,4230],{},[133,47567,136],{"stretchy":135},[138,47569,47570,47572],{},[59,47571,142],{},[133,47573,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,47575,161],{"stretchy":135},[133,47577,4280],{"fence":70},[133,47579,155],{"separator":70},[63,47581,47582],{"encoding":65},"\\pi(s)\\in\\arg\\max_{a}\\sum_{s'}P(s'\\mid s,a)\n\\left[r(s,a,s')+\\gamma V(s')\\right],",[39,47584,47586,47613,47803],{"className":47585,"ariaHidden":70},[69],[39,47587,47589,47592,47595,47598,47601,47604,47607,47610],{"className":47588},[74],[39,47590],{"className":47591,"style":245},[78],[39,47593,2098],{"className":47594,"style":2114},[83,84],[39,47596,136],{"className":47597},[182],[39,47599,142],{"className":47600},[83,84],[39,47602,161],{"className":47603},[263],[39,47605],{"className":47606,"style":231},[230],[39,47608,534],{"className":47609},[235],[39,47611],{"className":47612,"style":231},[230],[39,47614,47616,47619,47624,47627,47674,47677,47753,47756,47759,47762,47794,47797,47800],{"className":47615},[74],[39,47617],{"className":47618,"style":22284},[78],[39,47620,39206,47622],{"className":47621},[1744],[39,47623,39209],{"style":35078},[39,47625],{"className":47626,"style":256},[230],[39,47628,47630],{"className":47629},[1744,3294],[39,47631,47633,47666],{"className":47632},[196,649],[39,47634,47636,47663],{"className":47635},[200],[39,47637,47639,47653],{"className":47638,"style":290},[204],[39,47640,47641,47644],{"style":22299},[39,47642],{"className":47643,"style":935},[212],[39,47645,47647],{"className":47646},[217,218,219,220],[39,47648,47650],{"className":47649},[83,220],[39,47651,158],{"className":47652},[83,84,220],[39,47654,47655,47658],{"style":21599},[39,47656],{"className":47657,"style":935},[212],[39,47659,47660],{},[39,47661,1687],{"className":47662},[1744],[39,47664,673],{"className":47665},[672],[39,47667,47669],{"className":47668},[200],[39,47670,47672],{"className":47671,"style":22331},[204],[39,47673],{},[39,47675],{"className":47676,"style":256},[230],[39,47678,47680],{"className":47679},[1744,3294],[39,47681,47683,47745],{"className":47682},[196,649],[39,47684,47686,47742],{"className":47685},[200],[39,47687,47689,47732],{"className":47688,"style":6670},[204],[39,47690,47691,47694],{"style":7097},[39,47692],{"className":47693,"style":3311},[212],[39,47695,47697],{"className":47696},[217,218,219,220],[39,47698,47700],{"className":47699},[83,220],[39,47701,47703,47706],{"className":47702},[83,220],[39,47704,142],{"className":47705},[83,84,220],[39,47707,47709],{"className":47708},[192],[39,47710,47712],{"className":47711},[196],[39,47713,47715],{"className":47714},[200],[39,47716,47718],{"className":47717,"style":3997},[204],[39,47719,47720,47723],{"style":4000},[39,47721],{"className":47722,"style":4004},[212],[39,47724,47726],{"className":47725},[217,4008,4009,220],[39,47727,47729],{"className":47728},[83,220],[39,47730,147],{"className":47731},[83,220],[39,47733,47734,47737],{"style":3330},[39,47735],{"className":47736,"style":3311},[212],[39,47738,47739],{},[39,47740,3191],{"className":47741},[1744,3339,3340],[39,47743,673],{"className":47744},[672],[39,47746,47748],{"className":47747},[200],[39,47749,47751],{"className":47750,"style":7158},[204],[39,47752],{},[39,47754],{"className":47755,"style":256},[230],[39,47757,131],{"className":47758,"style":178},[83,84],[39,47760,136],{"className":47761},[182],[39,47763,47765,47768],{"className":47764},[83],[39,47766,142],{"className":47767},[83,84],[39,47769,47771],{"className":47770},[192],[39,47772,47774],{"className":47773},[196],[39,47775,47777],{"className":47776},[200],[39,47778,47780],{"className":47779,"style":1024},[204],[39,47781,47782,47785],{"style":1027},[39,47783],{"className":47784,"style":213},[212],[39,47786,47788],{"className":47787},[217,218,219,220],[39,47789,47791],{"className":47790},[83,220],[39,47792,147],{"className":47793},[83,220],[39,47795],{"className":47796,"style":231},[230],[39,47798,150],{"className":47799},[235],[39,47801],{"className":47802,"style":231},[230],[39,47804,47806,47809,47812,47815,47818,47821,47824,47827,47948,47951],{"className":47805},[74],[39,47807],{"className":47808,"style":22469},[78],[39,47810,142],{"className":47811},[83,84],[39,47813,155],{"className":47814},[252],[39,47816],{"className":47817,"style":256},[230],[39,47819,158],{"className":47820},[83,84],[39,47822,161],{"className":47823},[263],[39,47825],{"className":47826,"style":256},[230],[39,47828,47830,47833,47836,47839,47842,47845,47848,47851,47854,47857,47889,47892,47895,47898,47901,47904,47907,47910,47942,47945],{"className":47829},[2842],[39,47831,537],{"className":47832,"style":4592},[182,4591],[39,47834,401],{"className":47835,"style":438},[83,84],[39,47837,136],{"className":47838},[182],[39,47840,142],{"className":47841},[83,84],[39,47843,155],{"className":47844},[252],[39,47846],{"className":47847,"style":256},[230],[39,47849,158],{"className":47850},[83,84],[39,47852,155],{"className":47853},[252],[39,47855],{"className":47856,"style":256},[230],[39,47858,47860,47863],{"className":47859},[83],[39,47861,142],{"className":47862},[83,84],[39,47864,47866],{"className":47865},[192],[39,47867,47869],{"className":47868},[196],[39,47870,47872],{"className":47871},[200],[39,47873,47875],{"className":47874,"style":1024},[204],[39,47876,47877,47880],{"style":1027},[39,47878],{"className":47879,"style":213},[212],[39,47881,47883],{"className":47882},[217,218,219,220],[39,47884,47886],{"className":47885},[83,220],[39,47887,147],{"className":47888},[83,220],[39,47890,161],{"className":47891},[263],[39,47893],{"className":47894,"style":461},[230],[39,47896,788],{"className":47897},[465],[39,47899],{"className":47900,"style":461},[230],[39,47902,531],{"className":47903,"style":565},[83,84],[39,47905,4230],{"className":47906,"style":461},[83,84],[39,47908,136],{"className":47909},[182],[39,47911,47913,47916],{"className":47912},[83],[39,47914,142],{"className":47915},[83,84],[39,47917,47919],{"className":47918},[192],[39,47920,47922],{"className":47921},[196],[39,47923,47925],{"className":47924},[200],[39,47926,47928],{"className":47927,"style":1024},[204],[39,47929,47930,47933],{"style":1027},[39,47931],{"className":47932,"style":213},[212],[39,47934,47936],{"className":47935},[217,218,219,220],[39,47937,47939],{"className":47938},[83,220],[39,47940,147],{"className":47941},[83,220],[39,47943,161],{"className":47944},[263],[39,47946,4280],{"className":47947,"style":4592},[263,4591],[39,47949],{"className":47950,"style":256},[230],[39,47952,155],{"className":47953},[252],[11,47955,14173,47956,15508,48037,48065],{},[39,47957,47959,47981],{"className":47958,"translate":43},[42],[39,47960,47962],{"className":47961},[47],[49,47963,47964],{"xmlns":51},[53,47965,47966,47978],{},[56,47967,47968,47974,47976],{},[138,47969,47970,47972],{},[59,47971,4230],{},[59,47973,2098],{},[133,47975,793],{},[59,47977,4230],{},[63,47979,47980],{"encoding":65},"V^\\pi=V",[39,47982,47984,48028],{"className":47983,"ariaHidden":70},[69],[39,47985,47987,47990,48019,48022,48025],{"className":47986},[74],[39,47988],{"className":47989,"style":79},[78],[39,47991,47993,47996],{"className":47992},[83],[39,47994,4230],{"className":47995,"style":461},[83,84],[39,47997,47999],{"className":47998},[192],[39,48000,48002],{"className":48001},[196],[39,48003,48005],{"className":48004},[200],[39,48006,48008],{"className":48007,"style":6410},[204],[39,48009,48010,48013],{"style":208},[39,48011],{"className":48012,"style":213},[212],[39,48014,48016],{"className":48015},[217,218,219,220],[39,48017,2098],{"className":48018,"style":2114},[83,84,220],[39,48020],{"className":48021,"style":231},[230],[39,48023,793],{"className":48024},[235],[39,48026],{"className":48027,"style":231},[230],[39,48029,48031,48034],{"className":48030},[74],[39,48032],{"className":48033,"style":79},[78],[39,48035,4230],{"className":48036,"style":461},[83,84],[39,48038,48040,48053],{"className":48039,"translate":43},[42],[39,48041,48043],{"className":48042},[47],[49,48044,48045],{"xmlns":51},[53,48046,48047,48051],{},[56,48048,48049],{},[59,48050,2098],{},[63,48052,2101],{"encoding":65},[39,48054,48056],{"className":48055,"ariaHidden":70},[69],[39,48057,48059,48062],{"className":48058},[74],[39,48060],{"className":48061,"style":290},[78],[39,48063,2098],{"className":48064,"style":2114},[83,84]," 是最优策略。",[11,48067,48068,48070,48071,48129,48130,47437,48158,48186,48187,42282,48330,742,48332,21228],{},[1489,48069,5595],{}," 第一式由 ",[39,48072,48074,48091],{"className":48073,"translate":43},[42],[39,48075,48077],{"className":48076},[47],[49,48078,48079],{"xmlns":51},[53,48080,48081,48089],{},[56,48082,48083],{},[138,48084,48085,48087],{},[59,48086,11772],{"mathvariant":617},[133,48088,21884],{},[63,48090,21887],{"encoding":65},[39,48092,48094],{"className":48093,"ariaHidden":70},[69],[39,48095,48097,48100],{"className":48096},[74],[39,48098],{"className":48099,"style":21897},[78],[39,48101,48103,48106],{"className":48102},[83],[39,48104,11772],{"className":48105,"style":11793},[83,641],[39,48107,48109],{"className":48108},[192],[39,48110,48112],{"className":48111},[196],[39,48113,48115],{"className":48114},[200],[39,48116,48118],{"className":48117,"style":21897},[204],[39,48119,48120,48123],{"style":208},[39,48121],{"className":48122,"style":213},[212],[39,48124,48126],{"className":48125},[217,218,219,220],[39,48127,21884],{"className":48128},[465,220]," 的不动点唯一性立即得到。若 ",[39,48131,48133,48146],{"className":48132,"translate":43},[42],[39,48134,48136],{"className":48135},[47],[49,48137,48138],{"xmlns":51},[53,48139,48140,48144],{},[56,48141,48142],{},[59,48143,2098],{},[63,48145,2101],{"encoding":65},[39,48147,48149],{"className":48148,"ariaHidden":70},[69],[39,48150,48152,48155],{"className":48151},[74],[39,48153],{"className":48154,"style":290},[78],[39,48156,2098],{"className":48157,"style":2114},[83,84],[39,48159,48161,48174],{"className":48160,"translate":43},[42],[39,48162,48164],{"className":48163},[47],[49,48165,48166],{"xmlns":51},[53,48167,48168,48172],{},[56,48169,48170],{},[59,48171,4230],{},[63,48173,4230],{"encoding":65},[39,48175,48177],{"className":48176,"ariaHidden":70},[69],[39,48178,48180,48183],{"className":48179},[74],[39,48181],{"className":48182,"style":79},[78],[39,48184,4230],{"className":48185,"style":461},[83,84]," 贪婪，则 ",[39,48188,48190,48224],{"className":48189,"translate":43},[42],[39,48191,48193],{"className":48192},[47],[49,48194,48195],{"xmlns":51},[53,48196,48197,48221],{},[56,48198,48199,48205,48207,48209,48215,48217,48219],{},[138,48200,48201,48203],{},[59,48202,11772],{"mathvariant":617},[59,48204,2098],{},[59,48206,4230],{},[133,48208,793],{},[138,48210,48211,48213],{},[59,48212,11772],{"mathvariant":617},[133,48214,21884],{},[59,48216,4230],{},[133,48218,793],{},[59,48220,4230],{},[63,48222,48223],{"encoding":65},"\\mathcal{T}^\\pi V=\\mathcal{T}^*V=V",[39,48225,48227,48274,48321],{"className":48226,"ariaHidden":70},[69],[39,48228,48230,48233,48262,48265,48268,48271],{"className":48229},[74],[39,48231],{"className":48232,"style":79},[78],[39,48234,48236,48239],{"className":48235},[83],[39,48237,11772],{"className":48238,"style":11793},[83,641],[39,48240,48242],{"className":48241},[192],[39,48243,48245],{"className":48244},[196],[39,48246,48248],{"className":48247},[200],[39,48249,48251],{"className":48250,"style":6410},[204],[39,48252,48253,48256],{"style":208},[39,48254],{"className":48255,"style":213},[212],[39,48257,48259],{"className":48258},[217,218,219,220],[39,48260,2098],{"className":48261,"style":2114},[83,84,220],[39,48263,4230],{"className":48264,"style":461},[83,84],[39,48266],{"className":48267,"style":231},[230],[39,48269,793],{"className":48270},[235],[39,48272],{"className":48273,"style":231},[230],[39,48275,48277,48280,48309,48312,48315,48318],{"className":48276},[74],[39,48278],{"className":48279,"style":21897},[78],[39,48281,48283,48286],{"className":48282},[83],[39,48284,11772],{"className":48285,"style":11793},[83,641],[39,48287,48289],{"className":48288},[192],[39,48290,48292],{"className":48291},[196],[39,48293,48295],{"className":48294},[200],[39,48296,48298],{"className":48297,"style":21897},[204],[39,48299,48300,48303],{"style":208},[39,48301],{"className":48302,"style":213},[212],[39,48304,48306],{"className":48305},[217,218,219,220],[39,48307,21884],{"className":48308},[465,220],[39,48310,4230],{"className":48311,"style":461},[83,84],[39,48313],{"className":48314,"style":231},[230],[39,48316,793],{"className":48317},[235],[39,48319],{"className":48320,"style":231},[230],[39,48322,48324,48327],{"className":48323},[74],[39,48325],{"className":48326,"style":79},[78],[39,48328,4230],{"className":48329,"style":461},[83,84],[158,48331,21294],{"href":21293},[39,48333,48335,48356],{"className":48334,"translate":43},[42],[39,48336,48338],{"className":48337},[47],[49,48339,48340],{"xmlns":51},[53,48341,48342,48354],{},[56,48343,48344,48350,48352],{},[138,48345,48346,48348],{},[59,48347,4230],{},[59,48349,2098],{},[133,48351,793],{},[59,48353,4230],{},[63,48355,47980],{"encoding":65},[39,48357,48359,48403],{"className":48358,"ariaHidden":70},[69],[39,48360,48362,48365,48394,48397,48400],{"className":48361},[74],[39,48363],{"className":48364,"style":79},[78],[39,48366,48368,48371],{"className":48367},[83],[39,48369,4230],{"className":48370,"style":461},[83,84],[39,48372,48374],{"className":48373},[192],[39,48375,48377],{"className":48376},[196],[39,48378,48380],{"className":48379},[200],[39,48381,48383],{"className":48382,"style":6410},[204],[39,48384,48385,48388],{"style":208},[39,48386],{"className":48387,"style":213},[212],[39,48389,48391],{"className":48390},[217,218,219,220],[39,48392,2098],{"className":48393,"style":2114},[83,84,220],[39,48395],{"className":48396,"style":231},[230],[39,48398,793],{"className":48399},[235],[39,48401],{"className":48402,"style":231},[230],[39,48404,48406,48409],{"className":48405},[74],[39,48407],{"className":48408,"style":79},[78],[39,48410,4230],{"className":48411,"style":461},[83,84],[11,48413,48414],{},"这个推论是贝尔曼方程「充分性」的精确表述。它说明，只要能够找到一个有界函数满足最优方程，就不需要再与其他策略比较，因为它必然等于最优值函数。反过来，最优值函数也必然满足该方程。因此，在折扣有界 MDP 中，贝尔曼最优方程既必要又充分。",[26,48416,48418],{"id":48417},"_45-例带决策的路径","4.5 例：带决策的路径",[11,48420,48421,48422,2183,48506,48534],{},"考虑如下路径选择问题。状态为 ",[39,48423,48425,48457],{"className":48424,"translate":43},[42],[39,48426,48428],{"className":48427},[47],[49,48429,48430],{"xmlns":51},[53,48431,48432,48454],{},[56,48433,48434,48436,48438,48440,48442,48445,48447,48450,48452],{},[59,48435,101],{},[133,48437,155],{"separator":70},[59,48439,19772],{},[133,48441,155],{"separator":70},[59,48443,48444],{},"C",[133,48446,155],{"separator":70},[59,48448,48449],{},"D",[133,48451,155],{"separator":70},[59,48453,4253],{},[63,48455,48456],{"encoding":65},"A,B,C,D,E",[39,48458,48460],{"className":48459,"ariaHidden":70},[69],[39,48461,48463,48466,48469,48472,48475,48478,48481,48484,48488,48491,48494,48497,48500,48503],{"className":48462},[74],[39,48464],{"className":48465,"style":6452},[78],[39,48467,101],{"className":48468},[83,84],[39,48470,155],{"className":48471},[252],[39,48473],{"className":48474,"style":256},[230],[39,48476,19772],{"className":48477,"style":19812},[83,84],[39,48479,155],{"className":48480},[252],[39,48482],{"className":48483,"style":256},[230],[39,48485,48444],{"className":48486,"style":48487},[83,84],"margin-right:0.0715em;",[39,48489,155],{"className":48490},[252],[39,48492],{"className":48493,"style":256},[230],[39,48495,48449],{"className":48496,"style":438},[83,84],[39,48498,155],{"className":48499},[252],[39,48501],{"className":48502,"style":256},[230],[39,48504,4253],{"className":48505,"style":85},[83,84],[39,48507,48509,48522],{"className":48508,"translate":43},[42],[39,48510,48512],{"className":48511},[47],[49,48513,48514],{"xmlns":51},[53,48515,48516,48520],{},[56,48517,48518],{},[59,48519,48444],{},[63,48521,48444],{"encoding":65},[39,48523,48525],{"className":48524,"ariaHidden":70},[69],[39,48526,48528,48531],{"className":48527},[74],[39,48529],{"className":48530,"style":79},[78],[39,48532,48444],{"className":48533,"style":48487},[83,84]," 为终止状态。转移关系为",[39,48536,48538],{"className":48537,"translate":43},[746],[39,48539,48541,48631],{"className":48540,"translate":43},[42],[39,48542,48544],{"className":48543},[47],[49,48545,48546],{"xmlns":51,"display":755},[53,48547,48548,48628],{},[759,48549,48550,48594],{"rowspacing":761,"columnalign":762,"columnspacing":146},[764,48551,48552,48558],{},[767,48553,48554],{},[770,48555,48556],{"scriptlevel":541,"displaystyle":70},[56,48557],{},[767,48559,48560],{},[770,48561,48562],{"scriptlevel":541,"displaystyle":70},[56,48563,48564,48566,48568,48570,48572,48574,48576,48578,48580,48582,48584,48586,48588,48590,48592],{},[56,48565],{},[59,48567,101],{},[133,48569,418],{},[59,48571,19772],{},[133,48573,155],{"separator":70},[230,48575],{"width":6891},[59,48577,19772],{},[133,48579,418],{},[59,48581,48444],{},[133,48583,155],{"separator":70},[230,48585],{"width":6891},[59,48587,101],{},[133,48589,418],{},[59,48591,48449],{},[133,48593,155],{"separator":70},[764,48595,48596,48602],{},[767,48597,48598],{},[770,48599,48600],{"scriptlevel":541,"displaystyle":70},[56,48601],{},[767,48603,48604],{},[770,48605,48606],{"scriptlevel":541,"displaystyle":70},[56,48607,48608,48610,48612,48614,48616,48618,48620,48622,48624,48626],{},[56,48609],{},[59,48611,48449],{},[133,48613,418],{},[59,48615,4253],{},[133,48617,155],{"separator":70},[230,48619],{"width":6891},[59,48621,4253],{},[133,48623,418],{},[59,48625,48444],{},[59,48627,2903],{"mathvariant":145},[63,48629,48630],{"encoding":65},"\\begin{aligned}\n&A\\to B,\\quad B\\to C,\\quad A\\to D,\\\\\n&D\\to E,\\quad E\\to C.\n\\end{aligned}",[39,48632,48634],{"className":48633,"ariaHidden":70},[69],[39,48635,48637,48640],{"className":48636},[74],[39,48638],{"className":48639,"style":4374},[78],[39,48641,48643],{"className":48642},[83],[39,48644,48646,48685],{"className":48645},[759],[39,48647,48649],{"className":48648},[918],[39,48650,48652,48677],{"className":48651},[196,649],[39,48653,48655,48674],{"className":48654},[200],[39,48656,48658,48666],{"className":48657,"style":4393},[204],[39,48659,48660,48663],{"style":28366},[39,48661],{"className":48662,"style":28370},[212],[39,48664],{"className":48665},[83],[39,48667,48668,48671],{"style":5834},[39,48669],{"className":48670,"style":28370},[212],[39,48672],{"className":48673},[83],[39,48675,673],{"className":48676},[672],[39,48678,48680],{"className":48679},[200],[39,48681,48683],{"className":48682,"style":4507},[204],[39,48684],{},[39,48686,48688],{"className":48687},[1223],[39,48689,48691,48830],{"className":48690},[196,649],[39,48692,48694,48827],{"className":48693},[200],[39,48695,48697,48774],{"className":48696,"style":4393},[204],[39,48698,48699,48702],{"style":4396},[39,48700],{"className":48701,"style":935},[212],[39,48703,48705,48708,48711,48714,48717,48720,48723,48726,48729,48732,48735,48738,48741,48744,48747,48750,48753,48756,48759,48762,48765,48768,48771],{"className":48704},[83],[39,48706],{"className":48707},[83],[39,48709,101],{"className":48710},[83,84],[39,48712],{"className":48713,"style":231},[230],[39,48715,418],{"className":48716},[235],[39,48718],{"className":48719,"style":231},[230],[39,48721,19772],{"className":48722,"style":19812},[83,84],[39,48724,155],{"className":48725},[252],[39,48727],{"className":48728,"style":7240},[230],[39,48730],{"className":48731,"style":256},[230],[39,48733,19772],{"className":48734,"style":19812},[83,84],[39,48736],{"className":48737,"style":231},[230],[39,48739,418],{"className":48740},[235],[39,48742],{"className":48743,"style":231},[230],[39,48745,48444],{"className":48746,"style":48487},[83,84],[39,48748,155],{"className":48749},[252],[39,48751],{"className":48752,"style":7240},[230],[39,48754],{"className":48755,"style":256},[230],[39,48757,101],{"className":48758},[83,84],[39,48760],{"className":48761,"style":231},[230],[39,48763,418],{"className":48764},[235],[39,48766],{"className":48767,"style":231},[230],[39,48769,48449],{"className":48770,"style":438},[83,84],[39,48772,155],{"className":48773},[252],[39,48775,48776,48779],{"style":4444},[39,48777],{"className":48778,"style":935},[212],[39,48780,48782,48785,48788,48791,48794,48797,48800,48803,48806,48809,48812,48815,48818,48821,48824],{"className":48781},[83],[39,48783],{"className":48784},[83],[39,48786,48449],{"className":48787,"style":438},[83,84],[39,48789],{"className":48790,"style":231},[230],[39,48792,418],{"className":48793},[235],[39,48795],{"className":48796,"style":231},[230],[39,48798,4253],{"className":48799,"style":85},[83,84],[39,48801,155],{"className":48802},[252],[39,48804],{"className":48805,"style":7240},[230],[39,48807],{"className":48808,"style":256},[230],[39,48810,4253],{"className":48811,"style":85},[83,84],[39,48813],{"className":48814,"style":231},[230],[39,48816,418],{"className":48817},[235],[39,48819],{"className":48820,"style":231},[230],[39,48822,48444],{"className":48823,"style":48487},[83,84],[39,48825,2903],{"className":48826},[83],[39,48828,673],{"className":48829},[672],[39,48831,48833],{"className":48832},[200],[39,48834,48836],{"className":48835,"style":4507},[204],[39,48837],{},[11,48839,48840,48841,48875,48876,48904,48905,48934,48935,48985,48986,1607],{},"每次转移获得奖励 ",[39,48842,48844,48860],{"className":48843,"translate":43},[42],[39,48845,48847],{"className":48846},[47],[49,48848,48849],{"xmlns":51},[53,48850,48851,48857],{},[56,48852,48853,48855],{},[133,48854,5378],{},[539,48856,546],{},[63,48858,48859],{"encoding":65},"-1",[39,48861,48863],{"className":48862,"ariaHidden":70},[69],[39,48864,48866,48869,48872],{"className":48865},[74],[39,48867],{"className":48868,"style":17882},[78],[39,48870,5378],{"className":48871},[83],[39,48873,546],{"className":48874},[83],"，进入终止状态 ",[39,48877,48879,48892],{"className":48878,"translate":43},[42],[39,48880,48882],{"className":48881},[47],[49,48883,48884],{"xmlns":51},[53,48885,48886,48890],{},[56,48887,48888],{},[59,48889,48444],{},[63,48891,48444],{"encoding":65},[39,48893,48895],{"className":48894,"ariaHidden":70},[69],[39,48896,48898,48901],{"className":48897},[74],[39,48899],{"className":48900,"style":79},[78],[39,48902,48444],{"className":48903,"style":48487},[83,84]," 时额外获得奖励 ",[39,48906,48908,48922],{"className":48907,"translate":43},[42],[39,48909,48911],{"className":48910},[47],[49,48912,48913],{"xmlns":51},[53,48914,48915,48920],{},[56,48916,48917],{},[539,48918,48919],{},"10",[63,48921,48919],{"encoding":65},[39,48923,48925],{"className":48924,"ariaHidden":70},[69],[39,48926,48928,48931],{"className":48927},[74],[39,48929],{"className":48930,"style":2473},[78],[39,48932,48919],{"className":48933},[83],"。设 ",[39,48936,48938,48955],{"className":48937,"translate":43},[42],[39,48939,48941],{"className":48940},[47],[49,48942,48943],{"xmlns":51},[53,48944,48945,48953],{},[56,48946,48947,48949,48951],{},[59,48948,531],{},[133,48950,793],{},[539,48952,17739],{},[63,48954,17742],{"encoding":65},[39,48956,48958,48976],{"className":48957,"ariaHidden":70},[69],[39,48959,48961,48964,48967,48970,48973],{"className":48960},[74],[39,48962],{"className":48963,"style":2654},[78],[39,48965,531],{"className":48966,"style":565},[83,84],[39,48968],{"className":48969,"style":231},[230],[39,48971,793],{"className":48972},[235],[39,48974],{"className":48975,"style":231},[230],[39,48977,48979,48982],{"className":48978},[74],[39,48980],{"className":48981,"style":2473},[78],[39,48983,17739],{"className":48984},[83],"，并令 ",[39,48987,48989,49013],{"className":48988,"translate":43},[42],[39,48990,48992],{"className":48991},[47],[49,48993,48994],{"xmlns":51},[53,48995,48996,49010],{},[56,48997,48998,49000,49002,49004,49006,49008],{},[59,48999,4230],{},[133,49001,136],{"stretchy":135},[59,49003,48444],{},[133,49005,161],{"stretchy":135},[133,49007,793],{},[539,49009,541],{},[63,49011,49012],{"encoding":65},"V(C)=0",[39,49014,49016,49043],{"className":49015,"ariaHidden":70},[69],[39,49017,49019,49022,49025,49028,49031,49034,49037,49040],{"className":49018},[74],[39,49020],{"className":49021,"style":245},[78],[39,49023,4230],{"className":49024,"style":461},[83,84],[39,49026,136],{"className":49027},[182],[39,49029,48444],{"className":49030,"style":48487},[83,84],[39,49032,161],{"className":49033},[263],[39,49035],{"className":49036,"style":231},[230],[39,49038,793],{"className":49039},[235],[39,49041],{"className":49042,"style":231},[230],[39,49044,49046,49049],{"className":49045},[74],[39,49047],{"className":49048,"style":2473},[78],[39,49050,541],{"className":49051},[83],[11,49053,49054,49055,1639,49083,49111,49112,49140],{},"在 ",[39,49056,49058,49071],{"className":49057,"translate":43},[42],[39,49059,49061],{"className":49060},[47],[49,49062,49063],{"xmlns":51},[53,49064,49065,49069],{},[56,49066,49067],{},[59,49068,19772],{},[63,49070,19772],{"encoding":65},[39,49072,49074],{"className":49073,"ariaHidden":70},[69],[39,49075,49077,49080],{"className":49076},[74],[39,49078],{"className":49079,"style":79},[78],[39,49081,19772],{"className":49082,"style":19812},[83,84],[39,49084,49086,49099],{"className":49085,"translate":43},[42],[39,49087,49089],{"className":49088},[47],[49,49090,49091],{"xmlns":51},[53,49092,49093,49097],{},[56,49094,49095],{},[59,49096,4253],{},[63,49098,4253],{"encoding":65},[39,49100,49102],{"className":49101,"ariaHidden":70},[69],[39,49103,49105,49108],{"className":49104},[74],[39,49106],{"className":49107,"style":79},[78],[39,49109,4253],{"className":49110,"style":85},[83,84],"，只有一条通往 ",[39,49113,49115,49128],{"className":49114,"translate":43},[42],[39,49116,49118],{"className":49117},[47],[49,49119,49120],{"xmlns":51},[53,49121,49122,49126],{},[56,49123,49124],{},[59,49125,48444],{},[63,49127,48444],{"encoding":65},[39,49129,49131],{"className":49130,"ariaHidden":70},[69],[39,49132,49134,49137],{"className":49133},[74],[39,49135],{"className":49136,"style":79},[78],[39,49138,48444],{"className":49139,"style":48487},[83,84]," 的动作，因此",[39,49142,49144],{"className":49143,"translate":43},[746],[39,49145,49147,49210],{"className":49146,"translate":43},[42],[39,49148,49150],{"className":49149},[47],[49,49151,49152],{"xmlns":51,"display":755},[53,49153,49154,49207],{},[56,49155,49156,49158,49160,49162,49164,49166,49168,49170,49172,49174,49176,49178,49180,49182,49184,49186,49188,49190,49192,49194,49196,49198,49200,49202,49204],{},[59,49157,4230],{},[133,49159,136],{"stretchy":135},[59,49161,19772],{},[133,49163,161],{"stretchy":135},[133,49165,793],{},[133,49167,5378],{},[539,49169,546],{},[133,49171,788],{},[539,49173,48919],{},[133,49175,788],{},[539,49177,17739],{},[59,49179,4230],{},[133,49181,136],{"stretchy":135},[59,49183,48444],{},[133,49185,161],{"stretchy":135},[133,49187,793],{},[539,49189,28873],{},[133,49191,155],{"separator":70},[230,49193],{"width":10251},[59,49195,4230],{},[133,49197,136],{"stretchy":135},[59,49199,4253],{},[133,49201,161],{"stretchy":135},[133,49203,793],{},[539,49205,49206],{},"9.",[63,49208,49209],{"encoding":65},"V(B)=-1+10+0.9V(C)=9,\n\\qquad\nV(E)=9.",[39,49211,49213,49240,49261,49279,49309,49348],{"className":49212,"ariaHidden":70},[69],[39,49214,49216,49219,49222,49225,49228,49231,49234,49237],{"className":49215},[74],[39,49217],{"className":49218,"style":245},[78],[39,49220,4230],{"className":49221,"style":461},[83,84],[39,49223,136],{"className":49224},[182],[39,49226,19772],{"className":49227,"style":19812},[83,84],[39,49229,161],{"className":49230},[263],[39,49232],{"className":49233,"style":231},[230],[39,49235,793],{"className":49236},[235],[39,49238],{"className":49239,"style":231},[230],[39,49241,49243,49246,49249,49252,49255,49258],{"className":49242},[74],[39,49244],{"className":49245,"style":17882},[78],[39,49247,5378],{"className":49248},[83],[39,49250,546],{"className":49251},[83],[39,49253],{"className":49254,"style":461},[230],[39,49256,788],{"className":49257},[465],[39,49259],{"className":49260,"style":461},[230],[39,49262,49264,49267,49270,49273,49276],{"className":49263},[74],[39,49265],{"className":49266,"style":17882},[78],[39,49268,48919],{"className":49269},[83],[39,49271],{"className":49272,"style":461},[230],[39,49274,788],{"className":49275},[465],[39,49277],{"className":49278,"style":461},[230],[39,49280,49282,49285,49288,49291,49294,49297,49300,49303,49306],{"className":49281},[74],[39,49283],{"className":49284,"style":245},[78],[39,49286,17739],{"className":49287},[83],[39,49289,4230],{"className":49290,"style":461},[83,84],[39,49292,136],{"className":49293},[182],[39,49295,48444],{"className":49296,"style":48487},[83,84],[39,49298,161],{"className":49299},[263],[39,49301],{"className":49302,"style":231},[230],[39,49304,793],{"className":49305},[235],[39,49307],{"className":49308,"style":231},[230],[39,49310,49312,49315,49318,49321,49324,49327,49330,49333,49336,49339,49342,49345],{"className":49311},[74],[39,49313],{"className":49314,"style":245},[78],[39,49316,28873],{"className":49317},[83],[39,49319,155],{"className":49320},[252],[39,49322],{"className":49323,"style":10787},[230],[39,49325],{"className":49326,"style":256},[230],[39,49328,4230],{"className":49329,"style":461},[83,84],[39,49331,136],{"className":49332},[182],[39,49334,4253],{"className":49335,"style":85},[83,84],[39,49337,161],{"className":49338},[263],[39,49340],{"className":49341,"style":231},[230],[39,49343,793],{"className":49344},[235],[39,49346],{"className":49347,"style":231},[230],[39,49349,49351,49354],{"className":49350},[74],[39,49352],{"className":49353,"style":2473},[78],[39,49355,49206],{"className":49356},[83],[11,49358,49054,49359,49387,49388,49416],{},[39,49360,49362,49375],{"className":49361,"translate":43},[42],[39,49363,49365],{"className":49364},[47],[49,49366,49367],{"xmlns":51},[53,49368,49369,49373],{},[56,49370,49371],{},[59,49372,48449],{},[63,49374,48449],{"encoding":65},[39,49376,49378],{"className":49377,"ariaHidden":70},[69],[39,49379,49381,49384],{"className":49380},[74],[39,49382],{"className":49383,"style":79},[78],[39,49385,48449],{"className":49386,"style":438},[83,84],"，唯一动作通向 ",[39,49389,49391,49404],{"className":49390,"translate":43},[42],[39,49392,49394],{"className":49393},[47],[49,49395,49396],{"xmlns":51},[53,49397,49398,49402],{},[56,49399,49400],{},[59,49401,4253],{},[63,49403,4253],{"encoding":65},[39,49405,49407],{"className":49406,"ariaHidden":70},[69],[39,49408,49410,49413],{"className":49409},[74],[39,49411],{"className":49412,"style":79},[78],[39,49414,4253],{"className":49415,"style":85},[83,84],"，故",[39,49418,49420],{"className":49419,"translate":43},[746],[39,49421,49423,49480],{"className":49422,"translate":43},[42],[39,49424,49426],{"className":49425},[47],[49,49427,49428],{"xmlns":51,"display":755},[53,49429,49430,49477],{},[56,49431,49432,49434,49436,49438,49440,49442,49444,49446,49448,49450,49452,49454,49456,49458,49460,49462,49464,49466,49468,49470,49472,49474],{},[59,49433,4230],{},[133,49435,136],{"stretchy":135},[59,49437,48449],{},[133,49439,161],{"stretchy":135},[133,49441,793],{},[133,49443,5378],{},[539,49445,546],{},[133,49447,788],{},[539,49449,17739],{},[59,49451,4230],{},[133,49453,136],{"stretchy":135},[59,49455,4253],{},[133,49457,161],{"stretchy":135},[133,49459,793],{},[133,49461,5378],{},[539,49463,546],{},[133,49465,788],{},[539,49467,17739],{},[133,49469,409],{},[539,49471,28873],{},[133,49473,793],{},[539,49475,49476],{},"7.1.",[63,49478,49479],{"encoding":65},"V(D)=-1+0.9V(E)=-1+0.9\\times9=7.1.",[39,49481,49483,49510,49531,49561,49582,49600,49618],{"className":49482,"ariaHidden":70},[69],[39,49484,49486,49489,49492,49495,49498,49501,49504,49507],{"className":49485},[74],[39,49487],{"className":49488,"style":245},[78],[39,49490,4230],{"className":49491,"style":461},[83,84],[39,49493,136],{"className":49494},[182],[39,49496,48449],{"className":49497,"style":438},[83,84],[39,49499,161],{"className":49500},[263],[39,49502],{"className":49503,"style":231},[230],[39,49505,793],{"className":49506},[235],[39,49508],{"className":49509,"style":231},[230],[39,49511,49513,49516,49519,49522,49525,49528],{"className":49512},[74],[39,49514],{"className":49515,"style":17882},[78],[39,49517,5378],{"className":49518},[83],[39,49520,546],{"className":49521},[83],[39,49523],{"className":49524,"style":461},[230],[39,49526,788],{"className":49527},[465],[39,49529],{"className":49530,"style":461},[230],[39,49532,49534,49537,49540,49543,49546,49549,49552,49555,49558],{"className":49533},[74],[39,49535],{"className":49536,"style":245},[78],[39,49538,17739],{"className":49539},[83],[39,49541,4230],{"className":49542,"style":461},[83,84],[39,49544,136],{"className":49545},[182],[39,49547,4253],{"className":49548,"style":85},[83,84],[39,49550,161],{"className":49551},[263],[39,49553],{"className":49554,"style":231},[230],[39,49556,793],{"className":49557},[235],[39,49559],{"className":49560,"style":231},[230],[39,49562,49564,49567,49570,49573,49576,49579],{"className":49563},[74],[39,49565],{"className":49566,"style":17882},[78],[39,49568,5378],{"className":49569},[83],[39,49571,546],{"className":49572},[83],[39,49574],{"className":49575,"style":461},[230],[39,49577,788],{"className":49578},[465],[39,49580],{"className":49581,"style":461},[230],[39,49583,49585,49588,49591,49594,49597],{"className":49584},[74],[39,49586],{"className":49587,"style":17882},[78],[39,49589,17739],{"className":49590},[83],[39,49592],{"className":49593,"style":461},[230],[39,49595,409],{"className":49596},[465],[39,49598],{"className":49599,"style":461},[230],[39,49601,49603,49606,49609,49612,49615],{"className":49602},[74],[39,49604],{"className":49605,"style":2473},[78],[39,49607,28873],{"className":49608},[83],[39,49610],{"className":49611,"style":231},[230],[39,49613,793],{"className":49614},[235],[39,49616],{"className":49617,"style":231},[230],[39,49619,49621,49624],{"className":49620},[74],[39,49622],{"className":49623,"style":2473},[78],[39,49625,49476],{"className":49626},[83],[11,49628,49054,49629,49657,49658,1639,49724,49790,49791,49819],{},[39,49630,49632,49645],{"className":49631,"translate":43},[42],[39,49633,49635],{"className":49634},[47],[49,49636,49637],{"xmlns":51},[53,49638,49639,49643],{},[56,49640,49641],{},[59,49642,101],{},[63,49644,101],{"encoding":65},[39,49646,49648],{"className":49647,"ariaHidden":70},[69],[39,49649,49651,49654],{"className":49650},[74],[39,49652],{"className":49653,"style":79},[78],[39,49655,101],{"className":49656},[83,84],"，有两条动作。记 ",[39,49659,49661,49685],{"className":49660,"translate":43},[42],[39,49662,49664],{"className":49663},[47],[49,49665,49666],{"xmlns":51},[53,49667,49668,49682],{},[56,49669,49670,49672,49674,49676,49678,49680],{},[59,49671,4295],{},[133,49673,136],{"stretchy":135},[59,49675,101],{},[133,49677,418],{},[59,49679,19772],{},[133,49681,161],{"stretchy":135},[63,49683,49684],{"encoding":65},"Q(A\\to B)",[39,49686,49688,49712],{"className":49687,"ariaHidden":70},[69],[39,49689,49691,49694,49697,49700,49703,49706,49709],{"className":49690},[74],[39,49692],{"className":49693,"style":245},[78],[39,49695,4295],{"className":49696},[83,84],[39,49698,136],{"className":49699},[182],[39,49701,101],{"className":49702},[83,84],[39,49704],{"className":49705,"style":231},[230],[39,49707,418],{"className":49708},[235],[39,49710],{"className":49711,"style":231},[230],[39,49713,49715,49718,49721],{"className":49714},[74],[39,49716],{"className":49717,"style":245},[78],[39,49719,19772],{"className":49720,"style":19812},[83,84],[39,49722,161],{"className":49723},[263],[39,49725,49727,49751],{"className":49726,"translate":43},[42],[39,49728,49730],{"className":49729},[47],[49,49731,49732],{"xmlns":51},[53,49733,49734,49748],{},[56,49735,49736,49738,49740,49742,49744,49746],{},[59,49737,4295],{},[133,49739,136],{"stretchy":135},[59,49741,101],{},[133,49743,418],{},[59,49745,48449],{},[133,49747,161],{"stretchy":135},[63,49749,49750],{"encoding":65},"Q(A\\to D)",[39,49752,49754,49778],{"className":49753,"ariaHidden":70},[69],[39,49755,49757,49760,49763,49766,49769,49772,49775],{"className":49756},[74],[39,49758],{"className":49759,"style":245},[78],[39,49761,4295],{"className":49762},[83,84],[39,49764,136],{"className":49765},[182],[39,49767,101],{"className":49768},[83,84],[39,49770],{"className":49771,"style":231},[230],[39,49773,418],{"className":49774},[235],[39,49776],{"className":49777,"style":231},[230],[39,49779,49781,49784,49787],{"className":49780},[74],[39,49782],{"className":49783,"style":245},[78],[39,49785,48449],{"className":49786,"style":438},[83,84],[39,49788,161],{"className":49789},[263]," 分别为在 ",[39,49792,49794,49807],{"className":49793,"translate":43},[42],[39,49795,49797],{"className":49796},[47],[49,49798,49799],{"xmlns":51},[53,49800,49801,49805],{},[56,49802,49803],{},[59,49804,101],{},[63,49806,101],{"encoding":65},[39,49808,49810],{"className":49809,"ariaHidden":70},[69],[39,49811,49813,49816],{"className":49812},[74],[39,49814],{"className":49815,"style":79},[78],[39,49817,101],{"className":49818},[83,84]," 选择这两条动作的动作值，则",[39,49821,49823],{"className":49822,"translate":43},[746],[39,49824,49826,49889],{"className":49825,"translate":43},[42],[39,49827,49829],{"className":49828},[47],[49,49830,49831],{"xmlns":51,"display":755},[53,49832,49833,49886],{},[56,49834,49835,49837,49839,49841,49843,49845,49847,49849,49851,49853,49855,49857,49859,49861,49863,49865,49867,49869,49871,49873,49875,49877,49879,49881,49884],{},[59,49836,4295],{},[133,49838,136],{"stretchy":135},[59,49840,101],{},[133,49842,418],{},[59,49844,19772],{},[133,49846,161],{"stretchy":135},[133,49848,793],{},[133,49850,5378],{},[539,49852,546],{},[133,49854,788],{},[539,49856,17739],{},[59,49858,4230],{},[133,49860,136],{"stretchy":135},[59,49862,19772],{},[133,49864,161],{"stretchy":135},[133,49866,793],{},[133,49868,5378],{},[539,49870,546],{},[133,49872,788],{},[539,49874,17739],{},[133,49876,409],{},[539,49878,28873],{},[133,49880,793],{},[539,49882,49883],{},"7.1",[133,49885,155],{"separator":70},[63,49887,49888],{"encoding":65},"Q(A\\to B)=-1+0.9V(B)=-1+0.9\\times9=7.1,",[39,49890,49892,49916,49937,49958,49988,50009,50027,50045],{"className":49891,"ariaHidden":70},[69],[39,49893,49895,49898,49901,49904,49907,49910,49913],{"className":49894},[74],[39,49896],{"className":49897,"style":245},[78],[39,49899,4295],{"className":49900},[83,84],[39,49902,136],{"className":49903},[182],[39,49905,101],{"className":49906},[83,84],[39,49908],{"className":49909,"style":231},[230],[39,49911,418],{"className":49912},[235],[39,49914],{"className":49915,"style":231},[230],[39,49917,49919,49922,49925,49928,49931,49934],{"className":49918},[74],[39,49920],{"className":49921,"style":245},[78],[39,49923,19772],{"className":49924,"style":19812},[83,84],[39,49926,161],{"className":49927},[263],[39,49929],{"className":49930,"style":231},[230],[39,49932,793],{"className":49933},[235],[39,49935],{"className":49936,"style":231},[230],[39,49938,49940,49943,49946,49949,49952,49955],{"className":49939},[74],[39,49941],{"className":49942,"style":17882},[78],[39,49944,5378],{"className":49945},[83],[39,49947,546],{"className":49948},[83],[39,49950],{"className":49951,"style":461},[230],[39,49953,788],{"className":49954},[465],[39,49956],{"className":49957,"style":461},[230],[39,49959,49961,49964,49967,49970,49973,49976,49979,49982,49985],{"className":49960},[74],[39,49962],{"className":49963,"style":245},[78],[39,49965,17739],{"className":49966},[83],[39,49968,4230],{"className":49969,"style":461},[83,84],[39,49971,136],{"className":49972},[182],[39,49974,19772],{"className":49975,"style":19812},[83,84],[39,49977,161],{"className":49978},[263],[39,49980],{"className":49981,"style":231},[230],[39,49983,793],{"className":49984},[235],[39,49986],{"className":49987,"style":231},[230],[39,49989,49991,49994,49997,50000,50003,50006],{"className":49990},[74],[39,49992],{"className":49993,"style":17882},[78],[39,49995,5378],{"className":49996},[83],[39,49998,546],{"className":49999},[83],[39,50001],{"className":50002,"style":461},[230],[39,50004,788],{"className":50005},[465],[39,50007],{"className":50008,"style":461},[230],[39,50010,50012,50015,50018,50021,50024],{"className":50011},[74],[39,50013],{"className":50014,"style":17882},[78],[39,50016,17739],{"className":50017},[83],[39,50019],{"className":50020,"style":461},[230],[39,50022,409],{"className":50023},[465],[39,50025],{"className":50026,"style":461},[230],[39,50028,50030,50033,50036,50039,50042],{"className":50029},[74],[39,50031],{"className":50032,"style":2473},[78],[39,50034,28873],{"className":50035},[83],[39,50037],{"className":50038,"style":231},[230],[39,50040,793],{"className":50041},[235],[39,50043],{"className":50044,"style":231},[230],[39,50046,50048,50051,50054],{"className":50047},[74],[39,50049],{"className":50050,"style":17578},[78],[39,50052,49883],{"className":50053},[83],[39,50055,155],{"className":50056},[252],[39,50058,50060],{"className":50059,"translate":43},[746],[39,50061,50063,50124],{"className":50062,"translate":43},[42],[39,50064,50066],{"className":50065},[47],[49,50067,50068],{"xmlns":51,"display":755},[53,50069,50070,50121],{},[56,50071,50072,50074,50076,50078,50080,50082,50084,50086,50088,50090,50092,50094,50096,50098,50100,50102,50104,50106,50108,50110,50112,50114,50116,50118],{},[59,50073,4295],{},[133,50075,136],{"stretchy":135},[59,50077,101],{},[133,50079,418],{},[59,50081,48449],{},[133,50083,161],{"stretchy":135},[133,50085,793],{},[133,50087,5378],{},[539,50089,546],{},[133,50091,788],{},[539,50093,17739],{},[59,50095,4230],{},[133,50097,136],{"stretchy":135},[59,50099,48449],{},[133,50101,161],{"stretchy":135},[133,50103,793],{},[133,50105,5378],{},[539,50107,546],{},[133,50109,788],{},[539,50111,17739],{},[133,50113,409],{},[539,50115,49883],{},[133,50117,793],{},[539,50119,50120],{},"5.39.",[63,50122,50123],{"encoding":65},"Q(A\\to D)=-1+0.9V(D)=-1+0.9\\times7.1=5.39.",[39,50125,50127,50151,50172,50193,50223,50244,50262,50280],{"className":50126,"ariaHidden":70},[69],[39,50128,50130,50133,50136,50139,50142,50145,50148],{"className":50129},[74],[39,50131],{"className":50132,"style":245},[78],[39,50134,4295],{"className":50135},[83,84],[39,50137,136],{"className":50138},[182],[39,50140,101],{"className":50141},[83,84],[39,50143],{"className":50144,"style":231},[230],[39,50146,418],{"className":50147},[235],[39,50149],{"className":50150,"style":231},[230],[39,50152,50154,50157,50160,50163,50166,50169],{"className":50153},[74],[39,50155],{"className":50156,"style":245},[78],[39,50158,48449],{"className":50159,"style":438},[83,84],[39,50161,161],{"className":50162},[263],[39,50164],{"className":50165,"style":231},[230],[39,50167,793],{"className":50168},[235],[39,50170],{"className":50171,"style":231},[230],[39,50173,50175,50178,50181,50184,50187,50190],{"className":50174},[74],[39,50176],{"className":50177,"style":17882},[78],[39,50179,5378],{"className":50180},[83],[39,50182,546],{"className":50183},[83],[39,50185],{"className":50186,"style":461},[230],[39,50188,788],{"className":50189},[465],[39,50191],{"className":50192,"style":461},[230],[39,50194,50196,50199,50202,50205,50208,50211,50214,50217,50220],{"className":50195},[74],[39,50197],{"className":50198,"style":245},[78],[39,50200,17739],{"className":50201},[83],[39,50203,4230],{"className":50204,"style":461},[83,84],[39,50206,136],{"className":50207},[182],[39,50209,48449],{"className":50210,"style":438},[83,84],[39,50212,161],{"className":50213},[263],[39,50215],{"className":50216,"style":231},[230],[39,50218,793],{"className":50219},[235],[39,50221],{"className":50222,"style":231},[230],[39,50224,50226,50229,50232,50235,50238,50241],{"className":50225},[74],[39,50227],{"className":50228,"style":17882},[78],[39,50230,5378],{"className":50231},[83],[39,50233,546],{"className":50234},[83],[39,50236],{"className":50237,"style":461},[230],[39,50239,788],{"className":50240},[465],[39,50242],{"className":50243,"style":461},[230],[39,50245,50247,50250,50253,50256,50259],{"className":50246},[74],[39,50248],{"className":50249,"style":17882},[78],[39,50251,17739],{"className":50252},[83],[39,50254],{"className":50255,"style":461},[230],[39,50257,409],{"className":50258},[465],[39,50260],{"className":50261,"style":461},[230],[39,50263,50265,50268,50271,50274,50277],{"className":50264},[74],[39,50266],{"className":50267,"style":2473},[78],[39,50269,49883],{"className":50270},[83],[39,50272],{"className":50273,"style":231},[230],[39,50275,793],{"className":50276},[235],[39,50278],{"className":50279,"style":231},[230],[39,50281,50283,50286],{"className":50282},[74],[39,50284],{"className":50285,"style":2473},[78],[39,50287,50120],{"className":50288},[83],[11,50290,18110],{},[39,50292,50294],{"className":50293,"translate":43},[746],[39,50295,50297,50340],{"className":50296,"translate":43},[42],[39,50298,50300],{"className":50299},[47],[49,50301,50302],{"xmlns":51,"display":755},[53,50303,50304,50337],{},[56,50305,50306,50308,50310,50312,50314,50316,50318,50320,50322,50324,50326,50329,50331,50333,50335],{},[59,50307,4230],{},[133,50309,136],{"stretchy":135},[59,50311,101],{},[133,50313,161],{"stretchy":135},[133,50315,793],{},[59,50317,1687],{},[133,50319,1690],{},[133,50321,12951],{"stretchy":135},[539,50323,49883],{},[133,50325,155],{"separator":70},[539,50327,50328],{},"5.39",[133,50330,12965],{"stretchy":135},[133,50332,793],{},[539,50334,49883],{},[133,50336,155],{"separator":70},[63,50338,50339],{"encoding":65},"V(A)=\\max\\{7.1,5.39\\}=7.1,",[39,50341,50343,50370,50406],{"className":50342,"ariaHidden":70},[69],[39,50344,50346,50349,50352,50355,50358,50361,50364,50367],{"className":50345},[74],[39,50347],{"className":50348,"style":245},[78],[39,50350,4230],{"className":50351,"style":461},[83,84],[39,50353,136],{"className":50354},[182],[39,50356,101],{"className":50357},[83,84],[39,50359,161],{"className":50360},[263],[39,50362],{"className":50363,"style":231},[230],[39,50365,793],{"className":50366},[235],[39,50368],{"className":50369,"style":231},[230],[39,50371,50373,50376,50379,50382,50385,50388,50391,50394,50397,50400,50403],{"className":50372},[74],[39,50374],{"className":50375,"style":245},[78],[39,50377,1687],{"className":50378},[1744],[39,50380,12951],{"className":50381},[182],[39,50383,49883],{"className":50384},[83],[39,50386,155],{"className":50387},[252],[39,50389],{"className":50390,"style":256},[230],[39,50392,50328],{"className":50393},[83],[39,50395,12965],{"className":50396},[263],[39,50398],{"className":50399,"style":231},[230],[39,50401,793],{"className":50402},[235],[39,50404],{"className":50405,"style":231},[230],[39,50407,50409,50412,50415],{"className":50408},[74],[39,50410],{"className":50411,"style":17578},[78],[39,50413,49883],{"className":50414},[83],[39,50416,155],{"className":50417},[252],[11,50419,50420,50421,50449,50450,50523,50524,50619,50620,50653],{},"最优动作是走向 ",[39,50422,50424,50437],{"className":50423,"translate":43},[42],[39,50425,50427],{"className":50426},[47],[49,50428,50429],{"xmlns":51},[53,50430,50431,50435],{},[56,50432,50433],{},[59,50434,19772],{},[63,50436,19772],{"encoding":65},[39,50438,50440],{"className":50439,"ariaHidden":70},[69],[39,50441,50443,50446],{"className":50442},[74],[39,50444],{"className":50445,"style":79},[78],[39,50447,19772],{"className":50448,"style":19812},[83,84],"。路线 ",[39,50451,50453,50475],{"className":50452,"translate":43},[42],[39,50454,50456],{"className":50455},[47],[49,50457,50458],{"xmlns":51},[53,50459,50460,50472],{},[56,50461,50462,50464,50466,50468,50470],{},[59,50463,101],{},[133,50465,418],{},[59,50467,19772],{},[133,50469,418],{},[59,50471,48444],{},[63,50473,50474],{"encoding":65},"A\\to B\\to C",[39,50476,50478,50496,50514],{"className":50477,"ariaHidden":70},[69],[39,50479,50481,50484,50487,50490,50493],{"className":50480},[74],[39,50482],{"className":50483,"style":79},[78],[39,50485,101],{"className":50486},[83,84],[39,50488],{"className":50489,"style":231},[230],[39,50491,418],{"className":50492},[235],[39,50494],{"className":50495,"style":231},[230],[39,50497,50499,50502,50505,50508,50511],{"className":50498},[74],[39,50500],{"className":50501,"style":79},[78],[39,50503,19772],{"className":50504,"style":19812},[83,84],[39,50506],{"className":50507,"style":231},[230],[39,50509,418],{"className":50510},[235],[39,50512],{"className":50513,"style":231},[230],[39,50515,50517,50520],{"className":50516},[74],[39,50518],{"className":50519,"style":79},[78],[39,50521,48444],{"className":50522,"style":48487},[83,84]," 虽然与 ",[39,50525,50527,50553],{"className":50526,"translate":43},[42],[39,50528,50530],{"className":50529},[47],[49,50531,50532],{"xmlns":51},[53,50533,50534,50550],{},[56,50535,50536,50538,50540,50542,50544,50546,50548],{},[59,50537,101],{},[133,50539,418],{},[59,50541,48449],{},[133,50543,418],{},[59,50545,4253],{},[133,50547,418],{},[59,50549,48444],{},[63,50551,50552],{"encoding":65},"A\\to D\\to E\\to C",[39,50554,50556,50574,50592,50610],{"className":50555,"ariaHidden":70},[69],[39,50557,50559,50562,50565,50568,50571],{"className":50558},[74],[39,50560],{"className":50561,"style":79},[78],[39,50563,101],{"className":50564},[83,84],[39,50566],{"className":50567,"style":231},[230],[39,50569,418],{"className":50570},[235],[39,50572],{"className":50573,"style":231},[230],[39,50575,50577,50580,50583,50586,50589],{"className":50576},[74],[39,50578],{"className":50579,"style":79},[78],[39,50581,48449],{"className":50582,"style":438},[83,84],[39,50584],{"className":50585,"style":231},[230],[39,50587,418],{"className":50588},[235],[39,50590],{"className":50591,"style":231},[230],[39,50593,50595,50598,50601,50604,50607],{"className":50594},[74],[39,50596],{"className":50597,"style":79},[78],[39,50599,4253],{"className":50600,"style":85},[83,84],[39,50602],{"className":50603,"style":231},[230],[39,50605,418],{"className":50606},[235],[39,50608],{"className":50609,"style":231},[230],[39,50611,50613,50616],{"className":50612},[74],[39,50614],{"className":50615,"style":79},[78],[39,50617,48444],{"className":50618,"style":48487},[83,84]," 具有相同的终止奖励，但后者多走一步，多承担一次 ",[39,50621,50623,50638],{"className":50622,"translate":43},[42],[39,50624,50626],{"className":50625},[47],[49,50627,50628],{"xmlns":51},[53,50629,50630,50636],{},[56,50631,50632,50634],{},[133,50633,5378],{},[539,50635,546],{},[63,50637,48859],{"encoding":65},[39,50639,50641],{"className":50640,"ariaHidden":70},[69],[39,50642,50644,50647,50650],{"className":50643},[74],[39,50645],{"className":50646,"style":17882},[78],[39,50648,5378],{"className":50649},[83],[39,50651,546],{"className":50652},[83]," 的即时奖励，并通过折扣因子进一步缩小了终止奖励的现值，因此其价值更低。",[11,50655,50656,50657,50685,50686,50714,50715,50743,50744,50795,50796,50824,50825,50743,50853,50926,50927,50955],{},"这个例子同时展示了最优子结构的两层含义。第一，从 ",[39,50658,50660,50673],{"className":50659,"translate":43},[42],[39,50661,50663],{"className":50662},[47],[49,50664,50665],{"xmlns":51},[53,50666,50667,50671],{},[56,50668,50669],{},[59,50670,101],{},[63,50672,101],{"encoding":65},[39,50674,50676],{"className":50675,"ariaHidden":70},[69],[39,50677,50679,50682],{"className":50678},[74],[39,50680],{"className":50681,"style":79},[78],[39,50683,101],{"className":50684},[83,84]," 出发的最优路线选择了 ",[39,50687,50689,50702],{"className":50688,"translate":43},[42],[39,50690,50692],{"className":50691},[47],[49,50693,50694],{"xmlns":51},[53,50695,50696,50700],{},[56,50697,50698],{},[59,50699,19772],{},[63,50701,19772],{"encoding":65},[39,50703,50705],{"className":50704,"ariaHidden":70},[69],[39,50706,50708,50711],{"className":50707},[74],[39,50709],{"className":50710,"style":79},[78],[39,50712,19772],{"className":50713,"style":19812},[83,84],"，而从 ",[39,50716,50718,50731],{"className":50717,"translate":43},[42],[39,50719,50721],{"className":50720},[47],[49,50722,50723],{"xmlns":51},[53,50724,50725,50729],{},[56,50726,50727],{},[59,50728,19772],{},[63,50730,19772],{"encoding":65},[39,50732,50734],{"className":50733,"ariaHidden":70},[69],[39,50735,50737,50740],{"className":50736},[74],[39,50738],{"className":50739,"style":79},[78],[39,50741,19772],{"className":50742,"style":19812},[83,84]," 出发的剩余路线 ",[39,50745,50747,50765],{"className":50746,"translate":43},[42],[39,50748,50750],{"className":50749},[47],[49,50751,50752],{"xmlns":51},[53,50753,50754,50762],{},[56,50755,50756,50758,50760],{},[59,50757,19772],{},[133,50759,418],{},[59,50761,48444],{},[63,50763,50764],{"encoding":65},"B\\to C",[39,50766,50768,50786],{"className":50767,"ariaHidden":70},[69],[39,50769,50771,50774,50777,50780,50783],{"className":50770},[74],[39,50772],{"className":50773,"style":79},[78],[39,50775,19772],{"className":50776,"style":19812},[83,84],[39,50778],{"className":50779,"style":231},[230],[39,50781,418],{"className":50782},[235],[39,50784],{"className":50785,"style":231},[230],[39,50787,50789,50792],{"className":50788},[74],[39,50790],{"className":50791,"style":79},[78],[39,50793,48444],{"className":50794,"style":48487},[83,84]," 本身是 ",[39,50797,50799,50812],{"className":50798,"translate":43},[42],[39,50800,50802],{"className":50801},[47],[49,50803,50804],{"xmlns":51},[53,50805,50806,50810],{},[56,50807,50808],{},[59,50809,19772],{},[63,50811,19772],{"encoding":65},[39,50813,50815],{"className":50814,"ariaHidden":70},[69],[39,50816,50818,50821],{"className":50817},[74],[39,50819],{"className":50820,"style":79},[78],[39,50822,19772],{"className":50823,"style":19812},[83,84]," 处子问题的最优解。第二，从 ",[39,50826,50828,50841],{"className":50827,"translate":43},[42],[39,50829,50831],{"className":50830},[47],[49,50832,50833],{"xmlns":51},[53,50834,50835,50839],{},[56,50836,50837],{},[59,50838,48449],{},[63,50840,48449],{"encoding":65},[39,50842,50844],{"className":50843,"ariaHidden":70},[69],[39,50845,50847,50850],{"className":50846},[74],[39,50848],{"className":50849,"style":79},[78],[39,50851,48449],{"className":50852,"style":438},[83,84],[39,50854,50856,50878],{"className":50855,"translate":43},[42],[39,50857,50859],{"className":50858},[47],[49,50860,50861],{"xmlns":51},[53,50862,50863,50875],{},[56,50864,50865,50867,50869,50871,50873],{},[59,50866,48449],{},[133,50868,418],{},[59,50870,4253],{},[133,50872,418],{},[59,50874,48444],{},[63,50876,50877],{"encoding":65},"D\\to E\\to C",[39,50879,50881,50899,50917],{"className":50880,"ariaHidden":70},[69],[39,50882,50884,50887,50890,50893,50896],{"className":50883},[74],[39,50885],{"className":50886,"style":79},[78],[39,50888,48449],{"className":50889,"style":438},[83,84],[39,50891],{"className":50892,"style":231},[230],[39,50894,418],{"className":50895},[235],[39,50897],{"className":50898,"style":231},[230],[39,50900,50902,50905,50908,50911,50914],{"className":50901},[74],[39,50903],{"className":50904,"style":79},[78],[39,50906,4253],{"className":50907,"style":85},[83,84],[39,50909],{"className":50910,"style":231},[230],[39,50912,418],{"className":50913},[235],[39,50915],{"className":50916,"style":231},[230],[39,50918,50920,50923],{"className":50919},[74],[39,50921],{"className":50922,"style":79},[78],[39,50924,48444],{"className":50925,"style":48487},[83,84]," 也是 ",[39,50928,50930,50943],{"className":50929,"translate":43},[42],[39,50931,50933],{"className":50932},[47],[49,50934,50935],{"xmlns":51},[53,50936,50937,50941],{},[56,50938,50939],{},[59,50940,48449],{},[63,50942,48449],{"encoding":65},[39,50944,50946],{"className":50945,"ariaHidden":70},[69],[39,50947,50949,50952],{"className":50948},[74],[39,50950],{"className":50951,"style":79},[78],[39,50953,48449],{"className":50954,"style":438},[83,84]," 处子问题的最优解。整体最优并不是把各段的局部最优任意拼接，而是沿着实际到达的后继状态，取该后继状态的最优续策略。",[11,50957,50958,50959,51025],{},"若改用 ",[39,50960,50962,50986],{"className":50961,"translate":43},[42],[39,50963,50965],{"className":50964},[47],[49,50966,50967],{"xmlns":51},[53,50968,50969,50983],{},[56,50970,50971,50973,50975,50977,50979,50981],{},[59,50972,4230],{},[133,50974,136],{"stretchy":135},[59,50976,48444],{},[133,50978,161],{"stretchy":135},[133,50980,793],{},[539,50982,48919],{},[63,50984,50985],{"encoding":65},"V(C)=10",[39,50987,50989,51016],{"className":50988,"ariaHidden":70},[69],[39,50990,50992,50995,50998,51001,51004,51007,51010,51013],{"className":50991},[74],[39,50993],{"className":50994,"style":245},[78],[39,50996,4230],{"className":50997,"style":461},[83,84],[39,50999,136],{"className":51000},[182],[39,51002,48444],{"className":51003,"style":48487},[83,84],[39,51005,161],{"className":51006},[263],[39,51008],{"className":51009,"style":231},[230],[39,51011,793],{"className":51012},[235],[39,51014],{"className":51015,"style":231},[230],[39,51017,51019,51022],{"className":51018},[74],[39,51020],{"className":51021,"style":2473},[78],[39,51023,48919],{"className":51024},[83]," 的约定，即把终止奖励直接记在终止状态上，而不是记在进入终止状态的转移上，则上述数值会改变，但最优动作不变。两种约定的差别只在于奖励被记在转移上还是被记在状态上，贝尔曼方程的形式必须与所采用的约定保持一致。",[21,51027,51029],{"id":51028},"五收敛性与误差界","五、收敛性与误差界",[26,51031,51033],{"id":51032},"_51-不动点迭代","5.1 不动点迭代",[11,51035,51036,51037,51144],{},"压缩映射不仅证明了解的存在唯一性，还给出了构造解的方法。任取初始函数 ",[39,51038,51040,51068],{"className":51039,"translate":43},[42],[39,51041,51043],{"className":51042},[47],[49,51044,51045],{"xmlns":51},[53,51046,51047,51065],{},[56,51048,51049,51055,51057,51059,51061,51063],{},[613,51050,51051,51053],{},[59,51052,4230],{},[539,51054,541],{},[133,51056,534],{},[59,51058,19772],{},[133,51060,136],{"stretchy":135},[59,51062,61],{},[133,51064,161],{"stretchy":135},[63,51066,51067],{"encoding":65},"V_0\\in B(S)",[39,51069,51071,51126],{"className":51070,"ariaHidden":70},[69],[39,51072,51074,51077,51117,51120,51123],{"className":51073},[74],[39,51075],{"className":51076,"style":634},[78],[39,51078,51080,51083],{"className":51079},[83],[39,51081,4230],{"className":51082,"style":461},[83,84],[39,51084,51086],{"className":51085},[192],[39,51087,51089,51109],{"className":51088},[196,649],[39,51090,51092,51106],{"className":51091},[200],[39,51093,51095],{"className":51094,"style":963},[204],[39,51096,51097,51100],{"style":13607},[39,51098],{"className":51099,"style":213},[212],[39,51101,51103],{"className":51102},[217,218,219,220],[39,51104,541],{"className":51105},[83,220],[39,51107,673],{"className":51108},[672],[39,51110,51112],{"className":51111},[200],[39,51113,51115],{"className":51114,"style":680},[204],[39,51116],{},[39,51118],{"className":51119,"style":231},[230],[39,51121,534],{"className":51122},[235],[39,51124],{"className":51125,"style":231},[230],[39,51127,51129,51132,51135,51138,51141],{"className":51128},[74],[39,51130],{"className":51131,"style":245},[78],[39,51133,19772],{"className":51134,"style":19812},[83,84],[39,51136,136],{"className":51137},[182],[39,51139,61],{"className":51140,"style":85},[83,84],[39,51142,161],{"className":51143},[263],"，定义迭代",[39,51146,51148],{"className":51147,"translate":43},[746],[39,51149,51151,51207],{"className":51150,"translate":43},[42],[39,51152,51154],{"className":51153},[47],[49,51155,51156],{"xmlns":51,"display":755},[53,51157,51158,51204],{},[56,51159,51160,51172,51174,51176,51182,51184,51186,51188,51190,51192,51194,51196,51198,51200,51202],{},[613,51161,51162,51164],{},[59,51163,4230],{},[56,51165,51166,51168,51170],{},[59,51167,3196],{},[133,51169,788],{},[539,51171,546],{},[133,51173,793],{},[59,51175,11772],{"mathvariant":617},[613,51177,51178,51180],{},[59,51179,4230],{},[59,51181,3196],{},[133,51183,155],{"separator":70},[230,51185],{"width":10251},[59,51187,3196],{},[133,51189,793],{},[539,51191,541],{},[133,51193,155],{"separator":70},[539,51195,546],{},[133,51197,155],{"separator":70},[539,51199,17359],{},[133,51201,155],{"separator":70},[133,51203,2641],{},[63,51205,51206],{"encoding":65},"V_{k+1}=\\mathcal{T}V_k,\n\\qquad k=0,1,2,\\dots",[39,51208,51210,51274,51345],{"className":51209,"ariaHidden":70},[69],[39,51211,51213,51216,51265,51268,51271],{"className":51212},[74],[39,51214],{"className":51215,"style":7584},[78],[39,51217,51219,51222],{"className":51218},[83],[39,51220,4230],{"className":51221,"style":461},[83,84],[39,51223,51225],{"className":51224},[192],[39,51226,51228,51257],{"className":51227},[196,649],[39,51229,51231,51254],{"className":51230},[200],[39,51232,51234],{"className":51233,"style":3419},[204],[39,51235,51236,51239],{"style":13607},[39,51237],{"className":51238,"style":213},[212],[39,51240,51242],{"className":51241},[217,218,219,220],[39,51243,51245,51248,51251],{"className":51244},[83,220],[39,51246,3196],{"className":51247,"style":3321},[83,84,220],[39,51249,788],{"className":51250},[465,220],[39,51252,546],{"className":51253},[83,220],[39,51255,673],{"className":51256},[672],[39,51258,51260],{"className":51259},[200],[39,51261,51263],{"className":51262,"style":994},[204],[39,51264],{},[39,51266],{"className":51267,"style":231},[230],[39,51269,793],{"className":51270},[235],[39,51272],{"className":51273,"style":231},[230],[39,51275,51277,51281,51284,51324,51327,51330,51333,51336,51339,51342],{"className":51276},[74],[39,51278],{"className":51279,"style":51280},[78],"height:0.8889em;vertical-align:-0.1944em;",[39,51282,11772],{"className":51283,"style":11793},[83,641],[39,51285,51287,51290],{"className":51286},[83],[39,51288,4230],{"className":51289,"style":461},[83,84],[39,51291,51293],{"className":51292},[192],[39,51294,51296,51316],{"className":51295},[196,649],[39,51297,51299,51313],{"className":51298},[200],[39,51300,51302],{"className":51301,"style":3419},[204],[39,51303,51304,51307],{"style":13607},[39,51305],{"className":51306,"style":213},[212],[39,51308,51310],{"className":51309},[217,218,219,220],[39,51311,3196],{"className":51312,"style":3321},[83,84,220],[39,51314,673],{"className":51315},[672],[39,51317,51319],{"className":51318},[200],[39,51320,51322],{"className":51321,"style":680},[204],[39,51323],{},[39,51325,155],{"className":51326},[252],[39,51328],{"className":51329,"style":10787},[230],[39,51331],{"className":51332,"style":256},[230],[39,51334,3196],{"className":51335,"style":3321},[83,84],[39,51337],{"className":51338,"style":231},[230],[39,51340,793],{"className":51341},[235],[39,51343],{"className":51344,"style":231},[230],[39,51346,51348,51351,51354,51357,51360,51363,51366,51369,51372,51375,51378],{"className":51347},[74],[39,51349],{"className":51350,"style":17578},[78],[39,51352,541],{"className":51353},[83],[39,51355,155],{"className":51356},[252],[39,51358],{"className":51359,"style":256},[230],[39,51361,546],{"className":51362},[83],[39,51364,155],{"className":51365},[252],[39,51367],{"className":51368,"style":256},[230],[39,51370,17359],{"className":51371},[83],[39,51373,155],{"className":51374},[252],[39,51376],{"className":51377,"style":256},[230],[39,51379,2641],{"className":51380},[2842],[11,51382,24575,51383,51411,51412,34856,51470,51528],{},[39,51384,51386,51399],{"className":51385,"translate":43},[42],[39,51387,51389],{"className":51388},[47],[49,51390,51391],{"xmlns":51},[53,51392,51393,51397],{},[56,51394,51395],{},[59,51396,11772],{"mathvariant":617},[63,51398,34785],{"encoding":65},[39,51400,51402],{"className":51401,"ariaHidden":70},[69],[39,51403,51405,51408],{"className":51404},[74],[39,51406],{"className":51407,"style":79},[78],[39,51409,11772],{"className":51410,"style":11793},[83,641]," 可以是 ",[39,51413,51415,51432],{"className":51414,"translate":43},[42],[39,51416,51418],{"className":51417},[47],[49,51419,51420],{"xmlns":51},[53,51421,51422,51430],{},[56,51423,51424],{},[138,51425,51426,51428],{},[59,51427,11772],{"mathvariant":617},[59,51429,2098],{},[63,51431,11777],{"encoding":65},[39,51433,51435],{"className":51434,"ariaHidden":70},[69],[39,51436,51438,51441],{"className":51437},[74],[39,51439],{"className":51440,"style":79},[78],[39,51442,51444,51447],{"className":51443},[83],[39,51445,11772],{"className":51446,"style":11793},[83,641],[39,51448,51450],{"className":51449},[192],[39,51451,51453],{"className":51452},[196],[39,51454,51456],{"className":51455},[200],[39,51457,51459],{"className":51458,"style":6410},[204],[39,51460,51461,51464],{"style":208},[39,51462],{"className":51463,"style":213},[212],[39,51465,51467],{"className":51466},[217,218,219,220],[39,51468,2098],{"className":51469,"style":2114},[83,84,220],[39,51471,51473,51490],{"className":51472,"translate":43},[42],[39,51474,51476],{"className":51475},[47],[49,51477,51478],{"xmlns":51},[53,51479,51480,51488],{},[56,51481,51482],{},[138,51483,51484,51486],{},[59,51485,11772],{"mathvariant":617},[133,51487,21884],{},[63,51489,21887],{"encoding":65},[39,51491,51493],{"className":51492,"ariaHidden":70},[69],[39,51494,51496,51499],{"className":51495},[74],[39,51497],{"className":51498,"style":21897},[78],[39,51500,51502,51505],{"className":51501},[83],[39,51503,11772],{"className":51504,"style":11793},[83,641],[39,51506,51508],{"className":51507},[192],[39,51509,51511],{"className":51510},[196],[39,51512,51514],{"className":51513},[200],[39,51515,51517],{"className":51516,"style":21897},[204],[39,51518,51519,51522],{"style":208},[39,51520],{"className":51521,"style":213},[212],[39,51523,51525],{"className":51524},[217,218,219,220],[39,51526,21884],{"className":51527},[465,220],"。在强化学习中，这一迭代称为值迭代。由 Banach 不动点定理，迭代序列必然收敛到相应的不动点。",[31,51530,51533,51927,52331],{"id":51531,"title":51532},"theorem-convergence","定理 5.1（几何收敛）",[11,51534,37,51535,742,51607,21402,51635,34856,51693,34915,51751,51820,51821,742],{},[39,51536,51538,51559],{"className":51537,"translate":43},[42],[39,51539,51541],{"className":51540},[47],[49,51542,51543],{"xmlns":51},[53,51544,51545,51557],{},[56,51546,51547,51549,51551,51553,51555],{},[539,51548,541],{},[133,51550,1824],{},[59,51552,531],{},[133,51554,1693],{},[539,51556,546],{},[63,51558,14577],{"encoding":65},[39,51560,51562,51580,51598],{"className":51561,"ariaHidden":70},[69],[39,51563,51565,51568,51571,51574,51577],{"className":51564},[74],[39,51566],{"className":51567,"style":14587},[78],[39,51569,541],{"className":51570},[83],[39,51572],{"className":51573,"style":231},[230],[39,51575,1824],{"className":51576},[235],[39,51578],{"className":51579,"style":231},[230],[39,51581,51583,51586,51589,51592,51595],{"className":51582},[74],[39,51584],{"className":51585,"style":561},[78],[39,51587,531],{"className":51588,"style":565},[83,84],[39,51590],{"className":51591,"style":231},[230],[39,51593,1693],{"className":51594},[235],[39,51596],{"className":51597,"style":231},[230],[39,51599,51601,51604],{"className":51600},[74],[39,51602],{"className":51603,"style":2473},[78],[39,51605,546],{"className":51606},[83],[39,51608,51610,51623],{"className":51609,"translate":43},[42],[39,51611,51613],{"className":51612},[47],[49,51614,51615],{"xmlns":51},[53,51616,51617,51621],{},[56,51618,51619],{},[59,51620,11772],{"mathvariant":617},[63,51622,34785],{"encoding":65},[39,51624,51626],{"className":51625,"ariaHidden":70},[69],[39,51627,51629,51632],{"className":51628},[74],[39,51630],{"className":51631,"style":79},[78],[39,51633,11772],{"className":51634,"style":11793},[83,641],[39,51636,51638,51655],{"className":51637,"translate":43},[42],[39,51639,51641],{"className":51640},[47],[49,51642,51643],{"xmlns":51},[53,51644,51645,51653],{},[56,51646,51647],{},[138,51648,51649,51651],{},[59,51650,11772],{"mathvariant":617},[59,51652,2098],{},[63,51654,11777],{"encoding":65},[39,51656,51658],{"className":51657,"ariaHidden":70},[69],[39,51659,51661,51664],{"className":51660},[74],[39,51662],{"className":51663,"style":79},[78],[39,51665,51667,51670],{"className":51666},[83],[39,51668,11772],{"className":51669,"style":11793},[83,641],[39,51671,51673],{"className":51672},[192],[39,51674,51676],{"className":51675},[196],[39,51677,51679],{"className":51678},[200],[39,51680,51682],{"className":51681,"style":6410},[204],[39,51683,51684,51687],{"style":208},[39,51685],{"className":51686,"style":213},[212],[39,51688,51690],{"className":51689},[217,218,219,220],[39,51691,2098],{"className":51692,"style":2114},[83,84,220],[39,51694,51696,51713],{"className":51695,"translate":43},[42],[39,51697,51699],{"className":51698},[47],[49,51700,51701],{"xmlns":51},[53,51702,51703,51711],{},[56,51704,51705],{},[138,51706,51707,51709],{},[59,51708,11772],{"mathvariant":617},[133,51710,21884],{},[63,51712,21887],{"encoding":65},[39,51714,51716],{"className":51715,"ariaHidden":70},[69],[39,51717,51719,51722],{"className":51718},[74],[39,51720],{"className":51721,"style":21897},[78],[39,51723,51725,51728],{"className":51724},[83],[39,51726,11772],{"className":51727,"style":11793},[83,641],[39,51729,51731],{"className":51730},[192],[39,51732,51734],{"className":51733},[196],[39,51735,51737],{"className":51736},[200],[39,51738,51740],{"className":51739,"style":21897},[204],[39,51741,51742,51745],{"style":208},[39,51743],{"className":51744,"style":213},[212],[39,51746,51748],{"className":51747},[217,218,219,220],[39,51749,21884],{"className":51750},[465,220],[39,51752,51754,51771],{"className":51753,"translate":43},[42],[39,51755,51757],{"className":51756},[47],[49,51758,51759],{"xmlns":51},[53,51760,51761,51769],{},[56,51762,51763],{},[613,51764,51765,51767],{},[59,51766,4230],{},[59,51768,1696],{"mathvariant":145},[63,51770,34936],{"encoding":65},[39,51772,51774],{"className":51773,"ariaHidden":70},[69],[39,51775,51777,51780],{"className":51776},[74],[39,51778],{"className":51779,"style":634},[78],[39,51781,51783,51786],{"className":51782},[83],[39,51784,4230],{"className":51785,"style":461},[83,84],[39,51787,51789],{"className":51788},[192],[39,51790,51792,51812],{"className":51791},[196,649],[39,51793,51795,51809],{"className":51794},[200],[39,51796,51798],{"className":51797,"style":1728},[204],[39,51799,51800,51803],{"style":13607},[39,51801],{"className":51802,"style":213},[212],[39,51804,51806],{"className":51805},[217,218,219,220],[39,51807,1696],{"className":51808},[83,220],[39,51810,673],{"className":51811},[672],[39,51813,51815],{"className":51814},[200],[39,51816,51818],{"className":51817,"style":680},[204],[39,51819],{},"。则对任意 ",[39,51822,51824,51851],{"className":51823,"translate":43},[42],[39,51825,51827],{"className":51826},[47],[49,51828,51829],{"xmlns":51},[53,51830,51831,51849],{},[56,51832,51833,51839,51841,51843,51845,51847],{},[613,51834,51835,51837],{},[59,51836,4230],{},[539,51838,541],{},[133,51840,534],{},[59,51842,19772],{},[133,51844,136],{"stretchy":135},[59,51846,61],{},[133,51848,161],{"stretchy":135},[63,51850,51067],{"encoding":65},[39,51852,51854,51909],{"className":51853,"ariaHidden":70},[69],[39,51855,51857,51860,51900,51903,51906],{"className":51856},[74],[39,51858],{"className":51859,"style":634},[78],[39,51861,51863,51866],{"className":51862},[83],[39,51864,4230],{"className":51865,"style":461},[83,84],[39,51867,51869],{"className":51868},[192],[39,51870,51872,51892],{"className":51871},[196,649],[39,51873,51875,51889],{"className":51874},[200],[39,51876,51878],{"className":51877,"style":963},[204],[39,51879,51880,51883],{"style":13607},[39,51881],{"className":51882,"style":213},[212],[39,51884,51886],{"className":51885},[217,218,219,220],[39,51887,541],{"className":51888},[83,220],[39,51890,673],{"className":51891},[672],[39,51893,51895],{"className":51894},[200],[39,51896,51898],{"className":51897,"style":680},[204],[39,51899],{},[39,51901],{"className":51902,"style":231},[230],[39,51904,534],{"className":51905},[235],[39,51907],{"className":51908,"style":231},[230],[39,51910,51912,51915,51918,51921,51924],{"className":51911},[74],[39,51913],{"className":51914,"style":245},[78],[39,51916,19772],{"className":51917,"style":19812},[83,84],[39,51919,136],{"className":51920},[182],[39,51922,61],{"className":51923,"style":85},[83,84],[39,51925,161],{"className":51926},[263],[39,51928,51930],{"className":51929,"translate":43},[746],[39,51931,51933,51999],{"className":51932,"translate":43},[42],[39,51934,51936],{"className":51935},[47],[49,51937,51938],{"xmlns":51,"display":755},[53,51939,51940,51996],{},[56,51941,51942,51944,51950,51952,51958,51964,51966,51972,51974,51980,51982,51988,51994],{},[59,51943,15389],{"mathvariant":145},[613,51945,51946,51948],{},[59,51947,4230],{},[59,51949,3196],{},[133,51951,5378],{},[613,51953,51954,51956],{},[59,51955,4230],{},[59,51957,1696],{"mathvariant":145},[613,51959,51960,51962],{},[59,51961,15389],{"mathvariant":145},[59,51963,1696],{"mathvariant":145},[133,51965,1824],{},[138,51967,51968,51970],{},[59,51969,531],{},[59,51971,3196],{},[59,51973,15389],{"mathvariant":145},[613,51975,51976,51978],{},[59,51977,4230],{},[539,51979,541],{},[133,51981,5378],{},[613,51983,51984,51986],{},[59,51985,4230],{},[59,51987,1696],{"mathvariant":145},[613,51989,51990,51992],{},[59,51991,15389],{"mathvariant":145},[59,51993,1696],{"mathvariant":145},[59,51995,2903],{"mathvariant":145},[63,51997,51998],{"encoding":65},"\\|V_k-V_\\infty\\|_\\infty\n\\le\\gamma^k\\|V_0-V_\\infty\\|_\\infty.",[39,52000,52002,52060,52155,52242],{"className":52001,"ariaHidden":70},[69],[39,52003,52005,52008,52011,52051,52054,52057],{"className":52004},[74],[39,52006],{"className":52007,"style":245},[78],[39,52009,15389],{"className":52010},[83],[39,52012,52014,52017],{"className":52013},[83],[39,52015,4230],{"className":52016,"style":461},[83,84],[39,52018,52020],{"className":52019},[192],[39,52021,52023,52043],{"className":52022},[196,649],[39,52024,52026,52040],{"className":52025},[200],[39,52027,52029],{"className":52028,"style":3419},[204],[39,52030,52031,52034],{"style":13607},[39,52032],{"className":52033,"style":213},[212],[39,52035,52037],{"className":52036},[217,218,219,220],[39,52038,3196],{"className":52039,"style":3321},[83,84,220],[39,52041,673],{"className":52042},[672],[39,52044,52046],{"className":52045},[200],[39,52047,52049],{"className":52048,"style":680},[204],[39,52050],{},[39,52052],{"className":52053,"style":461},[230],[39,52055,5378],{"className":52056},[465],[39,52058],{"className":52059,"style":461},[230],[39,52061,52063,52066,52106,52146,52149,52152],{"className":52062},[74],[39,52064],{"className":52065,"style":245},[78],[39,52067,52069,52072],{"className":52068},[83],[39,52070,4230],{"className":52071,"style":461},[83,84],[39,52073,52075],{"className":52074},[192],[39,52076,52078,52098],{"className":52077},[196,649],[39,52079,52081,52095],{"className":52080},[200],[39,52082,52084],{"className":52083,"style":1728},[204],[39,52085,52086,52089],{"style":13607},[39,52087],{"className":52088,"style":213},[212],[39,52090,52092],{"className":52091},[217,218,219,220],[39,52093,1696],{"className":52094},[83,220],[39,52096,673],{"className":52097},[672],[39,52099,52101],{"className":52100},[200],[39,52102,52104],{"className":52103,"style":680},[204],[39,52105],{},[39,52107,52109,52112],{"className":52108},[83],[39,52110,15389],{"className":52111},[83],[39,52113,52115],{"className":52114},[192],[39,52116,52118,52138],{"className":52117},[196,649],[39,52119,52121,52135],{"className":52120},[200],[39,52122,52124],{"className":52123,"style":1728},[204],[39,52125,52126,52129],{"style":966},[39,5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V_\\infty",[39,52364,52366,52421],{"className":52365,"ariaHidden":70},[69],[39,52367,52369,52372,52412,52415,52418],{"className":52368},[74],[39,52370],{"className":52371,"style":634},[78],[39,52373,52375,52378],{"className":52374},[83],[39,52376,4230],{"className":52377,"style":461},[83,84],[39,52379,52381],{"className":52380},[192],[39,52382,52384,52404],{"className":52383},[196,649],[39,52385,52387,52401],{"className":52386},[200],[39,52388,52390],{"className":52389,"style":3419},[204],[39,52391,52392,52395],{"style":13607},[39,52393],{"className":52394,"style":213},[212],[39,52396,52398],{"className":52397},[217,218,219,220],[39,52399,3196],{"className":52400,"style":3321},[83,84,220],[39,52402,673],{"className":52403},[672],[39,52405,52407],{"className":52406},[200],[39,52408,52410],{"className":52409,"style":680},[204],[39,52411],{},[39,52413],{"className":52414,"style":231},[230],[39,52416,418],{"className":52417},[235],[39,52419],{"className":52420,"style":231},[230],[39,52422,52424,52427],{"className":52423},[74],[39,52425],{"className":52426,"style":634},[78],[39,52428,52430,52433],{"className":52429},[83],[39,52431,4230],{"className":52432,"style":461},[83,84],[39,52434,52436],{"className":52435},[192],[39,52437,52439,52459],{"className":52438},[196,649],[39,52440,52442,52456],{"className":52441},[200],[39,52443,52445],{"className":52444,"style":1728},[204],[39,52446,52447,52450],{"style":13607},[39,52448],{"className":52449,"style":213},[212],[39,52451,52453],{"className":52452},[217,218,219,220],[39,52454,1696],{"className":52455},[83,220],[39,52457,673],{"className":52458},[672],[39,52460,52462],{"className":52461},[200],[39,52463,52465],{"className":52464,"style":680},[204],[39,52466],{}," 在 sup 范数下成立。",[11,52469,52470,52472],{},[1489,52471,5595],{}," 由压缩性，",[39,52474,52476],{"className":52475,"translate":43},[746],[39,52477,52479,52607],{"className":52478,"translate":43},[42],[39,52480,52482],{"className":52481},[47],[49,52483,52484],{"xmlns":51,"display":755},[53,52485,52486,52604],{},[759,52487,52488,52560],{"rowspacing":761,"columnalign":762,"columnspacing":146},[764,52489,52490,52524],{},[767,52491,52492],{},[770,52493,52494],{"scriptlevel":541,"displaystyle":70},[56,52495,52496,52498,52510,52512,52518],{},[59,52497,15389],{"mathvariant":145},[613,52499,52500,52502],{},[59,52501,4230],{},[56,52503,52504,52506,52508],{},[59,52505,3196],{},[133,52507,788],{},[539,52509,546],{},[133,52511,5378],{},[613,52513,52514,52516],{},[59,52515,4230],{},[59,52517,1696],{"mathvariant":145},[613,52519,52520,52522],{},[59,52521,15389],{"mathvariant":145},[59,52523,1696],{"mathvariant":145},[767,52525,52526],{},[770,52527,52528],{"scriptlevel":541,"displaystyle":70},[56,52529,52530,52532,52534,52536,52538,52544,52546,52548,52554],{},[56,52531],{},[133,52533,793],{},[59,52535,15389],{"mathvariant":145},[59,52537,11772],{"mathvariant":617},[613,52539,52540,52542],{},[59,52541,4230],{},[59,52543,3196],{},[133,52545,5378],{},[59,52547,11772],{"mathvariant":617},[613,52549,52550,52552],{},[59,52551,4230],{},[59,52553,1696],{"mathvariant":145},[613,52555,52556,52558],{},[59,52557,15389],{"mathvariant":145},[59,52559,1696],{"mathvariant":145},[764,52561,52562,52568],{},[767,52563,52564],{},[770,52565,52566],{"scriptlevel":541,"displaystyle":70},[56,52567],{},[767,52569,52570],{},[770,52571,52572],{"scriptlevel":541,"displaystyle":70},[56,52573,52574,52576,52578,52580,52582,52588,52590,52596,52602],{},[56,52575],{},[133,52577,1824],{},[59,52579,531],{},[59,52581,15389],{"mathvariant":145},[613,52583,52584,52586],{},[59,52585,4230],{},[59,52587,3196],{},[133,52589,5378],{},[613,52591,52592,52594],{},[59,52593,4230],{},[59,52595,1696],{"mathvariant":145},[613,52597,52598,52600],{},[59,52599,15389],{"mathvariant":145},[59,52601,1696],{"mathvariant":145},[59,52603,2903],{"mathvariant":145},[63,52605,52606],{"encoding":65},"\\begin{aligned}\n\\|V_{k+1}-V_\\infty\\|_\\infty\n&=\\|\\mathcal{T}V_k-\\mathcal{T}V_\\infty\\|_\\infty\\\\\n&\\le\\gamma\\|V_k-V_\\infty\\|_\\infty.\n\\end{aligned}",[39,52608,52610],{"className":52609,"ariaHidden":70},[69],[39,52611,52613,52616],{"className":52612},[74],[39,52614],{"className":52615,"style":4374},[78],[39,52617,52619],{"className":52618},[83],[39,52620,52622,52802],{"className":52621},[759],[39,52623,52625],{"className":52624},[918],[39,52626,52628,52794],{"className":52627},[196,649],[39,52629,52631,52791],{"className":52630},[200],[39,52632,52634,52783],{"className":52633,"style":4393},[204],[39,52635,52636,52639],{"style":4396},[39,52637],{"className":52638,"style":935},[212],[39,52640,52642,52645,52694,52697,52700,52703,52743],{"className":52641},[83],[39,52643,15389],{"className":52644},[83],[39,52646,52648,52651],{"className":52647},[83],[39,52649,4230],{"className":52650,"style":461},[83,84],[39,52652,52654],{"className":52653},[192],[39,52655,52657,52686],{"className":52656},[196,649],[39,52658,52660,52683],{"className":52659},[200],[39,52661,52663],{"className":52662,"style":3419},[204],[39,52664,52665,52668],{"style":13607},[39,52666],{"className":52667,"style":213},[212],[39,52669,52671],{"className":52670},[217,218,219,220],[39,52672,52674,52677,52680],{"className":52673},[83,220],[39,52675,3196],{"className":52676,"style":3321},[83,84,220],[39,52678,788],{"className":52679},[465,220],[39,52681,546],{"className":52682},[83,220],[39,52684,673],{"className":52685},[672],[39,52687,52689],{"className":52688},[200],[39,52690,52692],{"className":52691,"style":994},[204],[39,52693],{},[39,52695],{"className":52696,"style":461},[230],[39,52698,5378],{"className":52699},[465],[39,52701],{"className":52702,"style":461},[230],[39,52704,52706,52709],{"className":52705},[83],[39,52707,4230],{"className":52708,"style":461},[83,84],[39,52710,52712],{"className":52711},[192],[39,52713,52715,52735],{"className":52714},[196,649],[39,52716,52718,52732],{"className":52717},[200],[39,52719,52721],{"className":52720,"style":1728},[204],[39,52722,52723,52726],{"style":13607},[39,52724],{"className":52725,"style":213},[212],[39,52727,52729],{"className":52728},[217,218,219,220],[39,52730,1696],{"className":52731},[83,220],[39,52733,673],{"className":52734},[672],[39,52736,52738],{"className":52737},[200],[39,52739,52741],{"className":52740,"style":680},[204],[39,52742],{},[39,52744,52746,52749],{"className":52745},[83],[39,52747,15389],{"className":52748},[83],[39,52750,52752],{"className":52751},[192],[39,52753,52755,52775],{"className":52754},[196,649],[39,52756,52758,52772],{"className":52757},[200],[39,52759,52761],{"className":52760,"style":1728},[204],[39,52762,52763,52766],{"style":966},[39,52764],{"className":52765,"style":213},[212],[39,52767,52769],{"className":52768},[217,218,219,220],[39,52770,1696],{"className":52771},[83,220],[39,52773,673],{"className":52774},[672],[39,52776,52778],{"className":52777},[200],[39,52779,52781],{"className":52780,"style":680},[204],[39,52782],{},[39,52784,52785,52788],{"style":4444},[39,52786],{"className":52787,"style":935},[212],[39,52789],{"className":52790},[83],[39,52792,673],{"className":52793},[672],[39,52795,52797],{"className":52796},[200],[39,52798,52800],{"className":52799,"style":4507},[204],[39,52801],{},[39,52803,52805],{"className":52804},[1223],[39,52806,52808,53133],{"className":52807},[196,649],[39,52809,52811,53130],{"className":52810},[200],[39,52812,52814,52972],{"className":52813,"style":4393},[204],[39,52815,52816,52819],{"style":4396},[39,52817],{"className":52818,"style":935},[212],[39,52820,52822,52825,52828,52831,52834,52837,52840,52880,52883,52886,52889,52892,52932],{"className":52821},[83],[39,52823],{"className":52824},[83],[39,52826],{"className":52827,"style":231},[230],[39,52829,793],{"className":52830},[235],[39,52832],{"className":52833,"style":231},[230],[39,52835,15389],{"className":52836},[83],[39,52838,11772],{"className":52839,"style":11793},[83,641],[39,52841,52843,52846],{"className":52842},[83],[39,52844,4230],{"className":52845,"style":461},[83,84],[39,52847,52849],{"className":52848},[192],[39,52850,52852,52872],{"className":52851},[196,649],[39,52853,52855,52869],{"className":52854},[200],[39,52856,52858],{"className":52857,"style":3419},[204],[39,52859,52860,52863],{"style":13607},[39,52861],{"className":52862,"style":213},[212],[39,52864,52866],{"className":52865},[217,218,219,220],[39,52867,3196],{"className":52868,"style":3321},[83,84,220],[39,52870,673],{"className":52871},[672],[39,52873,52875],{"className":52874},[200],[39,52876,52878],{"className":52877,"style":680},[204],[39,52879],{},[39,52881],{"className":52882,"style":461},[230],[39,52884,5378],{"className":52885},[465],[39,52887],{"className":52888,"style":461},[230],[39,52890,11772],{"className":52891,"style":11793},[83,641],[39,52893,52895,52898],{"className":52894},[83],[39,52896,4230],{"className":52897,"style":461},[83,84],[39,52899,52901],{"className":52900},[192],[39,52902,52904,52924],{"className":52903},[196,649],[39,52905,52907,52921],{"className":52906},[200],[39,52908,52910],{"className":52909,"style":1728},[204],[39,52911,52912,52915],{"style":13607},[39,52913],{"className":52914,"style":213},[212],[39,52916,52918],{"className":52917},[217,218,219,220],[39,52919,1696],{"className":52920},[83,220],[39,52922,673],{"className":52923},[672],[39,52925,52927],{"className":52926},[200],[39,52928,52930],{"className":52929,"style":680},[204],[39,52931],{},[39,52933,52935,52938],{"className":52934},[83],[39,52936,15389],{"className":52937},[83],[39,52939,52941],{"className":52940},[192],[39,52942,52944,52964],{"className":52943},[196,649],[39,52945,52947,52961],{"className":52946},[200],[39,52948,52950],{"className":52949,"style":1728},[204],[39,52951,52952,52955],{"style":966},[39,52953],{"className":52954,"style":213},[212],[39,52956,52958],{"className":52957},[217,218,219,220],[39,52959,1696],{"className":52960},[83,220],[39,52962,673],{"className":52963},[672],[39,52965,52967],{"className":52966},[200],[39,52968,52970],{"className":52969,"style":680},[204],[39,52971],{},[39,52973,52974,52977],{"style":4444},[39,52975],{"className":52976,"style":935},[212],[39,52978,52980,52983,52986,52989,52992,52995,52998,53038,53041,53044,53047,53087,53127],{"className":52979},[83],[39,52981],{"className":52982},[83],[39,52984],{"className":52985,"style":231},[230],[39,52987,1824],{"className":52988},[235],[39,52990],{"className":52991,"style":231},[230],[39,52993,531],{"className":52994,"style":565},[83,84],[39,52996,15389],{"className":52997},[83],[39,52999,53001,53004],{"className":53000},[83],[39,53002,4230],{"className":53003,"style":461},[83,84],[39,53005,53007],{"className":53006},[192],[39,53008,53010,53030],{"className":53009},[196,649],[39,53011,53013,53027],{"className":53012},[200],[39,53014,53016],{"className":53015,"style":3419},[204],[39,53017,53018,53021],{"style":13607},[39,53019],{"className":53020,"style":213},[212],[39,53022,53024],{"className":53023},[217,218,219,220],[39,53025,3196],{"className":53026,"style":3321},[83,84,220],[39,53028,673],{"className":53029},[672],[39,53031,53033],{"className":53032},[200],[39,53034,53036],{"className":53035,"style":680},[204],[39,53037],{},[39,53039],{"className":53040,"style":461},[230],[39,53042,5378],{"className":53043},[465],[39,53045],{"className":53046,"style":461},[230],[39,53048,53050,53053],{"className":53049},[83],[39,53051,4230],{"className":53052,"style":461},[83,84],[39,53054,53056],{"className":53055},[192],[39,53057,53059,53079],{"className":53058},[196,649],[39,53060,53062,53076],{"className":53061},[200],[39,53063,53065],{"className":53064,"style":1728},[204],[39,53066,53067,53070],{"style":13607},[39,53068],{"className":53069,"style":213},[212],[39,53071,53073],{"className":53072},[217,218,219,220],[39,53074,1696],{"className":53075},[83,220],[39,53077,673],{"className":53078},[672],[39,53080,53082],{"className":53081},[200],[39,53083,53085],{"className":53084,"style":680},[204],[39,53086],{},[39,53088,53090,53093],{"className":53089},[83],[39,53091,15389],{"className":53092},[83],[39,53094,53096],{"className":53095},[192],[39,53097,53099,53119],{"className":53098},[196,649],[39,53100,53102,53116],{"className":53101},[200],[39,53103,53105],{"className":53104,"style":1728},[204],[39,53106,53107,53110],{"style":966},[39,53108],{"className":53109,"style":213},[212],[39,53111,53113],{"className":53112},[217,218,219,220],[39,53114,1696],{"className":53115},[83,220],[39,53117,673],{"className":53118},[672],[39,53120,53122],{"className":53121},[200],[39,53123,53125],{"className":53124,"style":680},[204],[39,53126],{},[39,53128,2903],{"className":53129},[83],[39,53131,673],{"className":53132},[672],[39,53134,53136],{"className":53135},[200],[39,53137,53139],{"className":53138,"style":4507},[204],[39,53140],{},[11,53142,26443,53143,53171],{},[39,53144,53146,53159],{"className":53145,"translate":43},[42],[39,53147,53149],{"className":53148},[47],[49,53150,53151],{"xmlns":51},[53,53152,53153,53157],{},[56,53154,53155],{},[59,53156,3196],{},[63,53158,3196],{"encoding":65},[39,53160,53162],{"className":53161,"ariaHidden":70},[69],[39,53163,53165,53168],{"className":53164},[74],[39,53166],{"className":53167,"style":28824},[78],[39,53169,3196],{"className":53170,"style":3321},[83,84]," 反复迭代即得。证毕。",[11,53173,53174,53175,53245,53246,53305,53306,53334,53335,53363],{},"这个定理说明，初始估计 ",[39,53176,53178,53196],{"className":53177,"translate":43},[42],[39,53179,53181],{"className":53180},[47],[49,53182,53183],{"xmlns":51},[53,53184,53185,53193],{},[56,53186,53187],{},[613,53188,53189,53191],{},[59,53190,4230],{},[539,53192,541],{},[63,53194,53195],{"encoding":65},"V_0",[39,53197,53199],{"className":53198,"ariaHidden":70},[69],[39,53200,53202,53205],{"className":53201},[74],[39,53203],{"className":53204,"style":634},[78],[39,53206,53208,53211],{"className":53207},[83],[39,53209,4230],{"className":53210,"style":461},[83,84],[39,53212,53214],{"className":53213},[192],[39,53215,53217,53237],{"className":53216},[196,649],[39,53218,53220,53234],{"className":53219},[200],[39,53221,53223],{"className":53222,"style":963},[204],[39,53224,53225,53228],{"style":13607},[39,53226],{"className":53227,"style":213},[212],[39,53229,53231],{"className":53230},[217,218,219,220],[39,53232,541],{"className":53233},[83,220],[39,53235,673],{"className":53236},[672],[39,53238,53240],{"className":53239},[200],[39,53241,53243],{"className":53242,"style":680},[204],[39,53244],{}," 不需要很准确。无论从哪个有界函数出发，误差都会按 ",[39,53247,53249,53267],{"className":53248,"translate":43},[42],[39,53250,53252],{"className":53251},[47],[49,53253,53254],{"xmlns":51},[53,53255,53256,53264],{},[56,53257,53258],{},[138,53259,53260,53262],{},[59,53261,531],{},[59,53263,3196],{},[63,53265,53266],{"encoding":65},"\\gamma^k",[39,53268,53270],{"className":53269,"ariaHidden":70},[69],[39,53271,53273,53276],{"className":53272},[74],[39,53274],{"className":53275,"style":15644},[78],[39,53277,53279,53282],{"className":53278},[83],[39,53280,531],{"className":53281,"style":565},[83,84],[39,53283,53285],{"className":53284},[192],[39,53286,53288],{"className":53287},[196],[39,53289,53291],{"className":53290},[200],[39,53292,53294],{"className":53293,"style":15577},[204],[39,53295,53296,53299],{"style":208},[39,53297],{"className":53298,"style":213},[212],[39,53300,53302],{"className":53301},[217,218,219,220],[39,53303,3196],{"className":53304,"style":3321},[83,84,220]," 的速度衰减。",[39,53307,53309,53322],{"className":53308,"translate":43},[42],[39,53310,53312],{"className":53311},[47],[49,53313,53314],{"xmlns":51},[53,53315,53316,53320],{},[56,53317,53318],{},[59,53319,531],{},[63,53321,26586],{"encoding":65},[39,53323,53325],{"className":53324,"ariaHidden":70},[69],[39,53326,53328,53331],{"className":53327},[74],[39,53329],{"className":53330,"style":2654},[78],[39,53332,531],{"className":53333,"style":565},[83,84]," 越接近 ",[39,53336,53338,53351],{"className":53337,"translate":43},[42],[39,53339,53341],{"className":53340},[47],[49,53342,53343],{"xmlns":51},[53,53344,53345,53349],{},[56,53346,53347],{},[539,53348,546],{},[63,53350,546],{"encoding":65},[39,53352,53354],{"className":53353,"ariaHidden":70},[69],[39,53355,53357,53360],{"className":53356},[74],[39,53358],{"className":53359,"style":2473},[78],[39,53361,546],{"className":53362},[83],"，衰减越慢，所需的迭代次数越多。这也是折扣因子在数值上既影响问题定义、又影响求解难度的体现。",[26,53365,53367],{"id":53366},"_52-误差上界","5.2 误差上界",[11,53369,53370,53374,53375,53444],{},[158,53371,53373],{"href":53372},"#theorem-convergence","定理 5.1"," 给出的误差界含有未知的 ",[39,53376,53378,53395],{"className":53377,"translate":43},[42],[39,53379,53381],{"className":53380},[47],[49,53382,53383],{"xmlns":51},[53,53384,53385,53393],{},[56,53386,53387],{},[613,53388,53389,53391],{},[59,53390,4230],{},[59,53392,1696],{"mathvariant":145},[63,53394,34936],{"encoding":65},[39,53396,53398],{"className":53397,"ariaHidden":70},[69],[39,53399,53401,53404],{"className":53400},[74],[39,53402],{"className":53403,"style":634},[78],[39,53405,53407,53410],{"className":53406},[83],[39,53408,4230],{"className":53409,"style":461},[83,84],[39,53411,53413],{"className":53412},[192],[39,53414,53416,53436],{"className":53415},[196,649],[39,53417,53419,53433],{"className":53418},[200],[39,53420,53422],{"className":53421,"style":1728},[204],[39,53423,53424,53427],{"style":13607},[39,53425],{"className":53426,"style":213},[212],[39,53428,53430],{"className":53429},[217,218,219,220],[39,53431,1696],{"className":53432},[83,220],[39,53434,673],{"className":53435},[672],[39,53437,53439],{"className":53438},[200],[39,53440,53442],{"className":53441,"style":680},[204],[39,53443],{},"，在实际计算中不能直接使用。下面的命题给出只依赖迭代差分与初始差分的界。",[31,53446,53449,53783,54270,54272],{"id":53447,"title":53448},"proposition-error-bounds","命题 5.2（先验与后验误差界）",[11,53450,37,53451,21402,53479,53507,53508,742,53661,53730,53731,742],{},[39,53452,53454,53467],{"className":53453,"translate":43},[42],[39,53455,53457],{"className":53456},[47],[49,53458,53459],{"xmlns":51},[53,53460,53461,53465],{},[56,53462,53463],{},[59,53464,11772],{"mathvariant":617},[63,53466,34785],{"encoding":65},[39,53468,53470],{"className":53469,"ariaHidden":70},[69],[39,53471,53473,53476],{"className":53472},[74],[39,53474],{"className":53475,"style":79},[78],[39,53477,11772],{"className":53478,"style":11793},[83,641],[39,53480,53482,53495],{"className":53481,"translate":43},[42],[39,53483,53485],{"className":53484},[47],[49,53486,53487],{"xmlns":51},[53,53488,53489,53493],{},[56,53490,53491],{},[59,53492,531],{},[63,53494,26586],{"encoding":65},[39,53496,53498],{"className":53497,"ariaHidden":70},[69],[39,53499,53501,53504],{"className":53500},[74],[39,53502],{"className":53503,"style":2654},[78],[39,53505,531],{"className":53506,"style":565},[83,84],"-压缩映射，",[39,53509,53511,53545],{"className":53510,"translate":43},[42],[39,53512,53514],{"className":53513},[47],[49,53515,53516],{"xmlns":51},[53,53517,53518,53542],{},[56,53519,53520,53532,53534,53536],{},[613,53521,53522,53524],{},[59,53523,4230],{},[56,53525,53526,53528,53530],{},[59,53527,3196],{},[133,53529,788],{},[539,53531,546],{},[133,53533,793],{},[59,53535,11772],{"mathvariant":617},[613,53537,53538,53540],{},[59,53539,4230],{},[59,53541,3196],{},[63,53543,53544],{"encoding":65},"V_{k+1}=\\mathcal{T}V_k",[39,53546,53548,53612],{"className":53547,"ariaHidden":70},[69],[39,53549,53551,53554,53603,53606,53609],{"className":53550},[74],[39,53552],{"className":53553,"style":7584},[78],[39,53555,53557,53560],{"className":53556},[83],[39,53558,4230],{"className":53559,"style":461},[83,84],[39,53561,53563],{"className":53562},[192],[39,53564,53566,53595],{"className":53565},[196,649],[39,53567,53569,53592],{"className":53568},[200],[39,53570,53572],{"className":53571,"style":3419},[204],[39,53573,53574,53577],{"style":13607},[39,53575],{"className":53576,"style":213},[212],[39,53578,53580],{"className":53579},[217,218,219,220],[39,53581,53583,53586,53589],{"className":53582},[83,220],[39,53584,3196],{"className":53585,"style":3321},[83,84,220],[39,53587,788],{"className":53588},[465,220],[39,53590,546],{"className":53591},[83,220],[39,53593,673],{"className":53594},[672],[39,53596,53598],{"className":53597},[200],[39,53599,53601],{"className":53600,"style":994},[204],[39,53602],{},[39,53604],{"className":53605,"style":231},[230],[39,53607,793],{"className":53608},[235],[39,53610],{"className":53611,"style":231},[230],[39,53613,53615,53618,53621],{"className":53614},[74],[39,53616],{"className":53617,"style":634},[78],[39,53619,11772],{"className":53620,"style":11793},[83,641],[39,53622,53624,53627],{"className":53623},[83],[39,53625,4230],{"className":53626,"style":461},[83,84],[39,53628,53630],{"className":53629},[192],[39,53631,53633,53653],{"className":53632},[196,649],[39,53634,53636,53650],{"className":53635},[200],[39,53637,53639],{"className":53638,"style":3419},[204],[39,53640,53641,53644],{"style":13607},[39,53642],{"className":53643,"style":213},[212],[39,53645,53647],{"className":53646},[217,218,219,220],[39,53648,3196],{"className":53649,"style":3321},[83,84,220],[39,53651,673],{"className":53652},[672],[39,53654,53656],{"className":53655},[200],[39,53657,53659],{"className":53658,"style":680},[204],[39,53660],{},[39,53662,53664,53681],{"className":53663,"translate":43},[42],[39,53665,53667],{"className":53666},[47],[49,53668,53669],{"xmlns":51},[53,53670,53671,53679],{},[56,53672,53673],{},[613,53674,53675,53677],{},[59,53676,4230],{},[59,53678,1696],{"mathvariant":145},[63,53680,34936],{"encoding":65},[39,53682,53684],{"className":53683,"ariaHidden":70},[69],[39,53685,53687,53690],{"className":53686},[74],[39,53688],{"className":53689,"style":634},[78],[39,53691,53693,53696],{"className":53692},[83],[39,53694,4230],{"className":53695,"style":461},[83,84],[39,53697,53699],{"className":53698},[192],[39,53700,53702,53722],{"className":53701},[196,649],[39,53703,53705,53719],{"className":53704},[200],[39,53706,53708],{"className":53707,"style":1728},[204],[39,53709,53710,53713],{"style":13607},[39,53711],{"className":53712,"style":213},[212],[39,53714,53716],{"className":53715},[217,218,219,220],[39,53717,1696],{"className":53718},[83,220],[39,53720,673],{"className":53721},[672],[39,53723,53725],{"className":53724},[200],[39,53726,53728],{"className":53727,"style":680},[204],[39,53729],{}," 为不动点。则对任意 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先证第一式。由三角不等式与压缩性，",[39,54750,54752],{"className":54751,"translate":43},[746],[39,54753,54755,54951],{"className":54754,"translate":43},[42],[39,54756,54758],{"className":54757},[47],[49,54759,54760],{"xmlns":51,"display":755},[53,54761,54762,54948],{},[759,54763,54764,54846],{"rowspacing":761,"columnalign":762,"columnspacing":146},[764,54765,54766,54794],{},[767,54767,54768],{},[770,54769,54770],{"scriptlevel":541,"displaystyle":70},[56,54771,54772,54774,54780,54782,54788],{},[59,54773,15389],{"mathvariant":145},[613,54775,54776,54778],{},[59,54777,4230],{},[59,54779,3196],{},[133,54781,5378],{},[613,54783,54784,54786],{},[59,54785,4230],{},[59,54787,1696],{"mathvariant":145},[613,54789,54790,54792],{},[59,54791,15389],{"mathvariant":145},[59,54793,1696],{"mathvariant":145},[767,54795,54796],{},[770,54797,54798],{"scriptlevel":541,"displaystyle":70},[56,54799,54800,54802,54804,54818,54820,54832,54834,54840],{},[56,54801],{},[133,54803,1824],{},[3187,54805,54806,54808,54816],{},[133,54807,3191],{},[56,54809,54810,54812,54814],{},[59,54811,13446],{},[133,54813,793],{},[59,54815,3196],{},[59,54817,1696],{"mathvariant":145},[59,54819,15389],{"mathvariant":145},[613,54821,54822,54824],{},[59,54823,4230],{},[56,54825,54826,54828,54830],{},[59,54827,13446],{},[133,54829,788],{},[539,54831,546],{},[133,54833,5378],{},[613,54835,54836,54838],{},[59,54837,4230],{},[59,54839,13446],{},[613,54841,54842,54844],{},[59,54843,15389],{"mathvariant":145},[59,54845,1696],{"mathvariant":145},[764,54847,54848,54854],{},[767,54849,54850],{},[770,54851,54852],{"scriptlevel":541,"displaystyle":70},[56,54853],{},[767,54855,54856],{},[770,54857,54858],{"scriptlevel":541,"displaystyle":70},[56,54859,54860,54862,54864,54878,54884,54886,54892,54894,54900,54906,54908,54924,54926,54932,54934,54940,54946],{},[56,54861],{},[133,54863,1824],{},[3187,54865,54866,54868,54876],{},[133,54867,3191],{},[56,54869,54870,54872,54874],{},[59,54871,13446],{},[133,54873,793],{},[59,54875,3196],{},[59,54877,1696],{"mathvariant":145},[138,54879,54880,54882],{},[59,54881,531],{},[59,54883,13446],{},[59,54885,15389],{"mathvariant":145},[613,54887,54888,54890],{},[59,54889,4230],{},[539,54891,546],{},[133,54893,5378],{},[613,54895,54896,54898],{},[59,54897,4230],{},[539,54899,541],{},[613,54901,54902,54904],{},[59,54903,15389],{"mathvariant":145},[59,54905,1696],{"mathvariant":145},[133,54907,793],{},[5362,54909,54910,54916],{},[138,54911,54912,54914],{},[59,54913,531],{},[59,54915,3196],{},[56,54917,54918,54920,54922],{},[539,54919,546],{},[133,54921,5378],{},[59,54923,531],{},[59,54925,15389],{"mathvariant":145},[613,54927,54928,54930],{},[59,54929,4230],{},[539,54931,546],{},[133,54933,5378],{},[613,54935,54936,54938],{},[59,54937,4230],{},[539,54939,541],{},[613,54941,54942,54944],{},[59,54943,15389],{"mathvariant":145},[59,54945,1696],{"mathvariant":145},[59,54947,2903],{"mathvariant":145},[63,54949,54950],{"encoding":65},"\\begin{aligned}\n\\|V_k-V_\\infty\\|_\\infty\n&\\le\\sum_{j=k}^{\\infty}\\|V_{j+1}-V_j\\|_\\infty\\\\\n&\\le\\sum_{j=k}^{\\infty}\\gamma^j\\|V_1-V_0\\|_\\infty\n=\\frac{\\gamma^k}{1-\\gamma}\\|V_1-V_0\\|_\\infty.\n\\end{aligned}",[39,54952,54954],{"className":54953,"ariaHidden":70},[69],[39,54955,54957,54961],{"className":54956},[74],[39,54958],{"className":54959,"style":54960},[78],"height:6.4792em;vertical-align:-2.9896em;",[39,54962,54964],{"className":54963},[83],[39,54965,54967,55141],{"className":54966},[759],[39,54968,54970],{"className":54969},[918],[39,54971,54973,55132],{"className":54972},[196,649],[39,54974,54976,55129],{"className":54975},[200],[39,54977,54980,55121],{"className":54978,"style":54979},[204],"height:3.4896em;",[39,54981,54983,54986],{"style":54982},"top:-5.4896em;",[39,54984],{"className":54985,"style":5782},[212],[39,54987,54989,54992,55032,55035,55038,55041,55081],{"className":54988},[83],[39,54990,15389],{"className":54991},[83],[39,54993,54995,54998],{"className":54994},[83],[39,54996,4230],{"className":54997,"style":461},[83,84],[39,54999,55001],{"className":55000},[192],[39,55002,55004,55024],{"className":55003},[196,649],[39,55005,55007,55021],{"className":55006},[200],[39,55008,55010],{"className":55009,"style":3419},[204],[39,55011,55012,55015],{"style":13607},[39,55013],{"className":55014,"style":213},[212],[39,55016,55018],{"className":55017},[217,218,219,220],[39,55019,3196],{"className":55020,"style":3321},[83,84,220],[39,55022,673],{"className":55023},[672],[39,55025,55027],{"className":55026},[200],[39,55028,55030],{"className":55029,"style":680},[204],[39,55031],{},[39,55033],{"className":55034,"style":461},[230],[39,55036,5378],{"className":55037},[465],[39,55039],{"className":55040,"style":461},[230],[39,55042,55044,55047],{"className":55043},[83],[39,55045,4230],{"className":55046,"style":461},[83,84],[39,55048,55050],{"className":55049},[192],[39,55051,55053,55073],{"className":55052},[196,649],[39,55054,55056,55070],{"className":55055},[200],[39,55057,55059],{"className":55058,"style"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220],[39,57810,3196],{"className":57811,"style":3321},[83,84,220],[39,57813,673],{"className":57814},[672],[39,57816,57818],{"className":57817},[200],[39,57819,57821],{"className":57820,"style":680},[204],[39,57822],{},[39,57824,57826,57829],{"className":57825},[83],[39,57827,15389],{"className":57828},[83],[39,57830,57832],{"className":57831},[192],[39,57833,57835,57855],{"className":57834},[196,649],[39,57836,57838,57852],{"className":57837},[200],[39,57839,57841],{"className":57840,"style":1728},[204],[39,57842,57843,57846],{"style":966},[39,57844],{"className":57845,"style":213},[212],[39,57847,57849],{"className":57848},[217,218,219,220],[39,57850,1696],{"className":57851},[83,220],[39,57853,673],{"className":57854},[672],[39,57856,57858],{"className":57857},[200],[39,57859,57861],{"className":57860,"style":680},[204],[39,57862],{},[39,57864,155],{"className":57865},[252],[11,57867,57868],{},"即第二式。证毕。",[11,57870,57871,57872,57923],{},"第一式称为先验界，因为它在迭代之前就可以用来估计所需步数。第二式称为后验界，因为它使用当前迭代的实际差分，可以在计算过程中判断是否已经足够接近不动点。由第二式，当 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时，若",[39,57925,57927],{"className":57926,"translate":43},[746],[39,57928,57930,58027],{"className":57929,"translate":43},[42],[39,57931,57933],{"className":57932},[47],[49,57934,57935],{"xmlns":51,"display":755},[53,57936,57937,58024],{},[759,57938,57939,57983],{"rowspacing":761,"columnalign":762,"columnspacing":146},[764,57940,57941,57947],{},[767,57942,57943],{},[770,57944,57945],{"scriptlevel":541,"displaystyle":70},[56,57946],{},[767,57948,57949],{},[770,57950,57951],{"scriptlevel":541,"displaystyle":70},[56,57952,57953,57955,57957,57963,57965,57977],{},[56,57954],{},[59,57956,15389],{"mathvariant":145},[613,57958,57959,57961],{},[59,57960,4230],{},[59,57962,3196],{},[133,57964,5378],{},[613,57966,57967,57969],{},[59,57968,4230],{},[56,57970,57971,57973,57975],{},[59,57972,3196],{},[133,57974,5378],{},[539,57976,546],{},[613,57978,57979,57981],{},[59,57980,15389],{"mathvariant":145},[59,57982,1696],{"mathvariant":145},[764,57984,57985,57991],{},[767,57986,57987],{},[770,57988,57989],{"scriptlevel":541,"displaystyle":70},[56,57990],{},[767,57992,57993],{},[770,57994,57995],{"scriptlevel":541,"displaystyle":70},[56,57996,57997,57999,58001,58003,58022],{},[56,57998],{},[230,58000],{"width":6891},[133,58002,1824],{},[5362,58004,58005,58020],{},[56,58006,58007,58010,58012,58014,58016,58018],{},[59,58008,58009],{},"ε",[133,58011,136],{"stretchy":135},[539,58013,546],{},[133,58015,5378],{},[59,58017,531],{},[133,58019,161],{"stretchy":135},[59,58021,531],{},[133,58023,155],{"separator":70},[63,58025,58026],{"encoding":65},"\\begin{aligned}\n&\\|V_k-V_{k-1}\\|_\\infty\\\\\n&\\quad\\le\\frac{\\varepsilon(1-\\gamma)}{\\gamma},\n\\end{aligned}",[39,58028,58030],{"className":58029,"ariaHidden":70},[69],[39,58031,58033,58037],{"className":58032},[74],[39,58034],{"className":58035,"style":58036},[78],"height:3.8074em;vertical-align:-1.6537em;",[39,58038,58040],{"className":58039},[83],[39,58041,58043,58087],{"className":58042},[759],[39,58044,58046],{"className":58045},[918],[39,58047,58049,58078],{"className":58048},[196,649],[39,58050,58052,58075],{"className":58051},[200],[39,58053,58056,58066],{"className":58054,"style":58055},[204],"height:2.1537em;",[39,58057,58059,58063],{"style":58058},"top:-4.7407em;",[39,58060],{"className":58061,"style":58062},[212],"height:3.427em;",[39,58064],{"className":58065},[83],[39,58067,58069,58072],{"style":58068},"top:-2.6537em;",[39,58070],{"className":58071,"style":58062},[212],[39,58073],{"className":58074},[83],[39,58076,673],{"className":58077},[672],[39,58079,58081],{"className":58080},[200],[39,58082,58085],{"className":58083,"style":58084},[204],"height:1.6537em;",[39,58086],{},[39,58088,58090],{"className":58089},[1223],[39,58091,58093,58364],{"className":58092},[196,649],[39,58094,58096,58361],{"className":58095},[200],[39,58097,58099,58251],{"className":58098,"style":58055},[204],[39,58100,58101,58104],{"style":58058},[39,58102],{"className":58103,"style":58062},[212],[39,58105,58107,58110,58113,58153,58156,58159,58162,58211],{"className":58106},[83],[39,58108],{"className":58109},[83],[39,58111,15389],{"className":58112},[83],[39,58114,58116,58119],{"className":58115},[83],[39,58117,4230],{"className":58118,"style":461},[83,84],[39,58120,58122],{"className":58121},[192],[39,58123,58125,58145],{"className":58124},[196,649],[39,58126,58128,58142],{"className":58127},[200],[39,58129,58131],{"className":58130,"style":3419},[204],[39,58132,58133,58136],{"style":13607},[39,58134],{"className":58135,"style":213},[212],[39,58137,58139],{"className":58138},[217,218,219,220],[39,58140,3196],{"className":58141,"style":3321},[83,84,220],[39,58143,673],{"className":58144},[672],[39,58146,58148],{"className":58147},[200],[39,58149,58151],{"className":58150,"style":680},[204],[39,58152],{},[39,58154],{"className":58155,"style":461},[230],[39,58157,5378],{"className":58158},[465],[39,58160],{"className":58161,"style":461},[230],[39,58163,58165,58168],{"className":58164},[83],[39,58166,4230],{"className":58167,"style":461},[83,84],[39,58169,58171],{"className":58170},[192],[39,58172,58174,58203],{"className":58173},[196,649],[39,58175,58177,58200],{"className":58176},[200],[39,58178,58180],{"className":58179,"style":3419},[204],[39,58181,58182,58185],{"style":13607},[39,58183],{"className":58184,"style":213},[212],[39,58186,58188],{"className":58187},[217,218,219,220],[39,58189,58191,58194,58197],{"className":58190},[83,220],[39,58192,3196],{"className":58193,"style":3321},[83,84,220],[39,58195,5378],{"className":58196},[465,220],[39,58198,546],{"className":58199},[83,220],[39,58201,673],{"className":58202},[672],[39,58204,58206],{"className":58205},[200],[39,58207,58209],{"className":58208,"style":994},[204],[39,58210],{},[39,58212,58214,58217],{"className":58213},[83],[39,58215,15389],{"className":58216},[83],[39,58218,58220],{"className":58219},[192],[39,58221,58223,58243],{"className":58222},[196,649],[39,58224,58226,58240],{"className":58225},[200],[39,58227,58229],{"className":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",[39,58376,58378,58416],{"className":58377,"translate":43},[42],[39,58379,58381],{"className":58380},[47],[49,58382,58383],{"xmlns":51},[53,58384,58385,58413],{},[56,58386,58387,58389,58395,58397,58403,58409,58411],{},[59,58388,15389],{"mathvariant":145},[613,58390,58391,58393],{},[59,58392,4230],{},[59,58394,3196],{},[133,58396,5378],{},[613,58398,58399,58401],{},[59,58400,4230],{},[59,58402,1696],{"mathvariant":145},[613,58404,58405,58407],{},[59,58406,15389],{"mathvariant":145},[59,58408,1696],{"mathvariant":145},[133,58410,1824],{},[59,58412,58009],{},[63,58414,58415],{"encoding":65},"\\|V_k-V_\\infty\\|_\\infty\\le\\varepsilon",[39,58417,58419,58477,58572],{"className":58418,"ariaHidden":70},[69],[39,58420,58422,58425,58428,58468,58471,58474],{"className":58421},[74],[39,58423],{"className":58424,"style":245},[78],[39,58426,15389],{"className":58427},[83],[39,58429,58431,58434],{"className":58430},[83],[39,58432,4230],{"className":58433,"style":461},[83,84],[39,58435,58437],{"className":58436},[192],[39,58438,58440,58460],{"className":58439},[196,649],[39,58441,58443,58457],{"className":58442},[200],[39,58444,58446],{"className":58445,"style":3419},[204],[39,58447,58448,58451],{"style":13607},[39,58449],{"className":58450,"style":213},[212],[39,58452,58454],{"className":58453},[217,218,219,220],[39,58455,3196],{"className":58456,"style":3321},[83,84,220],[39,58458,673],{"className":58459},[672],[39,58461,58463],{"className":58462},[200],[39,58464,58466],{"className":58465,"style":680},[204],[39,58467],{},[39,58469],{"className":58470,"style":461},[230],[39,58472,5378],{"className":58473},[465],[39,58475],{"className":58476,"style":461},[230],[39,58478,58480,58483,58523,58563,58566,58569],{"className":58479},[74],[39,58481],{"className":58482,"style":245},[78],[39,58484,58486,58489],{"className":58485},[83],[39,58487,4230],{"className":58488,"style":461},[83,84],[39,58490,58492],{"className":58491},[192],[39,58493,58495,58515],{"className":58494},[196,649],[39,58496,58498,58512],{"className":58497},[200],[39,58499,58501],{"className":58500,"style":1728},[204],[39,58502,58503,58506],{"style":13607},[39,58504],{"className":58505,"style":213},[212],[39,58507,58509],{"className":58508},[217,218,219,220],[39,58510,1696],{"className":58511},[83,220],[39,58513,673],{"className":58514},[672],[39,58516,58518],{"className":58517},[200],[39,58519,58521],{"className":58520,"style":680},[204],[39,58522],{},[39,58524,58526,58529],{"className":58525},[83],[39,58527,15389],{"className":58528},[83],[39,58530,58532],{"className":58531},[192],[39,58533,58535,58555],{"className":58534},[196,649],[39,58536,58538,58552],{"className":58537},[200],[39,58539,58541],{"className":58540,"style":1728},[204],[39,58542,58543,58546],{"style":966},[39,58544],{"className":58545,"style":213},[212],[39,58547,58549],{"className":58548},[217,218,219,220],[39,58550,1696],{"className":58551},[83,220],[39,58553,673],{"className":58554},[672],[39,58556,58558],{"className":58557},[200],[39,58559,58561],{"className":58560,"style":680},[204],[39,58562],{},[39,58564],{"className":58565,"style":231},[230],[39,58567,1824],{"className":58568},[235],[39,58570],{"className":58571,"style":231},[230],[39,58573,58575,58578],{"className":58574},[74],[39,58576],{"className":58577,"style":290},[78],[39,58579,58009],{"className":58580},[83,84],[26,58582,58584],{"id":58583},"_53-例显式迭代","5.3 例：显式迭代",[11,58586,58587,58588,58703],{},"回到 2.4 节的两状态马尔可夫链。令 ",[39,58589,58591,58621],{"className":58590,"translate":43},[42],[39,58592,58594],{"className":58593},[47],[49,58595,58596],{"xmlns":51},[53,58597,58598,58618],{},[56,58599,58600,58606,58608,58610,58612,58614,58616],{},[613,58601,58602,58604],{},[59,58603,4230],{},[539,58605,541],{},[133,58607,793],{},[133,58609,136],{"stretchy":135},[539,58611,541],{},[133,58613,155],{"separator":70},[539,58615,541],{},[133,58617,161],{"stretchy":135},[63,58619,58620],{"encoding":65},"V_0=(0,0)",[39,58622,58624,58679],{"className":58623,"ariaHidden":70},[69],[39,58625,58627,58630,58670,58673,58676],{"className":58626},[74],[39,58628],{"className":58629,"style":634},[78],[39,58631,58633,58636],{"className":58632},[83],[39,58634,4230],{"className":58635,"style":461},[83,84],[39,58637,58639],{"className":58638},[192],[39,58640,58642,58662],{"className":58641},[196,649],[39,58643,58645,58659],{"className":58644},[200],[39,58646,58648],{"className":58647,"style":963},[204],[39,58649,58650,58653],{"style":13607},[39,58651],{"className":58652,"style":213},[212],[39,58654,58656],{"className":58655},[217,218,219,220],[39,58657,541],{"className":58658},[83,220],[39,58660,673],{"className":58661},[672],[39,58663,58665],{"className":58664},[200],[39,58666,58668],{"className":58667,"style":680},[204],[39,58669],{},[39,58671],{"className":58672,"style":231},[230],[39,58674,793],{"className":58675},[235],[39,58677],{"className":58678,"style":231},[230],[39,58680,58682,58685,58688,58691,58694,58697,58700],{"className":58681},[74],[39,58683],{"className":58684,"style":245},[78],[39,58686,136],{"className":58687},[182],[39,58689,541],{"className":58690},[83],[39,58692,155],{"className":58693},[252],[39,58695],{"className":58696,"style":256},[230],[39,58698,541],{"className":58699},[83],[39,58701,161],{"className":58702},[263],"，迭代为",[39,58705,58707],{"className":58706,"translate":43},[746],[39,58708,58710,58800],{"className":58709,"translate":43},[42],[39,58711,58713],{"className":58712},[47],[49,58714,58715],{"xmlns":51,"display":755},[53,58716,58717,58797],{},[56,58718,58719,58731,58733,58735,58737,58739,58741,58743,58745,58751,58753,58755,58757,58759,58761,58773,58775,58777,58779,58781,58783,58789,58791,58793,58795],{},[613,58720,58721,58723],{},[59,58722,4230],{},[56,58724,58725,58727,58729],{},[59,58726,3196],{},[133,58728,788],{},[539,58730,546],{},[133,58732,136],{"stretchy":135},[539,58734,546],{},[133,58736,161],{"stretchy":135},[133,58738,793],{},[539,58740,546],{},[133,58742,788],{},[539,58744,17739],{},[613,58746,58747,58749],{},[59,58748,4230],{},[59,58750,3196],{},[133,58752,136],{"stretchy":135},[539,58754,17359],{},[133,58756,161],{"stretchy":135},[133,58758,155],{"separator":70},[230,58760],{"width":10251},[613,58762,58763,58765],{},[59,58764,4230],{},[56,58766,58767,58769,58771],{},[59,58768,3196],{},[133,58770,788],{},[539,58772,546],{},[133,58774,136],{"stretchy":135},[539,58776,17359],{},[133,58778,161],{"stretchy":135},[133,58780,793],{},[539,58782,17739],{},[613,58784,58785,58787],{},[59,58786,4230],{},[59,58788,3196],{},[133,58790,136],{"stretchy":135},[539,58792,546],{},[133,58794,161],{"stretchy":135},[59,58796,2903],{"mathvariant":145},[63,58798,58799],{"encoding":65},"V_{k+1}(1)=1+0.9V_k(2),\n\\qquad\nV_{k+1}(2)=0.9V_k(1).",[39,58801,58803,58876,58894,59028],{"className":58802,"ariaHidden":70},[69],[39,58804,58806,58809,58858,58861,58864,58867,58870,58873],{"className":58805},[74],[39,58807],{"className":58808,"style":245},[78],[39,58810,58812,58815],{"className":58811},[83],[39,58813,4230],{"className":58814,"style":461},[83,84],[39,58816,58818],{"className":58817},[192],[39,58819,58821,58850],{"className":58820},[196,649],[39,58822,58824,58847],{"className":58823},[200],[39,58825,58827],{"className":58826,"style":3419},[204],[39,58828,58829,58832],{"style":13607},[39,58830],{"className":58831,"style":213},[212],[39,58833,58835],{"className":58834},[217,218,219,220],[39,58836,58838,58841,58844],{"className":58837},[83,220],[39,58839,3196],{"className":58840,"style":3321},[83,84,220],[39,58842,788],{"className":58843},[465,220],[39,58845,546],{"className":58846},[83,220],[39,58848,673],{"className":58849},[672],[39,58851,58853],{"className":58852},[200],[39,58854,58856],{"className":58855,"style":994},[204],[39,58857],{},[39,58859,136],{"className":58860},[182],[39,58862,546],{"className":58863},[83],[39,58865,161],{"className":58866},[263],[39,58868],{"className":58869,"style":231},[230],[39,58871,793],{"className":58872},[235],[39,58874],{"className":58875,"style":231},[230],[39,58877,58879,58882,58885,58888,58891],{"className":58878},[74],[39,58880],{"className":58881,"style":17882},[78],[39,58883,546],{"className":58884},[83],[39,58886],{"className":58887,"style":461},[230],[39,58889,788],{"className":58890},[465],[39,58892],{"className":58893,"style":461},[230],[39,58895,58897,58900,58903,58943,58946,58949,58952,58955,58958,58961,59010,59013,59016,59019,59022,59025],{"className":58896},[74],[39,58898],{"className":58899,"style":245},[78],[39,58901,17739],{"className":58902},[83],[39,58904,58906,58909],{"className":58905},[83],[39,58907,4230],{"className":58908,"style":461},[83,84],[39,58910,58912],{"className":58911},[192],[39,58913,58915,58935],{"className":58914},[196,649],[39,58916,58918,58932],{"className":58917},[200],[39,58919,58921],{"className":58920,"style":3419},[204],[39,58922,58923,58926],{"style":13607},[39,58924],{"className":58925,"style":213},[212],[39,58927,58929],{"className":58928},[217,218,219,220],[39,58930,3196],{"className":58931,"style":3321},[83,84,220],[39,58933,673],{"className":58934},[672],[39,58936,58938],{"className":58937},[200],[39,58939,58941],{"className":58940,"style":680},[204],[39,58942],{},[39,58944,136],{"className":58945},[182],[39,58947,17359],{"className":58948},[83],[39,58950,161],{"className":58951},[263],[39,58953,155],{"className":58954},[252],[39,58956],{"className":58957,"style":10787},[230],[39,58959],{"className":58960,"style":256},[230],[39,58962,58964,58967],{"className":58963},[83],[39,58965,4230],{"className":58966,"style":461},[83,84],[39,58968,58970],{"className":58969},[192],[39,58971,58973,59002],{"className":58972},[196,649],[39,58974,58976,58999],{"className":58975},[200],[39,58977,58979],{"className":58978,"style":3419},[204],[39,58980,58981,58984],{"style":13607},[39,58982],{"className":58983,"style":213},[212],[39,58985,58987],{"className":58986},[217,218,219,220],[39,58988,58990,58993,58996],{"className":58989},[83,220],[39,58991,3196],{"className":58992,"style":3321},[83,84,220],[39,58994,788],{"className":58995},[465,220],[39,58997,546],{"className":58998},[83,220],[39,59000,673],{"className":59001},[672],[39,59003,59005],{"className":59004},[200],[39,59006,59008],{"className":59007,"style":994},[204],[39,59009],{},[39,59011,136],{"className":59012},[182],[39,59014,17359],{"className":59015},[83],[39,59017,161],{"className":59018},[263],[39,59020],{"className":59021,"style":231},[230],[39,59023,793],{"className":59024},[235],[39,59026],{"className":59027,"style":231},[230],[39,59029,59031,59034,59037,59077,59080,59083,59086],{"className":59030},[74],[39,59032],{"className":59033,"style":245},[78],[39,59035,17739],{"className":59036},[83],[39,59038,59040,59043],{"className":59039},[83],[39,59041,4230],{"className":59042,"style":461},[83,84],[39,59044,59046],{"className":59045},[192],[39,59047,59049,59069],{"className":59048},[196,649],[39,59050,59052,59066],{"className":59051},[200],[39,59053,59055],{"className":59054,"style":3419},[204],[39,59056,59057,59060],{"style":13607},[39,59058],{"className":59059,"style":213},[212],[39,59061,59063],{"className":59062},[217,218,219,220],[39,59064,3196],{"className":59065,"style":3321},[83,84,220],[39,59067,673],{"className":59068},[672],[39,59070,59072],{"className":59071},[200],[39,59073,59075],{"className":59074,"style":680},[204],[39,59076],{},[39,59078,136],{"className":59079},[182],[39,59081,546],{"className":59082},[83],[39,59084,161],{"className":59085},[263],[39,59087,2903],{"className":59088},[83],[11,59090,59091],{},"前几步为",[39,59093,59095],{"className":59094,"translate":43},[746],[39,59096,59098,59286],{"className":59097,"translate":43},[42],[39,59099,59101],{"className":59100},[47],[49,59102,59103],{"xmlns":51,"display":755},[53,59104,59105,59283],{},[759,59106,59107,59141,59175,59211,59247],{"rowspacing":761,"columnalign":762,"columnspacing":146},[764,59108,59109,59119],{},[767,59110,59111],{},[770,59112,59113],{"scriptlevel":541,"displaystyle":70},[613,59114,59115,59117],{},[59,59116,4230],{},[539,59118,546],{},[767,59120,59121],{},[770,59122,59123],{"scriptlevel":541,"displaystyle":70},[56,59124,59125,59127,59129,59131,59133,59135,59137,59139],{},[56,59126],{},[133,59128,793],{},[133,59130,136],{"stretchy":135},[539,59132,546],{},[133,59134,155],{"separator":70},[539,59136,541],{},[133,59138,161],{"stretchy":135},[133,59140,155],{"separator":70},[764,59142,59143,59153],{},[767,59144,59145],{},[770,59146,59147],{"scriptlevel":541,"displaystyle":70},[613,59148,59149,59151],{},[59,59150,4230],{},[539,59152,17359],{},[767,59154,59155],{},[770,59156,59157],{"scriptlevel":541,"displaystyle":70},[56,59158,59159,59161,59163,59165,59167,59169,59171,59173],{},[56,59160],{},[133,59162,793],{},[133,59164,136],{"stretchy":135},[539,59166,546],{},[133,59168,155],{"separator":70},[539,59170,17739],{},[133,59172,161],{"stretchy":135},[133,59174,155],{"separator":70},[764,59176,59177,59188],{},[767,59178,59179],{},[770,59180,59181],{"scriptlevel":541,"displaystyle":70},[613,59182,59183,59185],{},[59,59184,4230],{},[539,59186,59187],{},"3",[767,59189,59190],{},[770,59191,59192],{"scriptlevel":541,"displaystyle":70},[56,59193,59194,59196,59198,59200,59203,59205,59207,59209],{},[56,59195],{},[133,59197,793],{},[133,59199,136],{"stretchy":135},[539,59201,59202],{},"1.81",[133,59204,155],{"separator":70},[539,59206,17739],{},[133,59208,161],{"stretchy":135},[133,59210,155],{"separator":70},[764,59212,59213,59224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",[39,60387,60389,60407],{"className":60388,"translate":43},[42],[39,60390,60392],{"className":60391},[47],[49,60393,60394],{"xmlns":51},[53,60395,60396,60404],{},[56,60397,60398,60400,60402],{},[59,60399,1748],{},[133,60401,418],{},[59,60403,1696],{"mathvariant":145},[63,60405,60406],{"encoding":65},"m\\to\\infty",[39,60408,60410,60428],{"className":60409,"ariaHidden":70},[69],[39,60411,60413,60416,60419,60422,60425],{"className":60412},[74],[39,60414],{"className":60415,"style":290},[78],[39,60417,1748],{"className":60418},[83,84],[39,60420],{"className":60421,"style":231},[230],[39,60423,418],{"className":60424},[235],[39,60426],{"className":60427,"style":231},[230],[39,60429,60431,60434],{"className":60430},[74],[39,60432],{"className":60433,"style":290},[78],[39,60435,1696],{"className":60436},[83],"，得到",[39,60439,60441],{"className":60440,"translate":43},[746],[39,60442,60444,60520],{"className":60443,"translate":43},[42],[39,60445,60447],{"className":60446},[47],[49,60448,60449],{"xmlns":51,"display":755},[53,60450,60451,60517],{},[56,60452,60453,60455,60457,60459,60461,60463,60479,60481,60483,60485,60487,60489,60491,60493,60495,60497,60513,60515],{},[59,60454,4230],{},[133,60456,136],{"stretchy":135},[539,60458,546],{},[133,60460,161],{"stretchy":135},[133,60462,793],{},[5362,60464,60465,60467],{},[539,60466,546],{},[56,60468,60469,60471,60473],{},[539,60470,546],{},[133,60472,5378],{},[138,60474,60475,60477],{},[539,60476,17739],{},[539,60478,17359],{},[133,60480,793],{},[539,60482,18152],{},[133,60484,155],{"separator":70},[230,60486],{"width":10251},[59,60488,4230],{},[133,60490,136],{"stretchy":135},[539,60492,17359],{},[133,60494,161],{"stretchy":135},[133,60496,793],{},[5362,60498,60499,60501],{},[539,60500,17739],{},[56,60502,60503,60505,60507],{},[539,60504,546],{},[133,60506,5378],{},[138,60508,60509,60511],{},[539,60510,17739],{},[539,60512,17359],{},[133,60514,793],{},[539,60516,18181],{},[63,60518,60519],{"encoding":65},"V(1)=\\frac{1}{1-0.9^2}=5.2632,\n\\qquad\nV(2)=\\frac{0.9}{1-0.9^2}=4.7368.",[39,60521,60523,60550,60670,60709,60827],{"className":60522,"ariaHidden":70},[69],[39,60524,60526,60529,60532,60535,60538,60541,60544,60547],{"className":60525},[74],[39,60527],{"className":60528,"style":245},[78],[39,60530,4230],{"className":60531,"style":461},[83,84],[39,60533,136],{"className":60534},[182],[39,60536,546],{"className":60537},[83],[39,60539,161],{"className":60540},[263],[39,60542],{"className":60543,"style":231},[230],[39,60545,793],{"className":60546},[235],[39,60548],{"className":60549,"style":231},[230],[39,60551,60553,60556,60661,60664,60667],{"className":60552},[74],[39,60554],{"className":60555,"style":18221},[78],[39,60557,60559,60562,60658],{"className":60558},[83],[39,60560],{"className":60561},[182,5463],[39,60563,60565],{"className":60564},[5362],[39,60566,60568,60650],{"className":60567},[196,649],[39,60569,60571,60647],{"className":60570},[200],[39,60572,60574,60628,60636],{"className":60573,"style":18240},[204],[39,60575,60576,60579],{"style":5479},[39,60577],{"className":60578,"style":935},[212],[39,60580,60582,60585,60588,60591,60594,60597],{"className":60581},[83],[39,60583,546],{"className":60584},[83],[39,60586],{"className":60587,"style":461},[230],[39,60589,5378],{"className":60590},[465],[39,60592],{"className":60593,"style":461},[230],[39,60595,28866],{"className":60596},[83],[39,60598,60600,60603],{"className":60599},[83],[39,60601,28873],{"className":60602},[83],[39,60604,60606],{"className":60605},[192],[39,60607,60609],{"className":60608},[196],[39,60610,60612],{"className":60611},[200],[39,60613,60616],{"className":60614,"style":60615},[204],"height:0.7401em;",[39,60617,60619,60622],{"style":60618},"top:-2.989em;margin-right:0.05em;",[39,60620],{"className":60621,"style":213},[212],[39,60623,60625],{"className":60624},[217,218,219,220],[39,60626,17359],{"className":60627},[83,220],[39,60629,60630,60633],{"style":5503},[39,60631],{"className":60632,"style":935},[212],[39,60634],{"className":60635,"style":5511},[5510],[39,60637,60638,60641],{"style":5514},[39,60639],{"className":60640,"style":935},[212],[39,60642,60644],{"className":60643},[83],[39,60645,546],{"className":60646},[83],[39,60648,673],{"className":60649},[672],[39,60651,60653],{"className":60652},[200],[39,60654,60656],{"className":60655,"style":18292},[204],[39,60657],{},[39,60659],{"className":60660},[263,5463],[39,60662],{"className":60663,"style":231},[230],[39,60665,793],{"className":60666},[235],[39,60668],{"className":60669,"style":231},[230],[39,60671,60673,60676,60679,60682,60685,60688,60691,60694,60697,60700,60703,60706],{"className":60672},[74],[39,60674],{"className":60675,"style":245},[78],[39,60677,18152],{"className":60678},[83],[39,60680,155],{"className":60681},[252],[39,60683],{"className":60684,"style":10787},[230],[39,60686],{"className":60687,"style":256},[230],[39,60689,4230],{"className":60690,"style":461},[83,84],[39,60692,136],{"className":60693},[182],[39,60695,17359],{"className":60696},[83],[39,60698,161],{"className":60699},[263],[39,60701],{"className":60702,"style":231},[230],[39,60704,793],{"className":60705},[235],[39,60707],{"className":60708,"style":231},[230],[39,60710,60712,60715,60818,60821,60824],{"className":60711},[74],[39,60713],{"className":60714,"style":18221},[78],[39,60716,60718,60721,60815],{"className":60717},[83],[39,60719],{"className":60720},[182,5463],[39,60722,60724],{"className":60723},[5362],[39,60725,60727,60807],{"className":60726},[196,649],[39,60728,60730,60804],{"className":60729},[200],[39,60731,60733,60785,60793],{"className":60732,"style":18240},[204],[39,60734,60735,60738],{"style":5479},[39,60736],{"className":60737,"style":935},[212],[39,60739,60741,60744,60747,60750,60753,60756],{"className":60740},[83],[39,60742,546],{"className":60743},[83],[39,60745],{"className":60746,"style":461},[230],[39,60748,5378],{"className":60749},[465],[39,60751],{"className":60752,"style":461},[230],[39,60754,28866],{"className":60755},[83],[39,60757,60759,60762],{"className":60758},[83],[39,60760,28873],{"className":60761},[83],[39,60763,60765],{"className":60764},[192],[39,60766,60768],{"className":60767},[196],[39,60769,60771],{"className":60770},[200],[39,60772,60774],{"className":60773,"style":60615},[204],[39,60775,60776,60779],{"style":60618},[39,60777],{"className":60778,"style":213},[212],[39,60780,60782],{"className":60781},[217,218,219,220],[39,60783,17359],{"className":60784},[83,220],[39,60786,60787,60790],{"style":5503},[39,60788],{"className":60789,"style":935},[212],[39,60791],{"className":60792,"style":5511},[5510],[39,60794,60795,60798],{"style":5514},[39,60796],{"className":60797,"style":935},[212],[39,60799,60801],{"className":60800},[83],[39,60802,17739],{"className":60803},[83],[39,60805,673],{"className":60806},[672],[39,60808,60810],{"className":60809},[200],[39,60811,60813],{"className":60812,"style":18292},[204],[39,60814],{},[39,60816],{"className":60817},[263,5463],[39,60819],{"className":60820,"style":231},[230],[39,60822,793],{"className":60823},[235],[39,60825],{"className":60826,"style":231},[230],[39,60828,60830,60833],{"className":60829},[74],[39,60831],{"className":60832,"style":2473},[78],[39,60834,18181],{"className":60835},[83],[11,60837,60838,60839,60923,60924,60952],{},"这个例子的结构值得注意。若只看奇数步或偶数步的子序列，误差按 ",[39,60840,60842,60864],{"className":60841,"translate":43},[42],[39,60843,60845],{"className":60844},[47],[49,60846,60847],{"xmlns":51},[53,60848,60849,60861],{},[56,60850,60851,60857,60859],{},[138,60852,60853,60855],{},[539,60854,17739],{},[539,60856,17359],{},[133,60858,793],{},[539,60860,18022],{},[63,60862,60863],{"encoding":65},"0.9^2=0.81",[39,60865,60867,60914],{"className":60866,"ariaHidden":70},[69],[39,60868,60870,60873,60876,60905,60908,60911],{"className":60869},[74],[39,60871],{"className":60872,"style":19710},[78],[39,60874,28866],{"className":60875},[83],[39,60877,60879,60882],{"className":60878},[83],[39,60880,28873],{"className":60881},[83],[39,60883,60885],{"className":60884},[192],[39,60886,60888],{"className":60887},[196],[39,60889,60891],{"className":60890},[200],[39,60892,60894],{"className":60893,"style":19710},[204],[39,60895,60896,60899],{"style":208},[39,60897],{"className":60898,"style":213},[212],[39,60900,60902],{"className":60901},[217,218,219,220],[39,60903,17359],{"className":60904},[83,220],[39,60906],{"className":60907,"style":231},[230],[39,60909,793],{"className":60910},[235],[39,60912],{"className":60913,"style":231},[230],[39,60915,60917,60920],{"className":60916},[74],[39,60918],{"className":60919,"style":2473},[78],[39,60921,18022],{"className":60922},[83]," 的比例衰减，因为状态每两步回到自身。但逐步的 sup 范数误差仍然按 ",[39,60925,60927,60940],{"className":60926,"translate":43},[42],[39,60928,60930],{"className":60929},[47],[49,60931,60932],{"xmlns":51},[53,60933,60934,60938],{},[56,60935,60936],{},[539,60937,17739],{},[63,60939,17739],{"encoding":65},[39,60941,60943],{"className":60942,"ariaHidden":70},[69],[39,60944,60946,60949],{"className":60945},[74],[39,60947],{"className":60948,"style":2473},[78],[39,60950,17739],{"className":60951},[83]," 衰减，只是最大误差所在的分量在两步之间交替。一般理论给出的是逐步最坏情形下的统一上界，具体问题的子序列可能具有更快的收敛速度。",[11,60954,16423,60955,60957,60958,49416],{},[158,60956,53373],{"href":53372},"，初始误差为 ",[39,60959,60961,60999],{"className":60960,"translate":43},[42],[39,60962,60964],{"className":60963},[47],[49,60965,60966],{"xmlns":51},[53,60967,60968,60996],{},[56,60969,60970,60972,60978,60980,60986,60992,60994],{},[59,60971,15389],{"mathvariant":145},[613,60973,60974,60976],{},[59,60975,4230],{},[539,60977,541],{},[133,60979,5378],{},[613,60981,60982,60984],{},[59,60983,4230],{},[59,60985,1696],{"mathvariant":145},[613,60987,60988,60990],{},[59,60989,15389],{"mathvariant":145},[59,60991,1696],{"mathvariant":145},[133,60993,793],{},[539,60995,18152],{},[63,60997,60998],{"encoding":65},"\\|V_0-V_\\infty\\|_\\infty=5.2632",[39,61000,61002,61060,61155],{"className":61001,"ariaHidden":70},[69],[39,61003,61005,61008,61011,61051,61054,61057],{"className":61004},[74],[39,61006],{"className":61007,"style":245},[78],[39,61009,15389],{"className":61010},[83],[39,61012,61014,61017],{"className":61013},[83],[39,61015,4230],{"className":61016,"style":461},[83,84],[39,61018,61020],{"className":61019},[192],[39,61021,61023,61043],{"className":61022},[196,649],[39,61024,61026,61040],{"className":61025},[200],[39,61027,61029],{"className":61028,"style":963},[204],[39,61030,61031,61034],{"style":13607},[39,61032],{"className":61033,"style":213},[212],[39,61035,61037],{"className":61036},[217,218,219,220],[39,61038,541],{"className":61039},[83,220],[39,61041,673],{"className":61042},[672],[39,61044,61046],{"className":61045},[200],[39,61047,61049],{"className":61048,"style":680},[204],[39,61050],{},[39,61052],{"className":61053,"style":461},[230],[39,61055,5378],{"className":61056},[465],[39,61058],{"className":61059,"style":461},[230],[39,61061,61063,61066,61106,61146,61149,61152],{"className":61062},[74],[39,61064],{"className":61065,"style":245},[78],[39,61067,61069,61072],{"className":61068},[83],[39,61070,4230],{"className":61071,"style":461},[83,84],[39,61073,61075],{"className":61074},[192],[39,61076,61078,61098],{"className":61077},[196,649],[39,61079,61081,61095],{"className":61080},[200],[39,61082,61084],{"className":61083,"style":1728},[204],[39,61085,61086,61089],{"style":13607},[39,61087],{"className":61088,"style":213},[212],[39,61090,61092],{"className":61091},[217,218,219,220],[39,61093,1696],{"className":61094},[83,220],[39,61096,673],{"className":61097},[672],[39,61099,61101],{"className":61100},[200],[39,61102,61104],{"className":61103,"style":680},[204],[39,61105],{},[39,61107,61109,61112],{"className":61108},[83],[39,61110,15389],{"className":61111},[83],[39,61113,61115],{"className":61114},[192],[39,61116,61118,61138],{"className":61117},[196,649],[39,61119,61121,61135],{"className":61120},[200],[39,61122,61124],{"className":61123,"style":1728},[204],[39,61125,61126,61129],{"style":966},[39,61127],{"className":61128,"style":213},[212],[39,61130,61132],{"className":61131},[217,218,219,220],[39,61133,1696],{"className":61134},[83,220],[39,61136,673],{"className":61137},[672],[39,61139,61141],{"className":61140},[200],[39,61142,61144],{"className":61143,"style":680},[204],[39,61145],{},[39,61147],{"className":61148,"style":231},[230],[39,61150,793],{"className":61151},[235],[39,61153],{"className":61154,"style":231},[230],[39,61156,61158,61161],{"className":61157},[74],[39,61159],{"className":61160,"style":2473},[78],[39,61162,18152],{"className":61163},[83],[39,61165,61167],{"className":61166,"translate":43},[746],[39,61168,61170,61218],{"className":61169,"translate":43},[42],[39,61171,61173],{"className":61172},[47],[49,61174,61175],{"xmlns":51,"display":755},[53,61176,61177,61215],{},[56,61178,61179,61181,61187,61189,61195,61201,61203,61205,61207,61213],{},[59,61180,15389],{"mathvariant":145},[613,61182,61183,61185],{},[59,61184,4230],{},[59,61186,3196],{},[133,61188,5378],{},[613,61190,61191,61193],{},[59,61192,4230],{},[59,61194,1696],{"mathvariant":145},[613,61196,61197,61199],{},[59,61198,15389],{"mathvariant":145},[59,61200,1696],{"mathvariant":145},[133,61202,1824],{},[539,61204,18152],{},[133,61206,409],{},[138,61208,61209,61211],{},[539,61210,17739],{},[59,61212,3196],{},[59,61214,2903],{"mathvariant":145},[63,61216,61217],{"encoding":65},"\\|V_k-V_\\infty\\|_\\infty\\le5.2632\\times0.9^k.",[39,61219,61221,61279,61374,61392],{"className":61220,"ariaHidden":70},[69],[39,61222,61224,61227,61230,61270,61273,61276],{"className":61223},[74],[39,61225],{"className":61226,"style":245},[78],[39,61228,15389],{"className":61229},[83],[39,61231,61233,61236],{"className":61232},[83],[39,61234,4230],{"className":61235,"style":461},[83,84],[39,61237,61239],{"className":61238},[192],[39,61240,61242,61262],{"className":61241},[196,649],[39,61243,61245,61259],{"className":61244},[200],[39,61246,61248],{"className":61247,"style":3419},[204],[39,61249,61250,61253],{"style":13607},[39,61251],{"className":61252,"style":213},[212],[39,61254,61256],{"className":61255},[217,218,219,220],[39,61257,3196],{"className":61258,"style":3321},[83,84,220],[39,61260,673],{"className":61261},[672],[39,61263,61265],{"className":61264},[200],[39,61266,61268],{"className":61267,"style":680},[204],[39,61269],{},[39,61271],{"className":61272,"style":461},[230],[39,61274,5378],{"className":61275},[465],[39,61277],{"className":61278,"style":461},[230],[39,61280,61282,61285,61325,61365,61368,61371],{"className":61281},[74],[39,61283],{"className":61284,"style":245},[78],[39,61286,61288,61291],{"className":61287},[83],[39,61289,4230],{"className":61290,"style":461},[83,84],[39,61292,61294],{"className":61293},[192],[39,61295,61297,61317],{"className":61296},[196,649],[39,61298,61300,61314],{"className":61299},[200],[39,61301,61303],{"className":61302,"style":1728},[204],[39,61304,61305,61308],{"style":13607},[39,61306],{"className":61307,"style":213},[212],[39,61309,61311],{"className":61310},[217,218,219,220],[39,61312,1696],{"className":61313},[83,220],[39,61315,673],{"className":61316},[672],[39,61318,61320],{"className":61319},[200],[39,61321,61323],{"className":61322,"style":680},[204],[39,61324],{},[39,61326,61328,61331],{"className":61327},[83],[39,61329,15389],{"className":61330},[83],[39,61332,61334],{"className":61333},[192],[39,61335,61337,61357],{"className":61336},[196,649],[39,61338,61340,61354],{"className":61339},[200],[39,61341,61343],{"className":61342,"style":1728},[204],[39,61344,61345,61348],{"style":966},[39,61346],{"className":61347,"style":213},[212],[39,61349,61351],{"className":61350},[217,218,219,220],[39,61352,1696],{"className":61353},[83,220],[39,61355,673],{"className":61356},[672],[39,61358,61360],{"className":61359},[200],[39,61361,61363],{"className":61362,"style":680},[204],[39,61364],{},[39,61366],{"className":61367,"style":231},[230],[39,61369,1824],{"className":61370},[235],[39,61372],{"className":61373,"style":231},[230],[39,61375,61377,61380,61383,61386,61389],{"className":61376},[74],[39,61378],{"className":61379,"style":17882},[78],[39,61381,18152],{"className":61382},[83],[39,61384],{"className":61385,"style":461},[230],[39,61387,409],{"className":61388},[465],[39,61390],{"className":61391,"style":461},[230],[39,61393,61395,61398,61401,61430],{"className":61394},[74],[39,61396],{"className":61397,"style":3389},[78],[39,61399,28866],{"className":61400},[83],[39,61402,61404,61407],{"className":61403},[83],[39,61405,28873],{"className":61406},[83],[39,61408,61410],{"className":61409},[192],[39,61411,61413],{"className":61412},[196],[39,61414,61416],{"className":61415},[200],[39,61417,61419],{"className":61418,"style":3389},[204],[39,61420,61421,61424],{"style":1027},[39,61422],{"className":61423,"style":213},[212],[39,61425,61427],{"className":61426},[217,218,219,220],[39,61428,3196],{"className":61429,"style":3321},[83,84,220],[39,61431,2903],{"className":61432},[83],[11,61434,12932,61435,61487,61488,28969,61517,61487,61569,61598],{},[39,61436,61438,61457],{"className":61437,"translate":43},[42],[39,61439,61441],{"className":61440},[47],[49,61442,61443],{"xmlns":51},[53,61444,61445,61454],{},[56,61446,61447,61449,61451],{},[59,61448,3196],{},[133,61450,793],{},[539,61452,61453],{},"20",[63,61455,61456],{"encoding":65},"k=20",[39,61458,61460,61478],{"className":61459,"ariaHidden":70},[69],[39,61461,61463,61466,61469,61472,61475],{"className":61462},[74],[39,61464],{"className":61465,"style":28824},[78],[39,61467,3196],{"className":61468,"style":3321},[83,84],[39,61470],{"className":61471,"style":231},[230],[39,61473,793],{"className":61474},[235],[39,61476],{"className":61477,"style":231},[230],[39,61479,61481,61484],{"className":61480},[74],[39,61482],{"className":61483,"style":2473},[78],[39,61485,61453],{"className":61486},[83]," 时，右端约为 ",[39,61489,61491,61505],{"className":61490,"translate":43},[42],[39,61492,61494],{"className":61493},[47],[49,61495,61496],{"xmlns":51},[53,61497,61498,61503],{},[56,61499,61500],{},[539,61501,61502],{},"0.64",[63,61504,61502],{"encoding":65},[39,61506,61508],{"className":61507,"ariaHidden":70},[69],[39,61509,61511,61514],{"className":61510},[74],[39,61512],{"className":61513,"style":2473},[78],[39,61515,61502],{"className":61516},[83],[39,61518,61520,61539],{"className":61519,"translate":43},[42],[39,61521,61523],{"className":61522},[47],[49,61524,61525],{"xmlns":51},[53,61526,61527,61536],{},[56,61528,61529,61531,61533],{},[59,61530,3196],{},[133,61532,793],{},[539,61534,61535],{},"50",[63,61537,61538],{"encoding":65},"k=50",[39,61540,61542,61560],{"className":61541,"ariaHidden":70},[69],[39,61543,61545,61548,61551,61554,61557],{"className":61544},[74],[39,61546],{"className":61547,"style":28824},[78],[39,61549,3196],{"className":61550,"style":3321},[83,84],[39,61552],{"className":61553,"style":231},[230],[39,61555,793],{"className":61556},[235],[39,61558],{"className":61559,"style":231},[230],[39,61561,61563,61566],{"className":61562},[74],[39,61564],{"className":61565,"style":2473},[78],[39,61567,61535],{"className":61568},[83],[39,61570,61572,61586],{"className":61571,"translate":43},[42],[39,61573,61575],{"className":61574},[47],[49,61576,61577],{"xmlns":51},[53,61578,61579,61584],{},[56,61580,61581],{},[539,61582,61583],{},"0.027",[63,61585,61583],{"encoding":65},[39,61587,61589],{"className":61588,"ariaHidden":70},[69],[39,61590,61592,61595],{"className":61591},[74],[39,61593],{"className":61594,"style":2473},[78],[39,61596,61583],{"className":61597},[83],"。这与几何收敛的预期一致。",[26,61600,61602],{"id":61601},"_54-折扣因子的必要性","5.4 折扣因子的必要性",[11,61604,61605,61606,34986,61656,61706],{},"压缩性依赖 ",[39,61607,61609,61626],{"className":61608,"translate":43},[42],[39,61610,61612],{"className":61611},[47],[49,61613,61614],{"xmlns":51},[53,61615,61616,61624],{},[56,61617,61618,61620,61622],{},[59,61619,531],{},[133,61621,1693],{},[539,61623,546],{},[63,61625,17296],{"encoding":65},[39,61627,61629,61647],{"className":61628,"ariaHidden":70},[69],[39,61630,61632,61635,61638,61641,61644],{"className":61631},[74],[39,61633],{"className":61634,"style":561},[78],[39,61636,531],{"className":61637,"style":565},[83,84],[39,61639],{"className":61640,"style":231},[230],[39,61642,1693],{"className":61643},[235],[39,61645],{"className":61646,"style":231},[230],[39,61648,61650,61653],{"className":61649},[74],[39,61651],{"className":61652,"style":2473},[78],[39,61654,546],{"className":61655},[83],[39,61657,61659,61676],{"className":61658,"translate":43},[42],[39,61660,61662],{"className":61661},[47],[49,61663,61664],{"xmlns":51},[53,61665,61666,61674],{},[56,61667,61668,61670,61672],{},[59,61669,531],{},[133,61671,793],{},[539,61673,546],{},[63,61675,28990],{"encoding":65},[39,61677,61679,61697],{"className":61678,"ariaHidden":70},[69],[39,61680,61682,61685,61688,61691,61694],{"className":61681},[74],[39,61683],{"className":61684,"style":2654},[78],[39,61686,531],{"className":61687,"style":565},[83,84],[39,61689],{"className":61690,"style":231},[230],[39,61692,793],{"className":61693},[235],[39,61695],{"className":61696,"style":231},[230],[39,61698,61700,61703],{"className":61699},[74],[39,61701],{"className":61702,"style":2473},[78],[39,61704,546],{"className":61705},[83],"，贝尔曼算子一般不再是压缩映射，贝尔曼方程可能没有有限解，也可能有多个解。以下两个例子说明这两种失效方式。",[11,61708,61709,61710,61752,61753,61826,61827,17773],{},"第一个例子是零奖励的确定性循环。设两个状态 ",[39,61711,61713,61731],{"className":61712,"translate":43},[42],[39,61714,61716],{"className":61715},[47],[49,61717,61718],{"xmlns":51},[53,61719,61720,61728],{},[56,61721,61722,61724,61726],{},[539,61723,546],{},[133,61725,155],{"separator":70},[539,61727,17359],{},[63,61729,61730],{"encoding":65},"1,2",[39,61732,61734],{"className":61733,"ariaHidden":70},[69],[39,61735,61737,61740,61743,61746,61749],{"className":61736},[74],[39,61738],{"className":61739,"style":17578},[78],[39,61741,546],{"className":61742},[83],[39,61744,155],{"className":61745},[252],[39,61747],{"className":61748,"style":256},[230],[39,61750,17359],{"className":61751},[83],"，转移为 ",[39,61754,61756,61778],{"className":61755,"translate":43},[42],[39,61757,61759],{"className":61758},[47],[49,61760,61761],{"xmlns":51},[53,61762,61763,61775],{},[56,61764,61765,61767,61769,61771,61773],{},[539,61766,546],{},[133,61768,418],{},[539,61770,17359],{},[133,61772,418],{},[539,61774,546],{},[63,61776,61777],{"encoding":65},"1\\to2\\to1",[39,61779,61781,61799,61817],{"className":61780,"ariaHidden":70},[69],[39,61782,61784,61787,61790,61793,61796],{"className":61783},[74],[39,61785],{"className":61786,"style":2473},[78],[39,61788,546],{"className":61789},[83],[39,61791],{"className":61792,"style":231},[230],[39,61794,418],{"className":61795},[235],[39,61797],{"className":61798,"style":231},[230],[39,61800,61802,61805,61808,61811,61814],{"className":61801},[74],[39,61803],{"className":61804,"style":2473},[78],[39,61806,17359],{"className":61807},[83],[39,61809],{"className":61810,"style":231},[230],[39,61812,418],{"className":61813},[235],[39,61815],{"className":61816,"style":231},[230],[39,61818,61820,61823],{"className":61819},[74],[39,61821],{"className":61822,"style":2473},[78],[39,61824,546],{"className":61825},[83],"，奖励恒为零，",[39,61828,61830,61847],{"className":61829,"translate":43},[42],[39,61831,61833],{"className":61832},[47],[49,61834,61835],{"xmlns":51},[53,61836,61837,61845],{},[56,61838,61839,61841,61843],{},[59,61840,531],{},[133,61842,793],{},[539,61844,546],{},[63,61846,28990],{"encoding":65},[39,61848,61850,61868],{"className":61849,"ariaHidden":70},[69],[39,61851,61853,61856,61859,61862,61865],{"className":61852},[74],[39,61854],{"className":61855,"style":2654},[78],[39,61857,531],{"className":61858,"style":565},[83,84],[39,61860],{"className":61861,"style":231},[230],[39,61863,793],{"className":61864},[235],[39,61866],{"className":61867,"style":231},[230],[39,61869,61871,61874],{"className":61870},[74],[39,61872],{"className":61873,"style":2473},[78],[39,61875,546],{"className":61876},[83],[39,61878,61880],{"className":61879,"translate":43},[746],[39,61881,61883,61937],{"className":61882,"translate":43},[42],[39,61884,61886],{"className":61885},[47],[49,61887,61888],{"xmlns":51,"display":755},[53,61889,61890,61934],{},[56,61891,61892,61894,61896,61898,61900,61902,61904,61906,61908,61910,61912,61914,61916,61918,61920,61922,61924,61926,61928,61930,61932],{},[59,61893,4230],{},[133,61895,136],{"stretchy":135},[539,61897,546],{},[133,61899,161],{"stretchy":135},[133,61901,793],{},[59,61903,4230],{},[133,61905,136],{"stretchy":135},[539,61907,17359],{},[133,61909,161],{"stretchy":135},[133,61911,155],{"separator":70},[230,61913],{"width":10251},[59,61915,4230],{},[133,61917,136],{"stretchy":135},[539,61919,17359],{},[133,61921,161],{"stretchy":135},[133,61923,793],{},[59,61925,4230],{},[133,61927,136],{"stretchy":135},[539,61929,546],{},[133,61931,161],{"stretchy":135},[59,61933,2903],{"mathvariant":145},[63,61935,61936],{"encoding":65},"V(1)=V(2),\\qquad V(2)=V(1).",[39,61938,61940,61967,62015],{"className":61939,"ariaHidden":70},[69],[39,61941,61943,61946,61949,61952,61955,61958,61961,61964],{"className":61942},[74],[39,61944],{"className":61945,"style":245},[78],[39,61947,4230],{"className":61948,"style":461},[83,84],[39,61950,136],{"className":61951},[182],[39,61953,546],{"className":61954},[83],[39,61956,161],{"className":61957},[263],[39,61959],{"className":61960,"style":231},[230],[39,61962,793],{"className":61963},[235],[39,61965],{"className":61966,"style":231},[230],[39,61968,61970,61973,61976,61979,61982,61985,61988,61991,61994,61997,62000,62003,62006,62009,62012],{"className":61969},[74],[39,61971],{"className":61972,"style":245},[78],[39,61974,4230],{"className":61975,"style":461},[83,84],[39,61977,136],{"className":61978},[182],[39,61980,17359],{"className":61981},[83],[39,61983,161],{"className":61984},[263],[39,61986,155],{"className":61987},[252],[39,61989],{"className":61990,"style":10787},[230],[39,61992],{"className":61993,"style":256},[230],[39,61995,4230],{"className":61996,"style":461},[83,84],[39,61998,136],{"className":61999},[182],[39,62001,17359],{"className":62002},[83],[39,62004,161],{"className":62005},[263],[39,62007],{"className":62008,"style":231},[230],[39,62010,793],{"className":62011},[235],[39,62013],{"className":62014,"style":231},[230],[39,62016,62018,62021,62024,62027,62030,62033],{"className":62017},[74],[39,62019],{"className":62020,"style":245},[78],[39,62022,4230],{"className":62023,"style":461},[83,84],[39,62025,136],{"className":62026},[182],[39,62028,546],{"className":62029},[83],[39,62031,161],{"className":62032},[263],[39,62034,2903],{"className":62035},[83],[11,62037,62038,62039,62142],{},"任意常数函数 ",[39,62040,62042,62076],{"className":62041,"translate":43},[42],[39,62043,62045],{"className":62044},[47],[49,62046,62047],{"xmlns":51},[53,62048,62049,62073],{},[56,62050,62051,62053,62055,62057,62059,62061,62063,62065,62067,62069,62071],{},[59,62052,4230],{},[133,62054,136],{"stretchy":135},[539,62056,546],{},[133,62058,161],{"stretchy":135},[133,62060,793],{},[59,62062,4230],{},[133,62064,136],{"stretchy":135},[539,62066,17359],{},[133,62068,161],{"stretchy":135},[133,62070,793],{},[59,62072,24077],{},[63,62074,62075],{"encoding":65},"V(1)=V(2)=c",[39,62077,62079,62106,62133],{"className":62078,"ariaHidden":70},[69],[39,62080,62082,62085,62088,62091,62094,62097,62100,62103],{"className":62081},[74],[39,62083],{"className":62084,"style":245},[78],[39,62086,4230],{"className":62087,"style":461},[83,84],[39,62089,136],{"className":62090},[182],[39,62092,546],{"className":62093},[83],[39,62095,161],{"className":62096},[263],[39,62098],{"className":62099,"style":231},[230],[39,62101,793],{"className":62102},[235],[39,62104],{"className":62105,"style":231},[230],[39,62107,62109,62112,62115,62118,62121,62124,62127,62130],{"className":62108},[74],[39,62110],{"className":62111,"style":245},[78],[39,62113,4230],{"className":62114,"style":461},[83,84],[39,62116,136],{"className":62117},[182],[39,62119,17359],{"className":62120},[83],[39,62122,161],{"className":62123},[263],[39,62125],{"className":62126,"style":231},[230],[39,62128,793],{"className":62129},[235],[39,62131],{"className":62132,"style":231},[230],[39,62134,62136,62139],{"className":62135},[74],[39,62137],{"className":62138,"style":290},[78],[39,62140,24077],{"className":62141},[83,84]," 都是解，因此不动点不唯一。Banach 定理的唯一性结论失效。",[11,62144,62145,62146,62174],{},"第二个例子是每步奖励为 ",[39,62147,62149,62162],{"className":62148,"translate":43},[42],[39,62150,62152],{"className":62151},[47],[49,62153,62154],{"xmlns":51},[53,62155,62156,62160],{},[56,62157,62158],{},[539,62159,546],{},[63,62161,546],{"encoding":65},[39,62163,62165],{"className":62164,"ariaHidden":70},[69],[39,62166,62168,62171],{"className":62167},[74],[39,62169],{"className":62170,"style":2473},[78],[39,62172,546],{"className":62173},[83]," 的同一循环。此时",[39,62176,62178],{"className":62177,"translate":43},[746],[39,62179,62181,62223],{"className":62180,"translate":43},[42],[39,62182,62184],{"className":62183},[47],[49,62185,62186],{"xmlns":51,"display":755},[53,62187,62188,62220],{},[56,62189,62190,62196,62198,62212,62214,62216,62218],{},[613,62191,62192,62194],{},[59,62193,3181],{},[59,62195,621],{},[133,62197,793],{},[3187,62199,62200,62202,62210],{},[133,62201,3191],{},[56,62203,62204,62206,62208],{},[59,62205,3196],{},[133,62207,793],{},[539,62209,541],{},[59,62211,1696],{"mathvariant":145},[539,62213,546],{},[133,62215,793],{},[59,62217,1696],{"mathvariant":145},[133,62219,155],{"separator":70},[63,62221,62222],{"encoding":65},"G_t=\\sum_{k=0}^{\\infty}1=\\infty,",[39,62224,62226,62281,62369],{"className":62225,"ariaHidden":70},[69],[39,62227,62229,62232,62272,62275,62278],{"className":62228},[74],[39,62230],{"className":62231,"style":634},[78],[39,62233,62235,62238],{"className":62234},[83],[39,62236,3181],{"className":62237},[83,84],[39,62239,62241],{"className":62240},[192],[39,62242,62244,62264],{"className":62243},[196,649],[39,62245,62247,62261],{"className":62246},[200],[39,62248,62250],{"className":62249,"style":656},[204],[39,62251,62252,62255],{"style":966},[39,62253],{"className":62254,"style":213},[212],[39,62256,62258],{"className":62257},[217,218,219,220],[39,62259,621],{"className":62260},[83,84,220],[39,62262,673],{"className":62263},[672],[39,62265,62267],{"className":62266},[200],[39,62268,62270],{"className":62269,"style":680},[204],[39,62271],{},[39,62273],{"className":62274,"style":231},[230],[39,62276,793],{"className":62277},[235],[39,62279],{"className":62280,"style":231},[230],[39,62282,62284,62287,62354,62357,62360,62363,62366],{"className":62283},[74],[39,62285],{"className":62286,"style":3290},[78],[39,62288,62290],{"className":62289},[1744,3294],[39,62291,62293,62346],{"className":62292},[196,649],[39,62294,62296,62343],{"className":62295},[200],[39,62297,62299,62319,62329],{"className":62298,"style":3304},[204],[39,62300,62301,62304],{"style":3307},[39,62302],{"className":62303,"style":3311},[212],[39,62305,62307],{"className":62306},[217,218,219,220],[39,62308,62310,62313,62316],{"className":62309},[83,220],[39,62311,3196],{"className":62312,"style":3321},[83,84,220],[39,62314,793],{"className":62315},[235,220],[39,62317,541],{"className":62318},[83,220],[39,62320,62321,62324],{"style":3330},[39,62322],{"className":62323,"style":3311},[212],[39,62325,62326],{},[39,62327,3191],{"className":62328},[1744,3339,3340],[39,62330,62331,62334],{"style":3343},[39,62332],{"className":62333,"style":3311},[212],[39,62335,62337],{"className":62336},[217,218,219,220],[39,62338,62340],{"className":62339},[83,220],[39,62341,1696],{"className":62342},[83,220],[39,62344,673],{"className":62345},[672],[39,62347,62349],{"className":62348},[200],[39,62350,62352],{"className":62351,"style":3365},[204],[39,62353],{},[39,62355],{"className":62356,"style":256},[230],[39,62358,546],{"className":62359},[83],[39,62361],{"className":62362,"style":231},[230],[39,62364,793],{"className":62365},[235],[39,62367],{"className":62368,"style":231},[230],[39,62370,62372,62375,62378],{"className":62371},[74],[39,62373],{"className":62374,"style":2654},[78],[39,62376,1696],{"className":62377},[83],[39,62379,155],{"className":62380},[252],[11,62382,62383,62384,62427],{},"折扣回报不再是有限实数，值函数不属于 ",[39,62385,62387,62406],{"className":62386,"translate":43},[42],[39,62388,62390],{"className":62389},[47],[49,62391,62392],{"xmlns":51},[53,62393,62394,62404],{},[56,62395,62396,62398,62400,62402],{},[59,62397,19772],{},[133,62399,136],{"stretchy":135},[59,62401,61],{},[133,62403,161],{"stretchy":135},[63,62405,21380],{"encoding":65},[39,62407,62409],{"className":62408,"ariaHidden":70},[69],[39,62410,62412,62415,62418,62421,62424],{"className":62411},[74],[39,62413],{"className":62414,"style":245},[78],[39,62416,19772],{"className":62417,"style":19812},[83,84],[39,62419,136],{"className":62420},[182],[39,62422,61],{"className":62423,"style":85},[83,84],[39,62425,161],{"className":62426},[263],"。对应的贝尔曼方程为",[39,62429,62431],{"className":62430,"translate":43},[746],[39,62432,62434,62496],{"className":62433,"translate":43},[42],[39,62435,62437],{"className":62436},[47],[49,62438,62439],{"xmlns":51,"display":755},[53,62440,62441,62493],{},[56,62442,62443,62445,62447,62449,62451,62453,62455,62457,62459,62461,62463,62465,62467,62469,62471,62473,62475,62477,62479,62481,62483,62485,62487,62489,62491],{},[59,62444,4230],{},[133,62446,136],{"stretchy":135},[539,62448,546],{},[133,62450,161],{"stretchy":135},[133,62452,793],{},[539,62454,546],{},[133,62456,788],{},[59,62458,4230],{},[133,62460,136],{"stretchy":135},[539,62462,17359],{},[133,62464,161],{"stretchy":135},[133,62466,155],{"separator":70},[230,62468],{"width":10251},[59,62470,4230],{},[133,62472,136],{"stretchy":135},[539,62474,17359],{},[133,62476,161],{"stretchy":135},[133,62478,793],{},[539,62480,546],{},[133,62482,788],{},[59,62484,4230],{},[133,62486,136],{"stretchy":135},[539,62488,546],{},[133,62490,161],{"stretchy":135},[133,62492,155],{"separator":70},[63,62494,62495],{"encoding":65},"V(1)=1+V(2),\\qquad V(2)=1+V(1),",[39,62497,62499,62526,62544,62592,62610],{"className":62498,"ariaHidden":70},[69],[39,62500,62502,62505,62508,62511,62514,62517,62520,62523],{"className":62501},[74],[39,62503],{"className":62504,"style":245},[78],[39,62506,4230],{"className":62507,"style":461},[83,84],[39,62509,136],{"className":62510},[182],[39,62512,546],{"className":62513},[83],[39,62515,161],{"className":62516},[263],[39,62518],{"className":62519,"style":231},[230],[39,62521,793],{"className":62522},[235],[39,62524],{"className":62525,"style":231},[230],[39,62527,62529,62532,62535,62538,62541],{"className":62528},[74],[39,62530],{"className":62531,"style":17882},[78],[39,62533,546],{"className":62534},[83],[39,62536],{"className":62537,"style":461},[230],[39,62539,788],{"className":62540},[465],[39,62542],{"className":62543,"style":461},[230],[39,62545,62547,62550,62553,62556,62559,62562,62565,62568,62571,62574,62577,62580,62583,62586,62589],{"className":62546},[74],[39,62548],{"className":62549,"style":245},[78],[39,62551,4230],{"className":62552,"style":461},[83,84],[39,62554,136],{"className":62555},[182],[39,62557,17359],{"className":62558},[83],[39,62560,161],{"className":62561},[263],[39,62563,155],{"className":62564},[252],[39,62566],{"className":62567,"style":10787},[230],[39,62569],{"className":62570,"style":256},[230],[39,62572,4230],{"className":62573,"style":461},[83,84],[39,62575,136],{"className":62576},[182],[39,62578,17359],{"className":62579},[83],[39,62581,161],{"className":62582},[263],[39,62584],{"className":62585,"style":231},[230],[39,62587,793],{"className":62588},[235],[39,62590],{"className":62591,"style":231},[230],[39,62593,62595,62598,62601,62604,62607],{"className":62594},[74],[39,62596],{"className":62597,"style":17882},[78],[39,62599,546],{"className":62600},[83],[39,62602],{"className":62603,"style":461},[230],[39,62605,788],{"className":62606},[465],[39,62608],{"className":62609,"style":461},[230],[39,62611,62613,62616,62619,62622,62625,62628],{"className":62612},[74],[39,62614],{"className":62615,"style":245},[78],[39,62617,4230],{"className":62618,"style":461},[83,84],[39,62620,136],{"className":62621},[182],[39,62623,546],{"className":62624},[83],[39,62626,161],{"className":62627},[263],[39,62629,155],{"className":62630},[252],[11,62632,62633,62634,62684],{},"它也没有有限解。这说明 ",[39,62635,62637,62654],{"className":62636,"translate":43},[42],[39,62638,62640],{"className":62639},[47],[49,62641,62642],{"xmlns":51},[53,62643,62644,62652],{},[56,62645,62646,62648,62650],{},[59,62647,531],{},[133,62649,1693],{},[539,62651,546],{},[63,62653,17296],{"encoding":65},[39,62655,62657,62675],{"className":62656,"ariaHidden":70},[69],[39,62658,62660,62663,62666,62669,62672],{"className":62659},[74],[39,62661],{"className":62662,"style":561},[78],[39,62664,531],{"className":62665,"style":565},[83,84],[39,62667],{"className":62668,"style":231},[230],[39,62670,1693],{"className":62671},[235],[39,62673],{"className":62674,"style":231},[230],[39,62676,62678,62681],{"className":62677},[74],[39,62679],{"className":62680,"style":2473},[78],[39,62682,546],{"className":62683},[83]," 不是技术上的方便假设，而是保证回报有界、算子压缩、解唯一的结构性条件。",[21,62686,62688],{"id":62687},"六最优子结构与最优性原理","六、最优子结构与最优性原理",[26,62690,62692],{"id":62691},"_61-有限期情形","6.1 有限期情形",[11,62694,62695,62696,62767,62768,2183,62956,63026,63027,63055],{},"最优子结构最清晰的表述出现在有限期问题中。设时间范围为 ",[39,62697,62699,62725],{"className":62698,"translate":43},[42],[39,62700,62702],{"className":62701},[47],[49,62703,62704],{"xmlns":51},[53,62705,62706,62722],{},[56,62707,62708,62710,62712,62714,62716,62718,62720],{},[539,62709,541],{},[133,62711,155],{"separator":70},[539,62713,546],{},[133,62715,155],{"separator":70},[133,62717,2641],{},[133,62719,155],{"separator":70},[59,62721,11772],{},[63,62723,62724],{"encoding":65},"0,1,\\dots,T",[39,62726,62728],{"className":62727,"ariaHidden":70},[69],[39,62729,62731,62734,62737,62740,62743,62746,62749,62752,62755,62758,62761,62764],{"className":62730},[74],[39,62732],{"className":62733,"style":6452},[78],[39,62735,541],{"className":62736},[83],[39,62738,155],{"className":62739},[252],[39,62741],{"className":62742,"style":256},[230],[39,62744,546],{"className":62745},[83],[39,62747,155],{"className":62748},[252],[39,62750],{"className":62751,"style":256},[230],[39,62753,2641],{"className":62754},[2842],[39,62756],{"className":62757,"style":256},[230],[39,62759,155],{"className":62760},[252],[39,62762],{"className":62763,"style":256},[230],[39,62765,11772],{"className":62766,"style":178},[83,84],"，策略为一个序列 ",[39,62769,62771,62815],{"className":62770,"translate":43},[42],[39,62772,62774],{"className":62773},[47],[49,62775,62776],{"xmlns":51},[53,62777,62778,62812],{},[56,62779,62780,62782,62784,62786,62792,62794,62796,62798,62810],{},[59,62781,2098],{},[133,62783,793],{},[133,62785,136],{"stretchy":135},[613,62787,62788,62790],{},[59,62789,2098],{},[539,62791,541],{},[133,62793,155],{"separator":70},[133,62795,2641],{},[133,62797,155],{"separator":70},[613,62799,62800,62802],{},[59,62801,2098],{},[56,62803,62804,62806,62808],{},[59,62805,11772],{},[133,62807,5378],{},[539,62809,546],{},[133,62811,161],{"stretchy":135},[63,62813,62814],{"encoding":65},"\\pi=(\\pi_0,\\dots,\\pi_{T-1})",[39,62816,62818,62836],{"className":62817,"ariaHidden":70},[69],[39,62819,62821,62824,62827,62830,62833],{"className":62820},[74],[39,62822],{"className":62823,"style":290},[78],[39,62825,2098],{"className":62826,"style":2114},[83,84],[39,62828],{"className":62829,"style":231},[230],[39,62831,793],{"className":62832},[235],[39,62834],{"className":62835,"style":231},[230],[39,62837,62839,62842,62845,62885,62888,62891,62894,62897,62900,62903,62953],{"className":62838},[74],[39,62840],{"className":62841,"style":245},[78],[39,62843,136],{"className":62844},[182],[39,62846,62848,62851],{"className":62847},[83],[39,62849,2098],{"className":62850,"style":2114},[83,84],[39,62852,62854],{"className":62853},[192],[39,62855,62857,62877],{"className":62856},[196,649],[39,62858,62860,62874],{"className":62859},[200],[39,62861,62863],{"className":62862,"style":963},[204],[39,62864,62865,62868],{"style":41316},[39,62866],{"className":62867,"style":213},[212],[39,62869,62871],{"className":62870},[217,218,219,220],[39,62872,541],{"className":62873},[83,220],[39,62875,673],{"className":62876},[672],[39,62878,62880],{"className":62879},[200],[39,62881,62883],{"className":62882,"style":680},[204],[39,62884],{},[39,62886,155],{"className":62887},[252],[39,62889],{"className":62890,"style":256},[230],[39,62892,2641],{"className":62893},[2842],[39,62895],{"className":62896,"style":256},[230],[39,62898,155],{"className":62899},[252],[39,62901],{"className":62902,"style":256},[230],[39,62904,62906,62909],{"className":62905},[83],[39,62907,2098],{"className":62908,"style":2114},[83,84],[39,62910,62912],{"className":62911},[192],[39,62913,62915,62945],{"className":62914},[196,649],[39,62916,62918,62942],{"className":62917},[200],[39,62919,62922],{"className":62920,"style":62921},[204],"height:0.3283em;",[39,62923,62924,62927],{"style":41316},[39,62925],{"className":62926,"style":213},[212],[39,62928,62930],{"className":62929},[217,218,219,220],[39,62931,62933,62936,62939],{"className":62932},[83,220],[39,62934,11772],{"className":62935,"style":178},[83,84,220],[39,62937,5378],{"className":62938},[465,220],[39,62940,546],{"className":62941},[83,220],[39,62943,673],{"className":62944},[672],[39,62946,62948],{"className":62947},[200],[39,62949,62951],{"className":62950,"style":994},[204],[39,62952],{},[39,62954,161],{"className":62955},[263],[39,62957,62959,62977],{"className":62958,"translate":43},[42],[39,62960,62962],{"className":62961},[47],[49,62963,62964],{"xmlns":51},[53,62965,62966,62974],{},[56,62967,62968],{},[613,62969,62970,62972],{},[59,62971,2098],{},[59,62973,621],{},[63,62975,62976],{"encoding":65},"\\pi_t",[39,62978,62980],{"className":62979,"ariaHidden":70},[69],[39,62981,62983,62986],{"className":62982},[74],[39,62984],{"className":62985,"style":2916},[78],[39,62987,62989,62992],{"className":62988},[83],[39,62990,2098],{"className":62991,"style":2114},[83,84],[39,62993,62995],{"className":62994},[192],[39,62996,62998,63018],{"className":62997},[196,649],[39,62999,63001,63015],{"className":63000},[200],[39,63002,63004],{"className":63003,"style":656},[204],[39,63005,63006,63009],{"style":41316},[39,63007],{"className":63008,"style":213},[212],[39,63010,63012],{"className":63011},[217,218,219,220],[39,63013,621],{"className":63014},[83,84,220],[39,63016,673],{"className":63017},[672],[39,63019,63021],{"className":63020},[200],[39,63022,63024],{"className":63023,"style":680},[204],[39,63025],{}," 是时刻 ",[39,63028,63030,63043],{"className":63029,"translate":43},[42],[39,63031,63033],{"className":63032},[47],[49,63034,63035],{"xmlns":51},[53,63036,63037,63041],{},[56,63038,63039],{},[59,63040,621],{},[63,63042,621],{"encoding":65},[39,63044,63046],{"className":63045,"ariaHidden":70},[69],[39,63047,63049,63052],{"className":63048},[74],[39,63050],{"className":63051,"style":709},[78],[39,63053,621],{"className":63054},[83,84]," 的策略。定义",[39,63057,63059],{"className":63058,"translate":43},[746],[39,63060,63062,63221],{"className":63061,"translate":43},[42],[39,63063,63065],{"className":63064},[47],[49,63066,63067],{"xmlns":51,"display":755},[53,63068,63069,63218],{},[759,63070,63071,63150,63184],{"rowspacing":761,"columnalign":762,"columnspacing":146},[764,63072,63073,63093],{},[767,63074,63075],{},[770,63076,63077],{"scriptlevel":541,"displaystyle":70},[56,63078,63079,63087,63089,63091],{},[9932,63080,63081,63083,63085],{},[59,63082,4230],{},[59,63084,621],{},[59,63086,2098],{},[133,63088,136],{"stretchy":135},[59,63090,142],{},[133,63092,161],{"stretchy":135},[767,63094,63095],{},[770,63096,63097],{"scriptlevel":541,"displaystyle":70},[56,63098,63099,63101,63103,63109,63112,63132,63144],{},[56,63100],{},[133,63102,793],{},[613,63104,63105,63107],{},[59,63106,4253],{"mathvariant":421},[59,63108,2098],{},[133,63110,537],{"fence":135,"stretchy":70,"minsize":63111,"maxsize":63111},"3em",[3187,63113,63114,63116,63124],{},[133,63115,3191],{},[56,63117,63118,63120,63122],{},[59,63119,3196],{},[133,63121,793],{},[59,63123,621],{},[56,63125,63126,63128,63130],{},[59,63127,11772],{},[133,63129,5378],{},[539,63131,546],{},[138,63133,63134,63136],{},[59,63135,531],{},[56,63137,63138,63140,63142],{},[59,63139,3196],{},[133,63141,5378],{},[59,63143,621],{},[613,63145,63146,63148],{},[59,63147,401],{},[59,63149,3196],{},[764,63151,63152,63158],{},[767,63153,63154],{},[770,63155,63156],{"scriptlevel":541,"displaystyle":70},[56,63157],{},[767,63159,63160],{},[770,63161,63162],{"scriptlevel":541,"displaystyle":70},[56,63163,63164,63166,63168,63170,63176,63178,63180,63182],{},[56,63165],{},[230,63167],{"width":10251},[133,63169,150],{},[613,63171,63172,63174],{},[59,63173,142],{},[59,63175,621],{},[133,63177,793],{},[59,63179,142],{},[133,63181,4280],{"fence":135,"stretchy":70,"minsize":63111,"maxsize":63111},[133,63183,155],{"separator":70},[764,63185,63186,63206],{},[767,63187,63188],{},[770,63189,63190],{"scriptlevel":541,"displaystyle":70},[56,63191,63192,63200,63202,63204],{},[9932,63193,63194,63196,63198],{},[59,63195,4230],{},[59,63197,11772],{},[59,63199,2098],{},[133,63201,136],{"stretchy":135},[59,63203,142],{},[133,63205,161],{"stretchy":135},[767,63207,63208],{},[770,63209,63210],{"scriptlevel":541,"displaystyle":70},[56,63211,63212,63214,63216],{},[56,63213],{},[133,63215,793],{},[539,63217,28866],{},[63,63219,63220],{"encoding":65},"\\begin{aligned}\nV_t^\\pi(s)\n&=\\mathbb{E}_\\pi\\Bigg[\\sum_{k=t}^{T-1}\\gamma^{k-t}r_k\\\\\n&\\qquad\\mid s_t=s\\Bigg],\\\\\nV_T^\\pi(s)&=0.\n\\end{aligned}",[39,63222,63224],{"className":63223,"ariaHidden":70},[69],[39,63225,63227,63231],{"className":63226},[74],[39,63228],{"className":63229,"style":63230},[78],"height:7.9305em;vertical-align:-3.7152em;",[39,63232,63234],{"className":63233},[83],[39,63235,63237,63410],{"className":63236},[759],[39,63238,63240],{"className":63239},[918],[39,63241,63243,63401],{"className":63242},[196,649],[39,63244,63246,63398],{"className":63245},[200],[39,63247,63250,63320,63329],{"className":63248,"style":63249},[204],"height:4.2152em;",[39,63251,63253,63257],{"style":63252},"top:-6.2152em;",[39,63254],{"className":63255,"style":63256},[212],"height:3.8283em;",[39,63258,63260,63311,63314,63317],{"className":63259},[83],[39,63261,63263,63266],{"className":63262},[83],[39,63264,4230],{"className":63265,"style":461},[83,84],[39,63267,63269],{"className":63268},[192],[39,63270,63272,63303],{"className":63271},[196,649],[39,63273,63275,63300],{"className":63274},[200],[39,63276,63278,63289],{"className":63277,"style":4421},[204],[39,63279,63280,63283],{"style":9973},[39,63281],{"className":63282,"style":213},[212],[39,63284,63286],{"className":63285},[217,218,219,220],[39,63287,621],{"className":63288},[83,84,220],[39,63290,63291,63294],{"style":1027},[39,63292],{"className":63293,"style":213},[212],[39,63295,63297],{"className":63296},[217,218,219,220],[39,63298,2098],{"className":63299,"style":2114},[83,84,220],[39,63301,673],{"className":63302},[672],[39,63304,63306],{"className":63305},[200],[39,63307,63309],{"className":63308,"style":10003},[204],[39,63310],{},[39,63312,136],{"className":63313},[182],[39,63315,142],{"className":63316},[83,84],[39,63318,161],{"className":63319},[263],[39,63321,63323,63326],{"style":63322},"top:-2.8631em;",[39,63324],{"className":63325,"style":63256},[212],[39,63327],{"className":63328},[83],[39,63330,63332,63335],{"style":63331},"top:-0.4731em;",[39,63333],{"className":63334,"style":63256},[212],[39,63336,63338,63389,63392,63395],{"className":63337},[83],[39,63339,63341,63344],{"className":63340},[83],[39,63342,4230],{"className":63343,"style":461},[83,84],[39,63345,63347],{"className":63346},[192],[39,63348,63350,63381],{"className":63349},[196,649],[39,63351,63353,63378],{"className":63352},[200],[39,63354,63356,63367],{"className":63355,"style":4421},[204],[39,63357,63358,63361],{"style":9973},[39,63359],{"className":63360,"style":213},[212],[39,63362,63364],{"className":63363},[217,218,219,220],[39,63365,11772],{"className":63366,"style":178},[83,84,220],[39,63368,63369,63372],{"style":1027},[39,63370],{"className":63371,"style":213},[212],[39,63373,63375],{"className":63374},[217,218,219,220],[39,63376,2098],{"className":63377,"style":2114},[83,84,220],[39,63379,673],{"className":63380},[672],[39,63382,63384],{"className":63383},[200],[39,63385,63387],{"className":63386,"style":10003},[204],[39,63388],{},[39,63390,136],{"className":63391},[182],[39,63393,142],{"className":63394},[83,84],[39,63396,161],{"className":63397},[263],[39,63399,673],{"className":63400},[672],[39,63402,63404],{"className":63403},[200],[39,63405,63408],{"className":63406,"style":63407},[204],"height:3.7152em;",[39,63409],{},[39,63411,63413],{"className":63412},[1223],[39,63414,63416,63757],{"className":63415},[196,649],[39,63417,63419,63754],{"className":63418},[200],[39,63420,63422,63647,63731],{"className":63421,"style":63249},[204],[39,63423,63424,63427],{"style":63252},[39,63425],{"className":63426,"style":63256},[212],[39,63428,63430,63433,63436,63439,63442,63482,63489,63492,63566,63569,63607],{"className":63429},[83],[39,63431],{"className":63432},[83],[39,63434],{"className":63435,"style":231},[230],[39,63437,793],{"className":63438},[235],[39,63440],{"className":63441,"style":231},[230],[39,63443,63445,63448],{"className":63444},[83],[39,63446,4253],{"className":63447},[83,515],[39,63449,63451],{"className":63450},[192],[39,63452,63454,63474],{"className":63453},[196,649],[39,63455,63457,63471],{"className":63456},[200],[39,63458,63460],{"className":63459,"style":1728},[204],[39,63461,63462,63465],{"style":966},[39,63463],{"className":63464,"style":213},[212],[39,63466,63468],{"className":63467},[217,218,219,220],[39,63469,2098],{"className":63470,"style":2114},[83,84,220],[39,63472,673],{"className":63473},[672],[39,63475,63477],{"className":63476},[200],[39,63478,63480],{"className":63479,"style":680},[204],[39,63481],{},[39,63483,63485],{"className":63484},[83],[39,63486,537],{"className":63487},[18520,63488],"size4",[39,63490],{"className":63491,"style":256},[230],[39,63493,63495],{"className":63494},[1744,3294],[39,63496,63498,63558],{"className":63497},[196,649],[39,63499,63501,63555],{"className":63500},[200],[39,63502,63505,63525,63535],{"className":63503,"style":63504},[204],"height:1.8283em;",[39,63506,63507,63510],{"style":3307},[39,63508],{"className":63509,"style":3311},[212],[39,63511,63513],{"className":63512},[217,218,219,220],[39,63514,635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V_t^\\pi(s)",[39,63824,63826,63901],{"className":63825,"ariaHidden":70},[69],[39,63827,63829,63832,63883,63886,63889,63892,63895,63898],{"className":63828},[74],[39,63830],{"className":63831,"style":245},[78],[39,63833,63835,63838],{"className":63834},[83],[39,63836,4230],{"className":63837,"style":461},[83,84],[39,63839,63841],{"className":63840},[192],[39,63842,63844,63875],{"className":63843},[196,649],[39,63845,63847,63872],{"className":63846},[200],[39,63848,63850,63861],{"className":63849,"style":21897},[204],[39,63851,63852,63855],{"style":9973},[39,63853],{"className":63854,"style":213},[212],[39,63856,63858],{"className":63857},[217,218,219,220],[39,63859,621],{"className":63860},[83,84,220],[39,63862,63863,63866],{"style":208},[39,63864],{"className":63865,"style":213},[212],[39,63867,63869],{"className":63868},[217,218,219,220],[39,63870,21884],{"className":63871},[465,220],[39,63873,673],{"className":63874},[672],[39,63876,63878],{"className":63877},[200],[39,63879,63881],{"className":63880,"style":10003},[204],[39,63882],{},[39,63884,136],{"className":63885},[182],[39,63887,142],{"className":63888},[83,84],[39,63890,161],{"className":63891},[263],[39,63893],{"className":63894,"style":231},[230],[39,63896,793],{"className":63897},[235],[39,63899],{"className":63900,"style":231},[230],[39,63902,63904,63907,63950,63953,64004,64007,64010],{"className":63903},[74],[39,63905],{"className":63906,"style":245},[78],[39,63908,63910,63913],{"className":63909},[1744],[39,63911,21465],{"className":63912},[1744],[39,63914,63916],{"className":63915},[192],[39,63917,63919,63941],{"className":63918},[196,649],[39,63920,63922,63938],{"className":63921},[200],[39,63923,63926],{"className":63924,"style":63925},[204],"height:0.0573em;",[39,63927,63929,63932],{"style":63928},"top:-2.4559em;margin-right:0.05em;",[39,63930],{"className":63931,"style":213},[212],[39,63933,63935],{"className":63934},[217,218,219,220],[39,63936,2098],{"className":63937,"style":2114},[83,84,220],[39,63939,673],{"className":63940},[672],[39,63942,63944],{"className":63943},[200],[39,63945,63948],{"className":63946,"style":63947},[204],"height:0.2441em;",[39,63949],{},[39,63951],{"className":63952,"style":256},[230],[39,63954,63956,63959],{"className":63955},[83],[39,63957,4230],{"className":63958,"style":461},[83,84],[39,63960,63962],{"className":63961},[192],[39,63963,63965,63996],{"className":63964},[196,649],[39,63966,63968,63993],{"className":63967},[200],[39,63969,63971,63982],{"className":63970,"style":6410},[204],[39,63972,63973,63976],{"style":9973},[39,63974],{"className":63975,"style":213},[212],[39,63977,63979],{"className":63978},[217,218,219,220],[39,63980,621],{"className":63981},[83,84,220],[39,63983,63984,63987],{"style":208},[39,63985],{"className":63986,"style":213},[212],[39,63988,63990],{"className":63989},[217,218,219,220],[39,63991,2098],{"className":63992,"style":2114},[83,84,220],[39,63994,673],{"className":63995},[672],[39,63997,63999],{"className":63998},[200],[39,64000,64002],{"className":64001,"style":10003},[204],[39,64003],{},[39,64005,136],{"className":64006},[182],[39,64008,142],{"className":64009},[83,84],[39,64011,161],{"className":64012},[263],[31,64014,64017,64156],{"id":64015,"title":64016},"theorem-finite-horizon","定理 6.1（有限期最优性原理）",[11,64018,26443,64019,6525],{},[39,64020,64022,64060],{"className":64021,"translate":43},[42],[39,64023,64025],{"className":64024},[47],[49,64026,64027],{"xmlns":51},[53,64028,64029,64057],{},[56,64030,64031,64033,64035,64037,64039,64041,64043,64045,64047,64049,64051,64053,64055],{},[59,64032,621],{},[133,64034,793],{},[59,64036,11772],{},[133,64038,5378],{},[539,64040,546],{},[133,64042,155],{"separator":70},[59,64044,11772],{},[133,64046,5378],{},[539,64048,17359],{},[133,64050,155],{"separator":70},[133,64052,2641],{},[133,64054,155],{"separator":70},[539,64056,541],{},[63,64058,64059],{"encoding":65},"t=T-1,T-2,\\dots,0",[39,64061,64063,64081,64099,64126],{"className":64062,"ariaHidden":70},[69],[39,64064,64066,64069,64072,64075,64078],{"className":64065},[74],[39,64067],{"className":64068,"style":709},[78],[39,64070,621],{"className":64071},[83,84],[39,64073],{"className":64074,"style":231},[230],[39,64076,793],{"className":64077},[235],[39,64079],{"className":64080,"style":231},[230],[39,64082,64084,64087,64090,64093,64096],{"className":64083},[74],[39,64085],{"className":64086,"style":454},[78],[39,64088,11772],{"className":64089,"style":178},[83,84],[39,64091],{"className":64092,"style":461},[230],[39,64094,5378],{"className":64095},[465],[39,64097],{"className":64098,"style":461},[230],[39,64100,64102,64105,64108,64111,64114,64117,64120,64123],{"className":64101},[74],[39,64103],{"className":64104,"style":6452},[78],[39,64106,546],{"className":64107},[83],[39,64109,155],{"className":64110},[252],[39,64112],{"className":64113,"style":256},[230],[39,64115,11772],{"className":64116,"style":178},[83,84],[39,64118],{"className":64119,"style":461},[230],[39,64121,5378],{"className":64122},[465],[39,64124],{"className":64125,"style":461},[230],[39,64127,64129,64132,64135,64138,64141,64144,64147,64150,64153],{"className":64128},[74],[39,64130],{"className":64131,"style":17578},[78],[39,64133,17359],{"className":64134},[83],[39,64136,155],{"className":64137},[252],[39,64139],{"className":64140,"style":256},[230],[39,64142,2641],{"className":64143},[2842],[39,64145],{"className":64146,"style":256},[230],[39,64148,155],{"className":64149},[252],[39,64151],{"className":64152,"style":256},[230],[39,64154,541],{"className":64155},[83],[39,64157,64159],{"className":64158,"translate":43},[746],[39,64160,64162,64244],{"className":64161,"translate":43},[42],[39,64163,64165],{"className":64164},[47],[49,64166,64167],{"xmlns":51,"display":755},[53,64168,64169,64241],{},[56,64170,64171,64179,64181,64191,64205,64207,64209,64217,64219,64225,64239],{},[9932,64172,64173,64175,64177],{},[59,64174,4230],{},[59,64176,621],{},[59,64178,2098],{},[133,64180,793],{},[138,64182,64183,64185],{},[59,64184,11772],{"mathvariant":617},[613,64186,64187,64189],{},[59,64188,2098],{},[59,64190,621],{},[9932,64192,64193,64195,64203],{},[59,64194,4230],{},[56,64196,64197,64199,64201],{},[59,64198,621],{},[133,64200,788],{},[539,64202,546],{},[59,64204,2098],{},[133,64206,155],{"separator":70},[230,64208],{"width":10251},[9932,64210,64211,64213,64215],{},[59,64212,4230],{},[59,64214,621],{},[133,64216,21884],{},[133,64218,793],{},[138,64220,64221,64223],{},[59,64222,11772],{"mathvariant":617},[133,64224,21884],{},[9932,64226,64227,64229,64237],{},[59,64228,4230],{},[56,64230,64231,64233,64235],{},[59,64232,621],{},[133,64234,788],{},[539,64236,546],{},[133,64238,21884],{},[59,64240,2903],{"mathvariant":145},[63,64242,64243],{"encoding":65},"V_t^\\pi=\\mathcal{T}^{\\pi_t} V_{t+1}^\\pi,\n\\qquad\nV_t^*=\\mathcal{T}^* V_{t+1}^*.",[39,64245,64247,64314,64522],{"className":64246,"ariaHidden":70},[69],[39,64248,64250,64254,64305,64308,64311],{"className":64249},[74],[39,64251],{"className":64252,"style":64253},[78],"height:0.9614em;vertical-align:-0.247em;",[39,64255,64257,64260],{"className":64256},[83],[39,64258,4230],{"className":64259,"style":461},[83,84],[39,64261,64263],{"className":64262},[192],[39,64264,64266,64297],{"className":64265},[196,649],[39,64267,64269,64294],{"className":64268},[200],[39,64270,64272,64283],{"className":64271,"style":4421},[204],[39,64273,64274,64277],{"style":9973},[39,64275],{"className":64276,"style":213},[212],[39,64278,64280],{"className":64279},[217,218,219,220],[39,64281,621],{"className":64282},[83,84,220],[39,64284,64285,64288],{"style":1027},[39,64286],{"className":64287,"style":213},[212],[39,64289,64291],{"className":64290},[217,218,219,220],[39,64292,2098],{"className":64293,"style":2114},[83,84,220],[39,64295,673],{"className":64296},[672],[39,64298,64300],{"className":64299},[200],[39,64301,64303],{"className":64302,"style":10003},[204],[39,64304],{},[39,64306],{"className":64307,"style":231},[230],[39,64309,793],{"className":64310},[235],[39,64312],{"className":64313,"style":231},[230],[39,64315,64317,64321,64392,64453,64456,64459,64462,64513,64516,64519],{"className":64316},[74],[39,64318],{"className":64319,"style":64320},[78],"height:1.044em;vertical-align:-0.3053em;",[39,64322,64324,64327],{"className":64323},[83],[39,64325,11772],{"className":64326,"style":11793},[83,641],[39,64328,64330],{"className":64329},[192],[39,64331,64333],{"className":64332},[196],[39,64334,64336],{"className":64335},[200],[39,64337,64339],{"className":64338,"style":4421},[204],[39,64340,64341,64344],{"style":1027},[39,64342],{"className":64343,"style":213},[212],[39,64345,64347],{"className":64346},[217,218,219,220],[39,64348,64350],{"className":64349},[83,220],[39,64351,64353,64356],{"className":64352},[83,220],[39,64354,2098],{"className":64355,"style":2114},[83,84,220],[39,64357,64359],{"className":64358},[192],[39,64360,64362,64383],{"className":64361},[196,649],[39,64363,64365,64380],{"className":64364},[200],[39,64366,64369],{"className":64367,"style":64368},[204],"height:0.2963em;",[39,64370,64371,64374],{"style":42108},[39,64372],{"className":64373,"style":4004},[212],[39,64375,64377],{"className":64376},[217,4008,4009,220],[39,64378,621],{"className":64379},[83,84,220],[39,64381,673],{"className":64382},[672],[39,64384,64386],{"className":64385},[200],[39,64387,64390],{"className":64388,"style":64389},[204],"height:0.143em;",[39,64391],{},[39,64393,64395,64398],{"className":64394},[83],[39,64396,4230],{"className":64397,"style":461},[83,84],[39,64399,64401],{"className":64400},[192],[39,64402,64404,64444],{"className":64403},[196,649],[39,64405,64407,64441],{"className":64406},[200],[39,64408,64410,64430],{"className":64409,"style":4421},[204],[39,64411,64412,64415],{"style":9973},[39,64413],{"className":64414,"style":213},[212],[39,64416,64418],{"className":64417},[217,218,219,220],[39,64419,64421,64424,64427],{"className":64420},[83,220],[39,64422,621],{"className":64423},[83,84,220],[39,64425,788],{"className":64426},[465,220],[39,64428,546],{"className":64429},[83,220],[39,64431,64432,64435],{"style":1027},[39,64433],{"className":64434,"style":213},[212],[39,64436,64438],{"className":64437},[217,218,219,220],[39,64439,2098],{"className":64440,"style":2114},[83,84,220],[39,64442,673],{"className":64443},[672],[39,64445,64447],{"className":64446},[200],[39,64448,64451],{"className":64449,"style":64450},[204],"height:0.3053em;",[39,64452],{},[39,64454,155],{"className":64455},[252],[39,64457],{"className":64458,"style":10787},[230],[39,64460],{"className":64461,"style":256},[230],[39,64463,64465,64468],{"className":64464},[83],[39,64466,4230],{"className":64467,"style":461},[83,84],[39,64469,64471],{"className":64470},[192],[39,64472,64474,64505],{"className":64473},[196,649],[39,64475,64477,64502],{"className":64476},[200],[39,64478,64480,64491],{"className":64479,"style":22242},[204],[39,64481,64482,64485],{"style":9973},[39,64483],{"className":64484,"style":213},[212],[39,64486,64488],{"className":64487},[217,218,219,220],[39,64489,621],{"className":64490},[83,84,220],[39,64492,64493,64496],{"style":1027},[39,64494],{"className":64495,"style":213},[212],[39,64497,64499],{"className":64498},[217,218,219,220],[39,64500,21884],{"className":64501},[465,220],[39,64503,673],{"className":64504},[672],[39,64506,64508],{"className":64507},[200],[39,64509,64511],{"className":64510,"style":10003},[204],[39,64512],{},[39,64514],{"className":64515,"style":231},[230],[39,64517,793],{"className":64518},[235],[39,64520],{"className":64521,"style":231},[230],[39,64523,64525,64528,64557,64617],{"className":64524},[74],[39,64526],{"className":64527,"style":64320},[78],[39,64529,64531,64534],{"className":64530},[83],[39,64532,11772],{"className":64533,"style":11793},[83,641],[39,64535,64537],{"classN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对 ",[39,64626,64628,64641],{"className":64627,"translate":43},[42],[39,64629,64631],{"className":64630},[47],[49,64632,64633],{"xmlns":51},[53,64634,64635,64639],{},[56,64636,64637],{},[59,64638,621],{},[63,64640,621],{"encoding":65},[39,64642,64644],{"className":64643,"ariaHidden":70},[69],[39,64645,64647,64650],{"className":64646},[74],[39,64648],{"className":64649,"style":709},[78],[39,64651,621],{"className":64652},[83,84]," 从 ",[39,64655,64657,64675],{"className":64656,"translate":43},[42],[39,64658,64660],{"className":64659},[47],[49,64661,64662],{"xmlns":51},[53,64663,64664,64672],{},[56,64665,64666,64668,64670],{},[59,64667,11772],{},[133,64669,5378],{},[539,64671,546],{},[63,64673,64674],{"encoding":65},"T-1",[39,64676,64678,64696],{"className":64677,"ariaHidden":70},[69],[39,64679,64681,64684,64687,64690,64693],{"className":64680},[74],[39,64682],{"className":64683,"style":454},[78],[39,64685,11772],{"className":64686,"style":178},[83,84],[39,64688],{"className":64689,"style":461},[230],[39,64691,5378],{"className":64692},[465],[39,64694],{"className":64695,"style":461},[230],[39,64697,64699,64702],{"className":64698},[74],[39,64700],{"className":64701,"style":2473},[78],[39,64703,546],{"className":64704},[83]," 到 ",[39,64707,64709,64722],{"className":64708,"translate":43},[42],[39,64710,64712],{"className":64711},[47],[49,64713,64714],{"xmlns":51},[53,64715,64716,64720],{},[56,64717,64718],{},[539,64719,541],{},[63,64721,541],{"encoding":65},[39,64723,64725],{"className":64724,"ariaHidden":70},[69],[39,64726,64728,64731],{"className":64727},[74],[39,64729],{"className":64730,"style":2473},[78],[39,64732,541],{"className":64733},[83]," 反向归纳。",[39,64736,64738,64756],{"className":64737,"translate":43},[42],[39,64739,64741],{"className":64740},[47],[49,64742,64743],{"xmlns":51},[53,64744,64745,64753],{},[56,64746,64747,64749,64751],{},[59,64748,621],{},[133,64750,793],{},[59,64752,11772],{},[63,64754,64755],{"encoding":65},"t=T",[39,64757,64759,64777],{"className":64758,"ariaHidden":70},[69],[39,64760,64762,64765,64768,64771,64774],{"className":64761},[74],[39,64763],{"className":64764,"style":709},[78],[39,64766,621],{"className":64767},[83,84],[39,64769],{"className":64770,"style":231},[230],[39,64772,793],{"className":64773},[235],[39,64775],{"className":64776,"style":231},[230],[39,64778,64780,64783],{"className":64779},[74],[39,64781],{"className":64782,"style":79},[78],[39,64784,11772],{"className":64785,"style":178},[83,84]," 时两端均为零。设结论对 ",[39,64788,64790,64808],{"className":64789,"translate":43},[42],[39,64791,64793],{"className":64792},[47],[49,64794,64795],{"xmlns":51},[53,64796,64797,64805],{},[56,64798,64799,64801,64803],{},[59,64800,621],{},[133,64802,788],{},[539,64804,546],{},[63,64806,64807],{"encoding":65},"t+1",[39,64809,64811,64830],{"className":64810,"ariaHidden":70},[69],[39,64812,64814,64818,64821,64824,64827],{"className":64813},[74],[39,64815],{"className":64816,"style":64817},[78],"height:0.6984em;vertical-align:-0.0833em;",[39,64819,621],{"className":64820},[83,84],[39,64822],{"className":64823,"style":461},[230],[39,64825,788],{"className":64826},[465],[39,64828],{"className":64829,"style":461},[230],[39,64831,64833,64836],{"className":64832},[74],[39,64834],{"className":64835,"style":2473},[78],[39,64837,546],{"className":64838},[83]," 成立。对任意策略 ",[39,64841,64843,64856],{"className":64842,"translate":43},[42],[39,64844,64846],{"className":64845},[47],[49,64847,64848],{"xmlns":51},[53,64849,64850,64854],{},[56,64851,64852],{},[59,64853,2098],{},[63,64855,2101],{"encoding":65},[39,64857,64859],{"className":64858,"ariaHidden":70},[69],[39,64860,64862,64865],{"className":64861},[74],[39,64863],{"className":64864,"style":290},[78],[39,64866,2098],{"className":64867,"style":2114},[83,84],"，在 ",[39,64870,64872,64894],{"className":64871,"translate":43},[42],[39,64873,64875],{"className":64874},[47],[49,64876,64877],{"xmlns":51},[53,64878,64879,64891],{},[56,64880,64881,64887,64889],{},[613,64882,64883,64885],{},[59,64884,142],{},[59,64886,621],{},[133,64888,793],{},[59,64890,142],{},[63,64892,64893],{"encoding":65},"s_t=s",[39,64895,64897,64952],{"className":64896,"ariaHidden":70},[69],[39,64898,64900,64903,64943,64946,64949],{"className":64899},[74],[39,64901],{"className":64902,"style":2916},[78],[39,64904,64906,64909],{"className":64905},[83],[39,64907,142],{"className":64908},[83,84],[39,64910,64912],{"className":64911},[192],[39,64913,64915,64935],{"className":64914},[196,649],[39,64916,64918,64932],{"className":64917},[200],[39,64919,64921],{"className":64920,"style":656},[204],[39,64922,64923,64926],{"style":966},[39,64924],{"className":64925,"style":213},[212],[39,64927,64929],{"className":64928},[217,218,219,220],[39,64930,621],{"className":64931},[83,84,220],[39,64933,673],{"className":64934},[672],[39,64936,64938],{"className":64937},[200],[39,64939,64941],{"className":64940,"style":680},[204],[39,64942],{},[39,64944],{"className":64945,"style":231},[230],[39,64947,793],{"className":64948},[235],[39,64950],{"className":64951,"style":231},[230],[39,64953,64955,64958],{"className":64954},[74],[39,64956],{"className":64957,"style":290},[78],[39,64959,142],{"className":64960},[83,84]," 处分解回报：",[39,64963,64965],{"className":64964,"translate":43},[746],[39,64966,64968,65095],{"className":64967,"translate":43},[42],[39,64969,64971],{"className":64970},[47],[49,64972,64973],{"xmlns":51,"display":755},[53,64974,64975,65092],{},[759,64976,64977,65058],{"rowspacing":761,"columnalign":762,"columnspacing":146},[764,64978,64979,64999],{},[767,64980,64981],{},[770,64982,64983],{"scriptlevel":541,"displaystyle":70},[56,64984,64985,64993,64995,64997],{},[9932,64986,64987,64989,64991],{},[59,64988,4230],{},[59,64990,621],{},[59,64992,2098],{},[133,64994,136],{"stretchy":135},[59,64996,142],{},[133,64998,161],{"stretchy":135},[767,65000,65001],{},[770,65002,65003],{"scriptlevel":541,"displaystyle":70},[56,65004,65005,65007,65009,65015,65018,65024,65026,65028,65042,65044,65056],{},[56,65006],{},[133,65008,793],{},[613,65010,65011,65013],{},[59,65012,4253],{"mathvariant":421},[59,65014,2098],{},[133,65016,537],{"fence":135,"stretchy":70,"minsize":65017,"maxsize":65017},"1.8em",[613,65019,65020,65022],{},[59,65021,401],{},[59,65023,621],{},[133,65025,788],{},[59,65027,531],{},[9932,65029,65030,65032,65040],{},[59,65031,4230],{},[56,65033,65034,65036,65038],{},[59,65035,621],{},[133,65037,788],{},[539,65039,546],{},[59,65041,2098],{},[133,65043,136],{"stretchy":135},[613,65045,65046,65048],{},[59,65047,142],{},[56,65049,65050,65052,65054],{},[59,65051,621],{},[133,65053,788],{},[539,65055,546],{},[133,65057,161],{"stretchy":135},[764,65059,65060,65066],{},[767,65061,65062],{},[770,65063,65064],{"scriptlevel":541,"displaystyle":70},[56,65065],{},[767,65067,65068],{},[770,65069,65070],{"scriptlevel":541,"displaystyle":70},[56,65071,65072,65074,65076,65078,65084,65086,65088,65090],{},[56,65073],{},[230,65075],{"width":10251},[133,65077,150],{},[613,65079,65080,65082],{},[59,65081,142],{},[59,65083,621],{},[133,65085,793],{},[59,65087,142],{},[133,65089,4280],{"fence":135,"stretchy":70,"minsize":65017,"maxsize":65017},[59,65091,2903],{"mathvariant":145},[63,65093,65094],{"encoding":65},"\\begin{aligned}\nV_t^\\pi(s)\n&=\\mathbb{E}_\\pi\\Big[r_t+\\gamma V_{t+1}^\\pi(s_{t+1})\\\\\n&\\qquad\\mid s_t=s\\Big].\n\\end{aligned}",[39,65096,65098],{"className":65097,"ariaHidden":70},[69],[39,65099,65101,65105],{"className":65100},[74],[39,65102],{"className":65103,"style":65104},[78],"height:3.9em;vertical-align:-1.7em;",[39,65106,65108],{"className":65107},[83],[39,65109,65111,65213],{"className":65110},[759],[39,65112,65114],{"className":65113},[918],[39,65115,65117,65204],{"className":65116},[196,649],[39,65118,65120,65201],{"className":65119},[200],[39,65121,65124,65193],{"className":65122,"style":65123},[204],"height:2.2em;",[39,65125,65127,65130],{"style":65126},"top:-4.2em;",[39,65128],{"className":65129,"style":33100},[212],[39,65131,65133,65184,65187,65190],{"className":65132},[83],[39,65134,65136,65139],{"className":65135},[83],[39,65137,4230],{"className":65138,"style":461},[83,84],[39,65140,65142],{"className":65141},[192],[39,65143,65145,65176],{"className":65144},[196,649],[39,65146,65148,65173],{"className":65147},[200],[39,65149,65151,65162],{"className":65150,"style":4421},[204],[39,65152,65153,65156],{"style":9973},[39,65154],{"className":65155,"style":213},[212],[39,65157,65159],{"className":65158},[217,218,219,220],[39,65160,621],{"className":65161},[83,84,220],[39,65163,65164,65167],{"style":1027},[39,65165],{"className":65166,"style":213},[212],[39,65168,65170],{"className":65169},[217,218,219,220],[39,65171,2098],{"className":65172,"style":2114},[83,84,220],[39,65174,673],{"className":65175},[672],[39,65177,65179],{"className":65178},[200],[39,65180,65182],{"className":65181,"style":10003},[204],[39,65183],{},[39,65185,136],{"className":65186},[182],[39,65188,142],{"className":65189},[83,84],[39,65191,161],{"className":65192},[263],[39,65194,65195,65198],{"style":5834},[39,65196],{"className":65197,"style":33100},[212],[39,65199],{"className":65200},[83],[39,65202,673],{"className":65203},[672],[39,65205,65207],{"className":65206},[200],[39,65208,65211],{"className":65209,"style":65210},[204],"height:1.7em;",[39,65212],{},[39,65214,65216],{"className":65215},[1223],[39,65217,65219,65545],{"className":65218},[196,649],[39,65220,65222,65542],{"className":65221},[200],[39,65223,65225,65458],{"className":65224,"style":65123},[204],[39,65226,65227,65230],{"style":65126},[39,65228],{"className":65229,"style":33100},[212],[39,65231,65233,65236,65239,65242,65245,65285,65291,65331,65334,65337,65340,65343,65403,65406,65455],{"className":65232},[83],[39,65234],{"className":65235},[83],[39,65237],{"className":65238,"style":231},[230],[39,65240,793],{"className":65241},[235],[39,65243],{"className":65244,"style":231},[230],[39,65246,65248,65251],{"className":65247},[83],[39,65249,4253],{"className":65250},[83,515],[39,65252,65254],{"className":65253},[192],[39,65255,65257,65277],{"className":65256},[196,649],[39,65258,65260,65274],{"className":65259},[200],[39,65261,65263],{"className":65262,"style":1728},[204],[39,65264,65265,65268],{"style":966},[39,65266],{"className":65267,"style":213},[212],[39,65269,65271],{"className":65270},[217,218,219,220],[39,65272,2098],{"className":65273,"style":2114},[83,84,220],[39,65275,673],{"className":65276},[672],[39,65278,65280],{"className":65279},[200],[39,65281,65283],{"className":65282,"style":680},[204],[39,65284],{},[39,65286,65288],{"className":65287},[83],[39,65289,537],{"className":65290},[18520,33204],[39,65292,65294,65297],{"className":65293},[83],[39,65295,401],{"className":65296,"style":438},[83,84],[39,65298,65300],{"className":65299},[192],[39,65301,65303,65323],{"className":65302},[196,649],[39,65304,65306,65320],{"className":65305},[200],[39,65307,65309],{"className":65308,"style":656},[204],[39,65310,65311,65314],{"style":2937},[39,65312],{"className":65313,"style":213},[212],[39,65315,65317],{"className":65316},[217,218,219,220],[39,65318,621],{"className":65319},[83,84,220],[39,65321,673],{"className":65322},[672],[39,65324,65326],{"className":65325},[200],[39,65327,65329],{"className":65328,"style":680},[204],[39,65330],{},[39,65332],{"className":65333,"style":461},[230],[39,65335,788],{"className":65336},[465],[39,65338],{"className":65339,"style":461},[230],[39,65341,531],{"className":65342,"style":565},[83,84],[39,65344,65346,65349],{"className":65345},[83],[39,65347,4230],{"className":65348,"style":461},[83,84],[39,65350,65352],{"className":65351},[192],[39,65353,65355,65395],{"className":65354},[196,649],[39,65356,65358,65392],{"className":65357},[200],[39,65359,65361,65381],{"className":65360,"style":4421},[204],[39,65362,65363,65366],{"style":9973},[39,65364],{"className":65365,"style":213},[212],[39,65367,65369],{"className":65368},[217,218,219,220],[39,65370,65372,65375,65378],{"className":65371},[83,220],[39,65373,621],{"className":65374},[83,84,220],[39,65376,788],{"className":65377},[465,220],[39,65379,546],{"className":65380},[83,220],[39,65382,65383,65386],{"style":1027},[39,65384],{"className":65385,"style":213},[212],[39,65387,65389],{"className":65388},[217,218,219,220],[39,65390,2098],{"className":65391,"style":2114},[83,84,220],[39,65393,673],{"className":65394},[672],[39,65396,65398],{"className":65397},[200],[39,65399,65401],{"className":65400,"style":64450},[204],[39,65402],{},[39,65404,136],{"className":65405},[182],[39,65407,65409,65412],{"className":65408},[83],[39,65410,142],{"className":65411},[83,84],[39,65413,65415],{"className":65414},[192],[39,65416,65418,65447],{"className":65417},[196,649],[39,65419,65421,65444],{"className":65420},[200],[39,65422,65424],{"className":65423,"style":963},[204],[39,65425,65426,65429],{"style":966},[39,65427],{"className":65428,"style":213},[212],[39,65430,65432],{"className":65431},[217,218,219,220],[39,65433,65435,65438,65441],{"className":65434},[83,220],[39,65436,621],{"className":65437},[83,84,220],[39,65439,788],{"className":65440},[465,220],[39,65442,546],{"className":65443},[83,220],[39,65445,673],{"className":65446},[672],[39,65448,65450],{"className":65449},[200],[39,65451,65453],{"className":65452,"style":994},[204],[39,65454],{},[39,65456,161],{"className":65457},[263],[39,65459,65460,65463],{"style":5834},[39,65461],{"className":65462,"style":33100},[212],[39,65464,65466,65469,65472,65475,65478,65481,65521,65524,65527,65530,65533,65539],{"className":65465},[83],[39,65467],{"className":65468},[83],[39,65470],{"className":65471,"style":10787},[230],[39,65473],{"className":65474,"style":231},[230],[39,65476,150],{"className":65477},[235],[39,65479],{"className":65480,"style":231},[230],[39,65482,65484,65487],{"className":65483},[83],[39,65485,142],{"className":65486},[83,84],[39,65488,65490],{"className":65489},[192],[39,65491,65493,65513],{"className":65492},[196,649],[39,65494,65496,65510],{"className":65495},[200],[39,65497,65499],{"className":65498,"style":656},[204],[39,65500,65501,65504],{"style":966},[39,65502],{"className":65503,"style":213},[212],[39,65505,65507],{"className":65506},[217,218,219,220],[39,65508,621],{"className":65509},[83,84,220],[39,65511,673],{"className":65512},[672],[39,65514,65516],{"className":65515},[200],[39,65517,65519],{"className":65518,"style":680},[204],[39,65520],{},[39,65522],{"className":65523,"style":231},[230],[39,65525,793],{"className":65526},[235],[39,65528],{"className":65529,"style":231},[230],[39,65531,142],{"className":65532},[83,84],[39,65534,65536],{"className":65535},[83],[39,65537,4280],{"className":65538},[18520,33204],[39,65540,2903],{"className":65541},[83],[39,65543,673],{"className":65544},[672],[39,65546,65548],{"className":65547},[200],[39,65549,65551],{"className":65550,"style":65210},[204],[39,65552],{},[11,65554,65555,65556,65657,65658,49416],{},"由 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的定义，",[39,65659,65661,65703],{"className":65660,"translate":43},[42],[39,65662,65664],{"className":65663},[47],[49,65665,65666],{"xmlns":51},[53,65667,65668,65700],{},[56,65669,65670,65684,65686],{},[9932,65671,65672,65674,65682],{},[59,65673,4230],{},[56,65675,65676,65678,65680],{},[59,65677,621],{},[133,65679,788],{},[539,65681,546],{},[59,65683,2098],{},[133,65685,1824],{},[9932,65687,65688,65690,65698],{},[59,65689,4230],{},[56,65691,65692,65694,65696],{},[59,65693,621],{},[133,65695,788],{},[539,65697,546],{},[133,65699,21884],{},[63,65701,65702],{"encoding":65},"V_{t+1}^\\pi\\le 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时刻可达的最优值。取 ",[39,67277,67279,67297],{"className":67278,"translate":43},[42],[39,67280,67282],{"className":67281},[47],[49,67283,67284],{"xmlns":51},[53,67285,67286,67294],{},[56,67287,67288],{},[613,67289,67290,67292],{},[59,67291,158],{},[59,67293,621],{},[63,67295,67296],{"encoding":65},"a_t",[39,67298,67300],{"className":67299,"ariaHidden":70},[69],[39,67301,67303,67306],{"className":67302},[74],[39,67304],{"className":67305,"style":2916},[78],[39,67307,67309,67312],{"className":67308},[83],[39,67310,158],{"className":67311},[83,84],[39,67313,67315],{"className":67314},[192],[39,67316,67318,67338],{"className":67317},[196,649],[39,67319,67321,67335],{"className":67320},[200],[39,67322,67324],{"className":67323,"style":656},[204],[39,67325,67326,67329],{"style":966},[39,67327],{"className":67328,"style":213},[212],[39,67330,67332],{"className":67331},[217,218,219,220],[39,67333,621],{"className":67334},[83,84,220],[39,67336,673],{"className":67337},[672],[39,67339,67341],{"className":67340},[200],[39,67342,67344],{"className":67343,"style":680},[204],[39,67345],{}," 为使上式右端最大的动作，并从 ",[39,67348,67350,67367],{"className":67349,"translate":43},[42],[39,67351,67353],{"className":67352},[47],[49,67354,67355],{"xmlns":51},[53,67356,67357,67365],{},[56,67358,67359,67361,67363],{},[59,67360,621],{},[133,67362,788],{},[539,67364,546],{},[63,67366,64807],{"encoding":65},[39,67368,67370,67388],{"className":67369,"ariaHidden":70},[69],[39,67371,67373,67376,67379,67382,67385],{"className":67372},[74],[39,67374],{"className":67375,"style":64817},[78],[39,67377,621],{"className":67378},[83,84],[39,67380],{"className":67381,"style":461},[230],[39,67383,788],{"className":67384},[465],[39,67386],{"className":67387,"style":461},[230],[39,67389,67391,67394],{"className":67390},[74],[39,67392],{"className":67393,"style":2473},[78],[39,67395,546],{"className":67396},[83]," 开始采用达到 ",[39,67399,67401,67426],{"className":67400,"translate":43},[42],[39,67402,67404],{"className":67403},[47],[49,67405,67406],{"xmlns":51},[53,67407,67408,67424],{},[56,67409,67410],{},[9932,67411,67412,67414,67422],{},[59,67413,4230],{},[56,67415,67416,67418,67420],{},[59,67417,621],{},[133,67419,788],{},[539,67421,546],{},[133,67423,21884],{},[63,67425,65584],{"encoding":65},[39,67427,67429],{"className":67428,"ariaHidden":70},[69],[39,67430,67432,67435],{"className":67431},[74],[39,67433],{"className":67434,"style":65594},[78],[39,67436,67438,67441],{"className":67437},[83],[39,67439,4230],{"className":67440,"style":461},[83,84],[39,67442,67444],{"className":67443},[192],[39,67445,67447,67487],{"className":67446},[196,649],[39,67448,67450,67484],{"className":67449},[200],[39,67451,67453,67473],{"className":67452,"style":21897},[204],[39,67454,67455,67458],{"style":65615},[39,67456],{"className":67457,"style":213},[212],[39,67459,67461],{"className":67460},[217,218,219,220],[39,67462,67464,67467,67470],{"className":67463},[83,220],[39,67465,621],{"className":67466},[83,84,220],[39,67468,788],{"className":67469},[465,220],[39,67471,546],{"className":67472},[83,220],[39,67474,67475,67478],{"style":208},[39,67476],{"className":67477,"style":213},[212],[39,67479,67481],{"className":67480},[217,218,219,220],[39,67482,21884],{"className":67483},[465,220],[39,67485,673],{"className":67486},[672],[39,67488,67490],{"className":67489},[200],[39,67491,67493],{"className":67492,"style":65654},[204],[39,67494],{}," 的最优续策略，则等号可以达到。在有限状态与有限动作下，可以逐状态选取最优续策略，因此不需要额外的可测选择条件。故 ",[39,67497,67499,67541],{"className":67498,"translate":43},[42],[39,67500,67502],{"className":67501},[47],[49,67503,67504],{"xmlns":51},[53,67505,67506,67538],{},[56,67507,67508,67516,67518,67524],{},[9932,67509,67510,67512,67514],{},[59,67511,4230],{},[59,67513,621],{},[133,67515,21884],{},[133,67517,793],{},[138,67519,67520,67522],{},[59,67521,11772],{"mathvariant":617},[133,67523,21884],{},[9932,67525,67526,67528,67536],{},[59,67527,4230],{},[56,67529,67530,67532,67534],{},[59,67531,621],{},[133,67533,788],{},[539,67535,546],{},[133,67537,21884],{},[63,67539,67540],{"encoding":65},"V_t^*=\\mathcal{T}^*V_{t+1}^*",[39,67542,67544,67611],{"className":67543,"ariaHidden":70},[69],[39,67545,67547,67551,67602,67605,67608],{"className":67546},[74],[39,67548],{"className":67549,"style":67550},[78],"height:0.9357em;vertical-align:-0.247em;",[39,67552,67554,67557],{"className":67553},[83],[39,67555,4230],{"className":67556,"style":461},[83,84],[39,67558,67560],{"className":67559},[192],[39,67561,67563,67594],{"className":67562},[196,649],[39,67564,67566,67591],{"className":67565},[200],[39,67567,67569,67580],{"className":67568,"style":21897},[204],[39,67570,67571,67574],{"style":9973},[39,67572],{"className":67573,"style":213},[212],[39,67575,67577],{"className":67576},[217,218,219,220],[39,67578,621],{"className":67579},[83,84,220],[39,67581,67582,67585],{"style":208},[39,67583],{"className":67584,"style":213},[212],[39,67586,67588],{"className":67587},[217,218,219,220],[39,67589,21884],{"className":67590},[465,220],[39,67592,673],{"className":67593},[672],[39,67595,67597],{"className":67596},[200],[39,67598,67600],{"className":67599,"style":10003},[204],[39,67601],{},[39,67603],{"className":67604,"style":231},[230],[39,67606,793],{"className":67607},[235],[39,67609],{"className":67610,"style":231},[230],[39,67612,67614,67617,67646],{"className":67613},[74],[39,67615],{"className":67616,"style":65594},[78],[39,67618,67620,67623],{"className":67619},[83],[39,67621,11772],{"className":67622,"style":11793},[83,641],[39,67624,67626],{"className":67625},[192],[39,67627,67629],{"className":67628},[196],[39,67630,67632],{"className":67631},[200],[39,67633,67635],{"className":67634,"style":21897},[204],[39,67636,67637,67640],{"style":208},[39,67638],{"className":67639,"style":213},[212],[39,67641,67643],{"className":67642},[217,218,219,220],[39,67644,21884],{"className":67645},[465,220],[39,67647,67649,67652],{"className":67648},[83],[39,67650,4230],{"className":67651,"style":461},[83,84],[39,67653,67655],{"className":67654},[192],[39,67656,67658,67698],{"className":67657},[196,649],[39,67659,67661,67695],{"className":67660},[200],[39,67662,67664,67684],{"className":67663,"style":21897},[204],[39,67665,67666,67669],{"style":65615},[39,67667],{"className":67668,"style":213},[212],[39,67670,67672],{"className":67671},[217,218,219,220],[39,67673,67675,67678,67681],{"className":67674},[83,220],[39,67676,621],{"className":67677},[83,84,220],[39,67679,788],{"className":67680},[465,220],[39,67682,546],{"className":67683},[83,220],[39,67685,67686,67689],{"style":208},[39,67687],{"className":67688,"style":213},[212],[39,67690,67692],{"className":67691},[217,218,219,220],[39,67693,21884],{"className":67694},[465,220],[39,67696,673],{"className":67697},[672],[39,67699,67701],{"className":67700},[200],[39,67702,67704],{"className":67703,"style":65654},[204],[39,67705],{},"。策略评估式同理。证毕。",[11,67708,67709,67710,67738,67739,67836,67837,67921,67922,68006],{},"这个定理就是反向归纳的数学形式。它说明，在时刻 ",[39,67711,67713,67726],{"className":67712,"translate":43},[42],[39,67714,67716],{"className":67715},[47],[49,67717,67718],{"xmlns":51},[53,67719,67720,67724],{},[56,67721,67722],{},[59,67723,621],{},[63,67725,621],{"encoding":65},[39,67727,67729],{"className":67728,"ariaHidden":70},[69],[39,67730,67732,67735],{"className":67731},[74],[39,67733],{"className":67734,"style":709},[78],[39,67736,621],{"className":67737},[83,84],"，最优决策只需要知道当前状态与下一时刻的最优值函数 ",[39,67740,67742,67767],{"className":67741,"translate":43},[42],[39,67743,67745],{"className":67744},[47],[49,67746,67747],{"xmlns":51},[53,67748,67749,67765],{},[56,67750,67751],{},[9932,67752,67753,67755,67763],{},[59,67754,4230],{},[56,67756,67757,67759,67761],{},[59,67758,621],{},[133,67760,788],{},[539,67762,546],{},[133,67764,21884],{},[63,67766,65584],{"encoding":65},[39,67768,67770],{"className":67769,"ariaHidden":70},[69],[39,67771,67773,67776],{"className":67772},[74],[39,67774],{"className":67775,"style":65594},[78],[39,67777,67779,67782],{"className":67778},[83],[39,67780,4230],{"className":67781,"style":461},[83,84],[39,67783,67785],{"className":67784},[192],[39,67786,67788,67828],{"className":67787},[196,649],[39,67789,67791,67825],{"className":67790},[200],[39,67792,67794,67814],{"className":67793,"style":21897},[204],[39,67795,67796,67799],{"style":65615},[39,67797],{"className":67798,"style":213},[212],[39,67800,67802],{"className":67801},[217,218,219,220],[39,67803,67805,67808,67811],{"className":67804},[83,220],[39,67806,621],{"className":67807},[83,84,220],[39,67809,788],{"className":67810},[465,220],[39,67812,546],{"className":67813},[83,220],[39,67815,67816,67819],{"style":208},[39,67817],{"className":67818,"style":213},[212],[39,67820,67822],{"className":67821},[217,218,219,220],[39,67823,21884],{"className":67824},[465,220],[39,67826,673],{"className":67827},[672],[39,67829,67831],{"className":67830},[200],[39,67832,67834],{"className":67833,"style":65654},[204],[39,67835],{},"。一旦后继状态 ",[39,67838,67840,67863],{"className":67839,"translate":43},[42],[39,67841,67843],{"className":67842},[47],[49,67844,67845],{"xmlns":51},[53,67846,67847,67861],{},[56,67848,67849],{},[613,67850,67851,67853],{},[59,67852,142],{},[56,67854,67855,67857,67859],{},[59,67856,621],{},[133,67858,788],{},[539,67860,546],{},[63,67862,3625],{"encoding":65},[39,67864,67866],{"className":67865,"ariaHidden":70},[69],[39,67867,67869,67872],{"className":67868},[74],[39,67870],{"className":67871,"style":3635},[78],[39,67873,67875,67878],{"className":67874},[83],[39,67876,142],{"className":67877},[83,84],[39,67879,67881],{"className":67880},[192],[39,67882,67884,67913],{"className":67883},[196,649],[39,67885,67887,67910],{"className":67886},[200],[39,67888,67890],{"className":67889,"style":963},[204],[39,67891,67892,67895],{"style":966},[39,67893],{"className":67894,"style":213},[212],[39,67896,67898],{"className":67897},[217,218,219,220],[39,67899,67901,67904,67907],{"className":67900},[83,220],[39,67902,621],{"className":67903},[83,84,220],[39,67905,788],{"className":67906},[465,220],[39,67908,546],{"className":67909},[83,220],[39,67911,673],{"className":67912},[672],[39,67914,67916],{"className":67915},[200],[39,67917,67919],{"className":67918,"style":994},[204],[39,67920],{}," 确定，从 ",[39,67923,67925,67948],{"className":67924,"translate":43},[42],[39,67926,67928],{"className":67927},[47],[49,67929,67930],{"xmlns":51},[53,67931,67932,67946],{},[56,67933,67934],{},[613,67935,67936,67938],{},[59,67937,142],{},[56,67939,67940,67942,67944],{},[59,67941,621],{},[133,67943,788],{},[539,67945,546],{},[63,67947,3625],{"encoding":65},[39,67949,67951],{"className":67950,"ariaHidden":70},[69],[39,67952,67954,67957],{"className":67953},[74],[39,67955],{"className":67956,"style":3635},[78],[39,67958,67960,67963],{"className":67959},[83],[39,67961,142],{"className":67962},[83,84],[39,67964,67966],{"className":67965},[192],[39,67967,67969,67998],{"className":67968},[196,649],[39,67970,67972,67995],{"className":67971},[200],[39,67973,67975],{"className":67974,"style":963},[204],[39,67976,67977,67980],{"style":966},[39,67978],{"className":67979,"style":213},[212],[39,67981,67983],{"className":67982},[217,218,219,220],[39,67984,67986,67989,67992],{"className":67985},[83,220],[39,67987,621],{"className":67988},[83,84,220],[39,67990,788],{"className":67991},[465,220],[39,67993,546],{"className":67994},[83,220],[39,67996,673],{"className":67997},[672],[39,67999,68001],{"className":68000},[200],[39,68002,68004],{"className":68003,"style":994},[204],[39,68005],{}," 开始的剩余决策必须构成该子问题的最优策略。否则，替换该续策略就可以严格改进原策略，与整体最优性矛盾。这就是贝尔曼最优性原理在最简单情形下的完整证明。",[26,68008,68010],{"id":68009},"_62-无限期情形","6.2 无限期情形",[11,68012,68013,68014,68016,68017,68075],{},"无限期问题不能直接使用反向归纳，因为不存在一个最后的时刻。但第 4 节的压缩映射论证给出了同样的结论。",[158,68015,37252],{"href":37251}," 表明，无限期最优贝尔曼方程就是 ",[39,68018,68020,68037],{"className":68019,"translate":43},[42],[39,68021,68023],{"className":68022},[47],[49,68024,68025],{"xmlns":51},[53,68026,68027,68035],{},[56,68028,68029],{},[138,68030,68031,68033],{},[59,68032,11772],{"mathvariant":617},[133,68034,21884],{},[63,68036,21887],{"encoding":65},[39,68038,68040],{"className":68039,"ariaHidden":70},[69],[39,68041,68043,68046],{"className":68042},[74],[39,68044],{"className":68045,"style":21897},[78],[39,68047,68049,68052],{"className":68048},[83],[39,68050,11772],{"className":68051,"style":11793},[83,641],[39,68053,68055],{"className":68054},[192],[39,68056,68058],{"className":68057},[196],[39,68059,68061],{"className":68060},[200],[39,68062,68064],{"className":68063,"style":21897},[204],[39,68065,68066,68069],{"style":208},[39,68067],{"className":68068,"style":213},[212],[39,68070,68072],{"className":68071},[217,218,219,220],[39,68073,21884],{"className":68074},[465,220]," 的不动点方程，而最优值函数正是该方程的唯一有界解。",[11,68077,68078,68079,68294],{},"因此，最优子结构在无限期情形下不是一条需要额外假设的公理，而是折扣、马尔可夫性与可加效用三者共同作用的结果。折扣保证了算子的压缩性，马尔可夫性保证后继状态足以概括历史，可加效用保证了回报可以分解为 ",[39,68080,68082,68123],{"className":68081,"translate":43},[42],[39,68083,68085],{"className":68084},[47],[49,68086,68087],{"xmlns":51},[53,68088,68089,68121],{},[56,68090,68091,68097,68099,68105,68107,68109],{},[613,68092,68093,68095],{},[59,68094,3181],{},[59,68096,621],{},[133,68098,793],{},[613,68100,68101,68103],{},[59,68102,401],{},[59,68104,621],{},[133,68106,788],{},[59,68108,531],{},[613,68110,68111,68113],{},[59,68112,3181],{},[56,68114,68115,68117,68119],{},[59,68116,621],{},[133,68118,788],{},[539,68120,546],{},[63,68122,7463],{"encoding":65},[39,68124,68126,68181,68236],{"className":68125,"ariaHidden":70},[69],[39,68127,68129,68132,68172,68175,68178],{"className":68128},[74],[39,68130],{"className":68131,"style":634},[78],[39,68133,68135,68138],{"className":68134},[83],[39,68136,3181],{"className":68137},[83,84],[39,68139,68141],{"className":68140},[192],[39,68142,68144,68164],{"className":68143},[196,649],[39,68145,68147,68161],{"className":68146},[200],[39,68148,68150],{"className":68149,"style":656},[204],[39,68151,68152,68155],{"style":966},[39,68153],{"className":68154,"style":213},[212],[39,68156,68158],{"className":68157},[217,218,219,220],[39,68159,621],{"className":68160},[83,84,220],[39,68162,673],{"className":68163},[672],[39,68165,68167],{"className":68166},[200],[39,68168,68170],{"className":68169,"style":680},[204],[39,68171],{},[39,68173],{"className":68174,"style":231},[230],[39,68176,793],{"className":68177},[235],[39,68179],{"className":68180,"style":231},[230],[39,68182,68184,68187,68227,68230,68233],{"className":68183},[74],[39,68185],{"className":68186,"style":7528},[78],[39,68188,68190,68193],{"className":68189},[83],[39,68191,401],{"className":68192,"style":438},[83,84],[39,68194,68196],{"className":68195},[192],[39,68197,68199,68219],{"className":68198},[196,649],[39,68200,68202,68216],{"className":68201},[200],[39,68203,68205],{"className":68204,"style":656},[204],[39,68206,68207,68210],{"style":2937},[39,68208],{"className":68209,"style":213},[212],[39,68211,68213],{"className":68212},[217,218,219,220],[39,68214,621],{"className":68215},[83,84,220],[39,68217,673],{"className":68218},[672],[39,68220,68222],{"className":68221},[200],[39,68223,68225],{"className":68224,"style":680},[204],[39,68226],{},[39,68228],{"className":68229,"style":461},[230],[39,68231,788],{"className":68232},[465],[39,68234],{"className":68235,"style":461},[230],[39,68237,68239,68242,68245],{"className":68238},[74],[39,68240],{"className":68241,"style":7584},[78],[39,68243,531],{"className":68244,"style":565},[83,84],[39,68246,68248,68251],{"className":68247},[83],[39,68249,3181],{"className":68250},[83,84],[39,68252,68254],{"className":68253},[192],[39,68255,68257,68286],{"className":68256},[196,649],[39,68258,68260,68283],{"className":68259},[200],[39,68261,68263],{"className":68262,"style":963},[204],[39,68264,68265,68268],{"style":966},[39,68266],{"className":68267,"style":213},[212],[39,68269,68271],{"className":68270},[217,218,219,220],[39,68272,68274,68277,68280],{"className":68273},[83,220],[39,68275,621],{"className":68276},[83,84,220],[39,68278,788],{"className":68279},[465,220],[39,68281,546],{"className":68282},[83,220],[39,68284,673],{"className":68285},[672],[39,68287,68289],{"className":68288},[200],[39,68290,68292],{"className":68291,"style":994},[204],[39,68293],{},"。缺少其中任何一个条件，贝尔曼方程的标准形式都可能失效。",[26,68296,68298],{"id":68297},"_63-成立条件","6.3 成立条件",[11,68300,68301],{},"最优子结构的成立依赖以下三个条件。",[11,68303,68304,68305,68505,68506,68721],{},"第一，效用必须是可加且按常数折扣的。也就是说，回报必须能够写成 ",[39,68306,68308,68352],{"className":68307,"translate":43},[42],[39,68309,68311],{"className":68310},[47],[49,68312,68313],{"xmlns":51},[53,68314,68315,68349],{},[56,68316,68317,68331,68337],{},[9932,68318,68319,68321,68329],{},[133,68320,3191],{},[56,68322,68323,68325,68327],{},[59,68324,3196],{},[133,68326,793],{},[539,68328,541],{},[59,68330,1696],{"mathvariant":145},[138,68332,68333,68335],{},[59,68334,531],{},[59,68336,3196],{},[613,68338,68339,68341],{},[59,68340,401],{},[56,68342,68343,68345,68347],{},[59,68344,621],{},[133,68346,788],{},[59,68348,3196],{},[63,68350,68351],{"encoding":65},"\\sum_{k=0}^{\\infty}\\gamma^k r_{t+k}",[39,68353,68355],{"className":68354,"ariaHidden":70},[69],[39,68356,68358,68361,68424,68427,68456],{"className":68357},[74],[39,68359],{"className":68360,"style":15719},[78],[39,68362,68364,68367],{"className":68363},[1744],[39,68365,3191],{"className":68366,"style":3953},[1744,3339,3952],[39,68368,68370],{"className":68369},[192],[39,68371,68373,68416],{"className":68372},[196,649],[39,68374,68376,68413],{"className":68375},[200],[39,68377,68379,68399],{"className":68378,"style":15738},[204],[39,68380,68381,68384],{"style":3969},[39,68382],{"className":68383,"style":213},[212],[39,68385,68387],{"className":68386},[217,218,219,220],[39,68388,68390,68393,68396],{"className":68389},[83,220],[39,68391,3196],{"className":68392,"style":3321},[83,84,220],[39,68394,793],{"className":68395},[235,220],[39,68397,541],{"className":68398},[83,220],[39,68400,68401,68404],{"style":15761},[39,68402],{"className":68403,"style":213},[212],[39,68405,68407],{"className":68406},[217,218,219,220],[39,68408,68410],{"className":68409},[83,220],[39,68411,1696],{"className":68412},[83,220],[39,68414,673],{"className":68415},[672],[39,68417,68419],{"className":68418},[200],[39,68420,68422],{"className":68421,"style":4025},[204],[39,68423],{},[39,68425],{"className":68426,"style":256},[230],[39,68428,68430,68433],{"className":68429},[83],[39,68431,531],{"className":68432,"style":565},[83,84],[39,68434,68436],{"className":68435},[192],[39,68437,68439],{"className":68438},[196],[39,68440,68442],{"className":68441},[200],[39,68443,68445],{"className":68444,"style":15577},[204],[39,68446,68447,68450],{"style":208},[39,68448],{"className":68449,"style":213},[212],[39,68451,68453],{"className":68452},[217,218,219,220],[39,68454,3196],{"className":68455,"style":3321},[83,84,220],[39,68457,68459,68462],{"className":68458},[83],[39,68460,401],{"className":68461,"style":438},[83,84],[39,68463,68465],{"className":68464},[192],[39,68466,68468,68497],{"className":68467},[196,649],[39,68469,68471,68494],{"className":68470},[200],[39,68472,68474],{"className":68473,"style":3419},[204],[39,68475,68476,68479],{"style":2937},[39,68477],{"className":68478,"style":213},[212],[39,68480,68482],{"className":68481},[217,218,219,220],[39,68483,68485,68488,68491],{"className":68484},[83,220],[39,68486,621],{"className":68487},[83,84,220],[39,68489,788],{"className":68490},[465,220],[39,68492,3196],{"className":68493,"style":3321},[83,84,220],[39,68495,673],{"className":68496},[672],[39,68498,68500],{"className":68499},[200],[39,68501,68503],{"className":68502,"style":994},[204],[39,68504],{},"。这个形式保证了 ",[39,68507,68509,68550],{"className":68508,"translate":43},[42],[39,68510,68512],{"className":68511},[47],[49,68513,68514],{"xmlns":51},[53,68515,68516,68548],{},[56,68517,68518,68524,68526,68532,68534,68536],{},[613,68519,68520,68522],{},[59,68521,3181],{},[59,68523,621],{},[133,68525,793],{},[613,68527,68528,68530],{},[59,68529,401],{},[59,68531,621],{},[133,68533,788],{},[59,68535,531],{},[613,68537,68538,68540],{},[59,68539,3181],{},[56,68541,68542,68544,68546],{},[59,68543,621],{},[133,68545,788],{},[539,68547,546],{},[63,68549,7463],{"encoding":65},[39,68551,68553,68608,68663],{"className":68552,"ariaHidden":70},[69],[39,68554,68556,68559,68599,68602,68605],{"className":68555},[74],[39,68557],{"className":68558,"style":634},[78],[39,68560,68562,68565],{"className":68561},[83],[39,68563,3181],{"className":68564},[83,84],[39,68566,68568],{"className":68567},[192],[39,68569,68571,68591],{"className":68570},[196,649],[39,68572,68574,68588],{"className":68573},[200],[39,68575,68577],{"className":68576,"style":656},[204],[39,68578,68579,68582],{"style":966},[39,68580],{"className":68581,"style":213},[212],[39,68583,68585],{"className":68584},[217,218,219,220],[39,68586,621],{"className":68587},[83,84,220],[39,68589,673],{"className":68590},[672],[39,68592,68594],{"className":68593},[200],[39,68595,68597],{"className":68596,"style":680},[204],[39,68598],{},[39,68600],{"className":68601,"style":231},[230],[39,68603,793],{"className":68604},[235],[39,68606],{"className":68607,"style":231},[230],[39,68609,68611,68614,68654,68657,68660],{"className":68610},[74],[39,68612],{"className":68613,"style":7528},[78],[39,68615,68617,68620],{"className":68616},[83],[39,68618,401],{"className":68619,"style":438},[83,84],[39,68621,68623],{"className":68622},[192],[39,68624,68626,68646],{"className":68625},[196,649],[39,68627,68629,68643],{"className":68628},[200],[39,68630,68632],{"className":68631,"style":656},[204],[39,68633,68634,68637],{"style":2937},[39,68635],{"className":68636,"style":213},[212],[39,68638,68640],{"className":68639},[217,218,219,220],[39,68641,621],{"className":68642},[83,84,220],[39,68644,673],{"className":68645},[672],[39,68647,68649],{"className":68648},[200],[39,68650,68652],{"className":68651,"style":680},[204],[39,68653],{},[39,68655],{"className":68656,"style":461},[230],[39,68658,788],{"className":68659},[465],[39,68661],{"className":68662,"style":461},[230],[39,68664,68666,68669,68672],{"className":68665},[74],[39,68667],{"className":68668,"style":7584},[78],[39,68670,531],{"className":68671,"style":565},[83,84],[39,68673,68675,68678],{"className":68674},[83],[39,68676,3181],{"className":68677},[83,84],[39,68679,68681],{"className":68680},[192],[39,68682,68684,68713],{"className":68683},[196,649],[39,68685,68687,68710],{"className":68686},[200],[39,68688,68690],{"className":68689,"style":963},[204],[39,68691,68692,68695],{"style":966},[39,68693],{"className":68694,"style":213},[212],[39,68696,68698],{"className":68697},[217,218,219,220],[39,68699,68701,68704,68707],{"className":68700},[83,220],[39,68702,621],{"className":68703},[83,84,220],[39,68705,788],{"className":68706},[465,220],[39,68708,546],{"className":68709},[83,220],[39,68711,673],{"className":68712},[672],[39,68714,68716],{"className":68715},[200],[39,68717,68719],{"className":68718,"style":994},[204],[39,68720],{},"。若效用是历史整体的非线性函数，例如某些非可加偏好，那么未来就不能被压缩成一个标量值函数。双曲折扣是一个典型例子。在双曲折扣下，今天的偏好与明天的偏好可能不一致，因而不存在一个与时间无关的递归值函数。",[11,68723,68724,68725,68794],{},"第二，状态必须满足马尔可夫性。若未来的转移依赖于更早的历史，当前状态 ",[39,68726,68728,68745],{"className":68727,"translate":43},[42],[39,68729,68731],{"className":68730},[47],[49,68732,68733],{"xmlns":51},[53,68734,68735,68743],{},[56,68736,68737],{},[613,68738,68739,68741],{},[59,68740,142],{},[59,68742,621],{},[63,68744,3548],{"encoding":65},[39,68746,68748],{"className":68747,"ariaHidden":70},[69],[39,68749,68751,68754],{"className":68750},[74],[39,68752],{"className":68753,"style":2916},[78],[39,68755,68757,68760],{"className":68756},[83],[39,68758,142],{"className":68759},[83,84],[39,68761,68763],{"className":68762},[192],[39,68764,68766,68786],{"className":68765},[196,649],[39,68767,68769,68783],{"className":68768},[200],[39,68770,68772],{"className":68771,"style":656},[204],[39,68773,68774,68777],{"style":966},[39,68775],{"className":68776,"style":213},[212],[39,68778,68780],{"className":68779},[217,218,219,220],[39,68781,621],{"className":68782},[83,84,220],[39,68784,673],{"className":68785},[672],[39,68787,68789],{"className":68788},[200],[39,68790,68792],{"className":68791,"style":680},[204],[39,68793],{}," 就不是充分的统计量。此时可以把状态扩充为包含必要历史信息的新状态，例如部分可观测问题中的信念状态。在扩充后的状态空间上，贝尔曼方程仍然可以保持相同的形式。",[11,68796,68797,68798,68848,68849,68899],{},"第三，回报必须有界，或者折扣必须足够强。有限期问题即使 ",[39,68799,68801,68818],{"className":68800,"translate":43},[42],[39,68802,68804],{"className":68803},[47],[49,68805,68806],{"xmlns":51},[53,68807,68808,68816],{},[56,68809,68810,68812,68814],{},[59,68811,531],{},[133,68813,793],{},[539,68815,546],{},[63,68817,28990],{"encoding":65},[39,68819,68821,68839],{"className":68820,"ariaHidden":70},[69],[39,68822,68824,68827,68830,68833,68836],{"className":68823},[74],[39,68825],{"className":68826,"style":2654},[78],[39,68828,531],{"className":68829,"style":565},[83,84],[39,68831],{"className":68832,"style":231},[230],[39,68834,793],{"className":68835},[235],[39,68837],{"className":68838,"style":231},[230],[39,68840,68842,68845],{"className":68841},[74],[39,68843],{"className":68844,"style":2473},[78],[39,68846,546],{"className":68847},[83]," 也可以有有界回报，因为求和的项数是有限的。无限期问题则需要 ",[39,68850,68852,68869],{"className":68851,"translate":43},[42],[39,68853,68855],{"className":68854},[47],[49,68856,68857],{"xmlns":51},[53,68858,68859,68867],{},[56,68860,68861,68863,68865],{},[59,68862,531],{},[133,68864,1693],{},[539,68866,546],{},[63,68868,17296],{"encoding":65},[39,68870,68872,68890],{"className":68871,"ariaHidden":70},[69],[39,68873,68875,68878,68881,68884,68887],{"className":68874},[74],[39,68876],{"className":68877,"style":561},[78],[39,68879,531],{"className":68880,"style":565},[83,84],[39,68882],{"className":68883,"style":231},[230],[39,68885,1693],{"className":68886},[235],[39,68888],{"className":68889,"style":231},[230],[39,68891,68893,68896],{"className":68892},[74],[39,68894],{"className":68895,"style":2473},[78],[39,68897,546],{"className":68898},[83],"，或者采用平均奖励、吸收态等替代准则。否则，值函数可能发散，或者贝尔曼方程的解不唯一。",[11,68901,68902],{},"这三个条件共同说明，最优子结构是一个有明确适用范围的定理，而不是一条无条件成立的哲学原则。",[21,68904,68906],{"id":68905},"七边界与常见误区","七、边界与常见误区",[26,68908,68910],{"id":68909},"_71-连续状态空间","7.1 连续状态空间",[11,68912,68913,68914,68957],{},"在标准的可测性假设下，连续状态空间中的 ",[39,68915,68917,68936],{"className":68916,"translate":43},[42],[39,68918,68920],{"className":68919},[47],[49,68921,68922],{"xmlns":51},[53,68923,68924,68934],{},[56,68925,68926,68928,68930,68932],{},[59,68927,19772],{},[133,68929,136],{"stretchy":135},[59,68931,61],{},[133,68933,161],{"stretchy":135},[63,68935,21380],{"encoding":65},[39,68937,68939],{"className":68938,"ariaHidden":70},[69],[39,68940,68942,68945,68948,68951,68954],{"className":68941},[74],[39,68943],{"className":68944,"style":245},[78],[39,68946,19772],{"className":68947,"style":19812},[83,84],[39,68949,136],{"className":68950},[182],[39,68952,61],{"className":68953,"style":85},[83,84],[39,68955,161],{"className":68956},[263]," 仍然是 Banach 空间，压缩映射的论证也仍然成立。因此，贝尔曼期望方程的唯一有界解依然存在，贝尔曼最优算子也依然有唯一有界不动点，值迭代的几何收敛性同样成立。在适当的可测选择条件下，该不动点等于最优值函数。",[11,68959,68960,68961,69019],{},"变化主要发生在 ",[39,68962,68964,68981],{"className":68963,"translate":43},[42],[39,68965,68967],{"className":68966},[47],[49,68968,68969],{"xmlns":51},[53,68970,68971,68979],{},[56,68972,68973],{},[138,68974,68975,68977],{},[59,68976,11772],{"mathvariant":617},[133,68978,21884],{},[63,68980,21887],{"encoding":65},[39,68982,68984],{"className":68983,"ariaHidden":70},[69],[39,68985,68987,68990],{"className":68986},[74],[39,68988],{"className":68989,"style":21897},[78],[39,68991,68993,68996],{"className":68992},[83],[39,68994,11772],{"className":68995,"style":11793},[83,641],[39,68997,68999],{"className":68998},[192],[39,69000,69002],{"className":69001},[196],[39,69003,69005],{"className":69004},[200],[39,69006,69008],{"className":69007,"style":21897},[204],[39,69009,69010,69013],{"style":208},[39,69011],{"className":69012,"style":213},[212],[39,69014,69016],{"className":69015},[217,218,219,220],[39,69017,21884],{"className":69018},[465,220]," 的定义上。此时最大值未必取到，应将其写成",[39,69021,69023],{"className":69022,"translate":43},[746],[39,69024,69026,69177],{"className":69025,"translate":43},[42],[39,69027,69029],{"className":69028},[47],[49,69030,69031],{"xmlns":51,"display":755},[53,69032,69033,69174],{},[759,69034,69035,69112],{"rowspacing":761,"columnalign":762,"columnspacing":146},[764,69036,69037,69061],{},[767,69038,69039],{},[770,69040,69041],{"scriptlevel":541,"displaystyle":70},[56,69042,69043,69045,69051,69053,69055,69057,69059],{},[133,69044,136],{"stretchy":135},[138,69046,69047,69049],{},[59,69048,11772],{"mathvariant":617},[133,69050,21884],{},[59,69052,4230],{},[133,69054,161],{"stretchy":135},[133,69056,136],{"stretchy":135},[59,69058,142],{},[133,69060,161],{"stretchy":135},[767,69062,69063],{},[770,69064,69065],{"scriptlevel":541,"displaystyle":70},[56,69066,69067,69069,69071,69081,69083,69090,69092,69094,69096,69098,69100,69102,69104,69110],{},[56,69068],{},[133,69070,793],{},[6556,69072,69073,69079],{},[56,69074,69075,69077],{},[59,69076,21465],{},[133,69078,1690],{},[59,69080,158],{},[133,69082,12951],{"fence":135,"stretchy":70,"minsize":63111,"maxsize":63111},[613,69084,69085,69088],{},[133,69086,69087],{},"∫",[59,69089,61],{},[133,69091,537],{"fence":135,"stretchy":70,"minsize":65017,"maxsize":65017},[59,69093,401],{},[133,69095,136],{"stretchy":135},[59,69097,142],{},[133,69099,155],{"separator":70},[59,69101,158],{},[133,69103,155],{"separator":70},[138,69105,69106,69108],{},[59,69107,142],{},[133,69109,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,69111,161],{"stretchy":135},[764,69113,69114,69120],{},[767,69115,69116],{},[770,69117,69118],{"scriptlevel":541,"displaystyle":70},[56,69119],{},[767,69121,69122],{},[770,69123,69124],{"scriptlevel":541,"displaystyle":70},[56,69125,69126,69128,69130,69132,69134,69136,69138,69144,69146,69148,69150,69152,69154,69160,69162,69164,69166,69168,69170,69172],{},[56,69127],{},[230,69129],{"width":10251},[133,69131,788],{},[59,69133,531],{},[59,69135,4230],{},[133,69137,136],{"stretchy":135},[138,69139,69140,69142],{},[59,69141,142],{},[133,69143,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,69145,161],{"stretchy":135},[133,69147,4280],{"fence":135,"stretchy":70,"minsize":65017,"maxsize":65017},[59,69149,131],{},[133,69151,136],{"stretchy":135},[59,69153,29425],{"mathvariant":145},[138,69155,69156,69158],{},[59,69157,142],{},[133,69159,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,69161,150],{},[59,69163,142],{},[133,69165,155],{"separator":70},[59,69167,158],{},[133,69169,161],{"stretchy":135},[133,69171,12965],{"fence":135,"stretchy":70,"minsize":63111,"maxsize":63111},[59,69173,2903],{"mathvariant":145},[63,69175,69176],{"encoding":65},"\\begin{aligned}\n(\\mathcal{T}^*V)(s)\n&=\\sup_{a}\\Bigg\\{\\int_S \\Big[r(s,a,s')\\\\\n&\\qquad+\\gamma V(s')\\Big]P(\\mathrm{d}s'\\mid s,a)\\Bigg\\}.\n\\end{aligned}",[39,69178,69180],{"className":69179,"ariaHidden":70},[69],[39,69181,69183,69187],{"className":69182},[74],[39,69184],{"className":69185,"style":69186},[78],"height:6.3001em;vertical-align:-2.9em;",[39,69188,69190],{"className":69189},[83],[39,69191,69193,69283],{"className":69192},[759],[39,69194,69196],{"className":69195},[918],[39,69197,69199,69274],{"className":69198},[196,649],[39,69200,69202,69271],{"className":69201},[200],[39,69203,69206,69263],{"className":69204,"style":69205},[204],"height:3.4em;",[39,69207,69209,69213],{"style":69208},"top:-5.4em;",[39,69210],{"className":69211,"style":69212},[212],"height:3.75em;",[39,69214,69216,69219,69248,69251,69254,69257,69260],{"className":69215},[83],[39,69217,136],{"className":69218},[182],[39,69220,69222,69225],{"className":69221},[83],[39,69223,11772],{"className":69224,"style":11793},[83,641],[39,69226,69228],{"className":69227},[192],[39,69229,69231],{"className":69230},[196],[39,69232,69234],{"className":69233},[200],[39,69235,69237],{"className":69236,"style":22242},[204],[39,69238,69239,69242],{"style":1027},[39,69240],{"className":69241,"style":213},[212],[39,69243,69245],{"className":69244},[217,218,219,220],[39,69246,21884],{"className":69247},[465,220],[39,69249,4230],{"className":69250,"style":461},[83,84],[39,69252,161],{"className":69253},[263],[39,69255,136],{"className":69256},[182],[39,69258,142],{"className":69259},[83,84],[39,69261,161],{"className":69262},[263],[39,69264,69265,69268],{"style":5834},[39,69266],{"className":69267,"style":69212},[212],[39,69269],{"className":69270},[83],[39,69272,673],{"className":69273},[672],[39,69275,69277],{"className":69276},[200],[39,69278,69281],{"className":69279,"style":69280},[204],"height:2.9em;",[39,69282],{},[39,69284,69286],{"className":69285},[1223],[39,69287,69289,69636],{"className":69288},[196,649],[39,69290,69292,69633],{"className":69291},[200],[39,69293,69295,69486],{"className":69294,"style":69205},[204],[39,69296,69297,69300],{"style":69208},[39,69298],{"className":69299,"style":69212},[212],[39,69301,69303,69306,69309,69312,69315,69362,69365,69371,69374,69418,69421,69427,69430,69433,69436,69439,69442,69445,69448,69451,69483],{"className":69302},[83],[39,69304],{"className":69305},[83],[39,69307],{"className":69308,"style":231},[230],[39,69310,793],{"className":69311},[235],[39,69313],{"className":69314,"style":231},[230],[39,69316,69318],{"className":69317},[1744,3294],[39,69319,69321,69354],{"className":69320},[196,649],[39,69322,69324,69351],{"className":69323},[200],[39,69325,69327,69341],{"className":69326,"style":290},[204],[39,69328,69329,69332],{"style":36729},[39,69330],{"className":69331,"style":935},[212],[39,69333,69335],{"className":69334},[217,218,219,220],[39,69336,69338],{"className":69337},[83,220],[39,69339,158],{"className":69340},[83,84,220],[39,69342,69343,69346],{"style":21599},[39,69344],{"className":69345,"style":935},[212],[39,69347,69348],{},[39,69349,21465],{"className":69350},[1744],[39,69352,673],{"className":69353},[672],[39,69355,69357],{"className":69356},[200],[39,69358,69360],{"className":69359,"style":36761},[204],[39,69361],{},[39,69363],{"className":69364,"style":256},[230],[39,69366,69368],{"className":69367},[83],[39,69369,12951],{"className":69370},[18520,63488],[39,69372],{"className":69373,"style":256},[230],[39,69375,69377,69381],{"className":69376},[1744],[39,69378,69087],{"className":69379,"style":69380},[1744,3339,3340],"margin-right:0.4445em;position:relative;top:-0.0011em;",[39,69382,69384],{"className":69383},[192],[39,69385,69387,69409],{"className":69386},[196,649],[39,69388,69390,69406],{"className":69389},[200],[39,69391,69394],{"className":69392,"style":69393},[204],"height:-0.4336em;",[39,69395,69397,69400],{"style":69396},"top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;",[39,69398],{"className":69399,"style":213},[212],[39,69401,69403],{"className":69402},[217,218,219,220],[39,69404,61],{"className":69405,"style":85},[83,84,220],[39,69407,673],{"className":69408},[672],[39,69410,69412],{"className":69411},[200],[39,69413,69416],{"className":69414,"style":69415},[204],"height:0.9119em;",[39,69417],{},[39,69419],{"className":69420,"style":256},[230],[39,69422,69424],{"className":69423},[83],[39,69425,537],{"className":69426},[18520,33204],[39,69428,401],{"className":69429,"style":438},[83,84],[39,69431,136],{"className":69432},[182],[39,69434,142],{"className":69435},[83,84],[39,69437,155],{"className":69438},[252],[39,69440],{"className":69441,"style":256},[230],[39,69443,158],{"className":69444},[83,84],[39,69446,155],{"className":69447},[252],[39,69449],{"className":69450,"style":256},[230],[39,69452,69454,69457],{"className":69453},[83],[39,69455,142],{"className":69456},[83,84],[39,69458,69460],{"className":69459},[192],[39,69461,69463],{"className":69462},[196],[39,69464,69466],{"className":69465},[200],[39,69467,69469],{"className":69468,"style":1024},[204],[39,69470,69471,69474],{"style":1027},[39,69472],{"className":69473,"style":213},[212],[39,69475,69477],{"className":69476},[217,218,219,220],[39,69478,69480],{"className":69479},[83,220],[39,69481,147],{"className":69482},[83,220],[39,69484,161],{"className":69485},[263],[39,69487,69488,69491],{"style":5834},[39,69489],{"className":69490,"style":69212},[212],[39,69492,69494,69497,69500,69503,69506,69509,69512,69515,69518,69550,69553,69559,69562,69565,69568,69600,69603,69606,69609,69612,69615,69618,69621,69624,69630],{"className":69493},[83],[39,69495],{"className":69496},[83],[39,69498],{"className":69499,"style":10787},[230],[39,69501],{"className":69502,"style":461},[230],[39,69504,788],{"className":69505},[465],[39,69507],{"className":69508,"style":461},[230],[39,69510,531],{"className":69511,"style":565},[83,84],[39,69513,4230],{"className":69514,"style":461},[83,84],[39,69516,136],{"className":69517},[182],[39,69519,69521,69524],{"className":69520},[83],[39,69522,142],{"className":69523},[83,84],[39,69525,69527],{"className":69526},[192],[39,69528,69530],{"className":69529},[196],[39,69531,69533],{"className":69532},[200],[39,69534,69536],{"className":69535,"style":1024},[204],[39,69537,69538,69541],{"style":1027},[39,69539],{"className":69540,"style":213},[212],[39,69542,69544],{"className":69543},[217,218,219,220],[39,69545,69547],{"className":69546},[83,220],[39,69548,147],{"className":69549},[83,220],[39,69551,161],{"className":69552},[263],[39,69554,69556],{"className":69555},[83],[39,69557,4280],{"className":69558},[18520,33204],[39,69560,131],{"className":69561,"style":178},[83,84],[39,69563,136],{"className":69564},[182],[39,69566,29425],{"className":69567},[83,35077],[39,69569,69571,69574],{"className":69570},[83],[39,69572,142],{"className":69573},[83,84],[39,69575,69577],{"className":69576},[192],[39,69578,69580],{"className":69579},[196],[39,69581,69583],{"className":69582},[200],[39,69584,69586],{"className":69585,"style":1024},[204],[39,69587,69588,69591],{"style":1027},[39,69589],{"className":69590,"style":213},[212],[39,69592,69594],{"className":69593},[217,218,219,220],[39,69595,69597],{"className":69596},[83,220],[39,69598,147],{"className":69599},[83,220],[39,69601],{"className":69602,"style":231},[230],[39,69604,150],{"className":69605},[235],[39,69607],{"className":69608,"style":231},[230],[39,69610,142],{"className":69611},[83,84],[39,69613,155],{"className":69614},[252],[39,69616],{"className":69617,"style":256},[230],[39,69619,158],{"className":69620},[83,84],[39,69622,161],{"className":69623},[263],[39,69625,69627],{"className":69626},[83],[39,69628,12965],{"className":69629},[18520,63488],[39,69631,2903],{"className":69632},[83],[39,69634,673],{"className":69635},[672],[39,69637,69639],{"className":69638},[200],[39,69640,69642],{"className":69641,"style":69280},[204],[39,69643],{},[11,69645,69646,69647,69705,69706,69734],{},"压缩性证明中的「最大值保持逐点序」可以替换为「上确界保持逐点序」，因此 ",[39,69648,69650,69667],{"className":69649,"translate":43},[42],[39,69651,69653],{"className":69652},[47],[49,69654,69655],{"xmlns":51},[53,69656,69657,69665],{},[56,69658,69659],{},[138,69660,69661,69663],{},[59,69662,11772],{"mathvariant":617},[133,69664,21884],{},[63,69666,21887],{"encoding":65},[39,69668,69670],{"className":69669,"ariaHidden":70},[69],[39,69671,69673,69676],{"className":69672},[74],[39,69674],{"className":69675,"style":21897},[78],[39,69677,69679,69682],{"className":69678},[83],[39,69680,11772],{"className":69681,"style":11793},[83,641],[39,69683,69685],{"className":69684},[192],[39,69686,69688],{"className":69687},[196],[39,69689,69691],{"className":69690},[200],[39,69692,69694],{"className":69693,"style":21897},[204],[39,69695,69696,69699],{"style":208},[39,69697],{"className":69698,"style":213},[212],[39,69700,69702],{"className":69701},[217,218,219,220],[39,69703,21884],{"className":69704},[465,220]," 仍然是 ",[39,69707,69709,69722],{"className":69708,"translate":43},[42],[39,69710,69712],{"className":69711},[47],[49,69713,69714],{"xmlns":51},[53,69715,69716,69720],{},[56,69717,69718],{},[59,69719,531],{},[63,69721,26586],{"encoding":65},[39,69723,69725],{"className":69724,"ariaHidden":70},[69],[39,69726,69728,69731],{"className":69727},[74],[39,69729],{"className":69730,"style":2654},[78],[39,69732,531],{"className":69733,"style":565},[83,84],"-压缩映射。唯一不动点仍然存在。",[11,69736,69737,69738,69796,69797,69895],{},"但是，最优策略的存在性需要额外条件。即使 ",[39,69739,69741,69758],{"className":69740,"translate":43},[42],[39,69742,69744],{"className":69743},[47],[49,69745,69746],{"xmlns":51},[53,69747,69748,69756],{},[56,69749,69750],{},[138,69751,69752,69754],{},[59,69753,11772],{"mathvariant":617},[133,69755,21884],{},[63,69757,21887],{"encoding":65},[39,69759,69761],{"className":69760,"ariaHidden":70},[69],[39,69762,69764,69767],{"className":69763},[74],[39,69765],{"className":69766,"style":21897},[78],[39,69768,69770,69773],{"className":69769},[83],[39,69771,11772],{"className":69772,"style":11793},[83,641],[39,69774,69776],{"className":69775},[192],[39,69777,69779],{"className":69778},[196],[39,69780,69782],{"className":69781},[200],[39,69783,69785],{"className":69784,"style":21897},[204],[39,69786,69787,69790],{"style":208},[39,69788],{"className":69789,"style":213},[212],[39,69791,69793],{"className":69792},[217,218,219,220],[39,69794,21884],{"className":69795},[465,220]," 有唯一不动点，也不一定存在一个可测的映射 ",[39,69798,69800,69829],{"className":69799,"translate":43},[42],[39,69801,69803],{"className":69802},[47],[49,69804,69805],{"xmlns":51},[53,69806,69807,69826],{},[56,69808,69809,69811,69814,69820,69822,69824],{},[59,69810,142],{},[133,69812,69813],{},"↦",[138,69815,69816,69818],{},[59,69817,2098],{},[133,69819,21884],{},[133,69821,136],{"stretchy":135},[59,69823,142],{},[133,69825,161],{"stretchy":135},[63,69827,69828],{"encoding":65},"s\\mapsto\\pi^*(s)",[39,69830,69832,69851],{"className":69831,"ariaHidden":70},[69],[39,69833,69835,69839,69842,69845,69848],{"className":69834},[74],[39,69836],{"className":69837,"style":69838},[78],"height:0.522em;vertical-align:-0.011em;",[39,69840,142],{"className":69841},[83,84],[39,69843],{"className":69844,"style":231},[230],[39,69846,69813],{"className":69847},[235],[39,69849],{"className":69850,"style":231},[230],[39,69852,69854,69857,69886,69889,69892],{"className":69853},[74],[39,69855],{"className":69856,"style":245},[78],[39,69858,69860,69863],{"className":69859},[83],[39,69861,2098],{"className":69862,"style":2114},[83,84],[39,69864,69866],{"className":69865},[192],[39,69867,69869],{"className":69868},[196],[39,69870,69872],{"className":69871},[200],[39,69873,69875],{"className":69874,"style":21897},[204],[39,69876,69877,69880],{"style":208},[39,69878],{"className":69879,"style":213},[212],[39,69881,69883],{"className":69882},[217,218,219,220],[39,69884,21884],{"className":69885},[465,220],[39,69887,136],{"className":69888},[182],[39,69890,142],{"className":69891},[83,84],[39,69893,161],{"className":69894},[263]," 在每个状态都取到上确界。通常需要动作集紧、奖励与转移核连续，或者使用可测选择定理。本文的有限状态与有限动作假设避开了这些技术问题。这也是为什么在一般情形下，贝尔曼最优方程常写成上确界形式，而不是最大值形式。",[26,69897,69899],{"id":69898},"_72-平均奖励与无折扣","7.2 平均奖励与无折扣",[11,69901,12932,69902,69952,69953,69983,69984,70013],{},[39,69903,69905,69922],{"className":69904,"translate":43},[42],[39,69906,69908],{"className":69907},[47],[49,69909,69910],{"xmlns":51},[53,69911,69912,69920],{},[56,69913,69914,69916,69918],{},[59,69915,531],{},[133,69917,793],{},[539,69919,546],{},[63,69921,28990],{"encoding":65},[39,69923,69925,69943],{"className":69924,"ariaHidden":70},[69],[39,69926,69928,69931,69934,69937,69940],{"className":69927},[74],[39,69929],{"className":69930,"style":2654},[78],[39,69932,531],{"className":69933,"style":565},[83,84],[39,69935],{"className":69936,"style":231},[230],[39,69938,793],{"className":69939},[235],[39,69941],{"className":69942,"style":231},[230],[39,69944,69946,69949],{"className":69945},[74],[39,69947],{"className":69948,"style":2473},[78],[39,69950,546],{"className":69951},[83]," 且时间范围为无限时，折扣回报可能发散，贝尔曼方程的标准形式不再适用。此时常用的替代准则是平均奖励准则。它引入增益 ",[39,69954,69956,69971],{"className":69955,"translate":43},[42],[39,69957,69959],{"className":69958},[47],[49,69960,69961],{"xmlns":51},[53,69962,69963,69968],{},[56,69964,69965],{},[59,69966,69967],{},"ρ",[63,69969,69970],{"encoding":65},"\\rho",[39,69972,69974],{"className":69973,"ariaHidden":70},[69],[39,69975,69977,69980],{"className":69976},[74],[39,69978],{"className":69979,"style":2654},[78],[39,69981,69967],{"className":69982},[83,84]," 与偏差函数 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是期望即时奖励。这个方程与折扣贝尔曼方程在结构上相似，但数学性质不同。平均奖励问题通常只保证偏差函数在相差一个常数的意义下唯一，因为若 ",[39,70964,70966,70979],{"className":70965,"translate":43},[42],[39,70967,70969],{"className":70968},[47],[49,70970,70971],{"xmlns":51},[53,70972,70973,70977],{},[56,70974,70975],{},[59,70976,69998],{},[63,70978,69998],{"encoding":65},[39,70980,70982],{"className":70981,"ariaHidden":70},[69],[39,70983,70985,70988],{"className":70984},[74],[39,70986],{"className":70987,"style":28824},[78],[39,70989,69998],{"className":70990},[83,84]," 是解，则 ",[39,70993,70995,71013],{"className":70994,"translate":43},[42],[39,70996,70998],{"className":70997},[47],[49,70999,71000],{"xmlns":51},[53,71001,71002,71010],{},[56,71003,71004,71006,71008],{},[59,71005,69998],{},[133,71007,788],{},[59,71009,24077],{},[63,71011,71012],{"encoding":65},"h+c",[39,71014,71016,71035],{"className":71015,"ariaHidden":70},[69],[39,71017,71019,71023,71026,71029,71032],{"className":71018},[74],[39,71020],{"className":71021,"style":71022},[78],"height:0.7778em;vertical-align:-0.0833em;",[39,71024,69998],{"className":71025},[83,84],[39,71027],{"className":71028,"style":461},[230],[39,71030,788],{"className":71031},[465],[39,71033],{"className":71034,"style":461},[230],[39,71036,71038,71041],{"className":71037},[74],[39,71039],{"className":71040,"style":290},[78],[39,71042,24077],{"className":71043},[83,84]," 也是解。折扣因子 ",[39,71046,71048,71065],{"className":71047,"translate":43},[42],[39,71049,71051],{"className":71050},[47],[49,71052,71053],{"xmlns":51},[53,71054,71055,71063],{},[56,71056,71057,71059,71061],{},[59,71058,531],{},[133,71060,1693],{},[539,71062,546],{},[63,71064,17296],{"encoding":65},[39,71066,71068,71086],{"className":71067,"ariaHidden":70},[69],[39,71069,71071,71074,71077,71080,71083],{"className":71070},[74],[39,71072],{"className":71073,"style":561},[78],[39,71075,531],{"className":71076,"style":565},[83,84],[39,71078],{"className":71079,"style":231},[230],[39,71081,1693],{"className":71082},[235],[39,71084],{"className":71085,"style":231},[230],[39,71087,71089,71092],{"className":71088},[74],[39,71090],{"className":71091,"style":2473},[78],[39,71093,546],{"className":71094},[83]," 消除了这种平移自由度，因此折扣情形的不动点是真正唯一的。",[11,71097,71098,71099,71149,71150,71200],{},"有限期问题是一个例外。即使 ",[39,71100,71102,71119],{"className":71101,"translate":43},[42],[39,71103,71105],{"className":71104},[47],[49,71106,71107],{"xmlns":51},[53,71108,71109,71117],{},[56,71110,71111,71113,71115],{},[59,71112,531],{},[133,71114,793],{},[539,71116,546],{},[63,71118,28990],{"encoding":65},[39,71120,71122,71140],{"className":71121,"ariaHidden":70},[69],[39,71123,71125,71128,71131,71134,71137],{"className":71124},[74],[39,71126],{"className":71127,"style":2654},[78],[39,71129,531],{"className":71130,"style":565},[83,84],[39,71132],{"className":71133,"style":231},[230],[39,71135,793],{"className":71136},[235],[39,71138],{"className":71139,"style":231},[230],[39,71141,71143,71146],{"className":71142},[74],[39,71144],{"className":71145,"style":2473},[78],[39,71147,546],{"className":71148},[83],"，只要时间范围有限，回报就是有限项之和，值函数仍然有界，反向归纳仍然成立。因此，",[39,71151,71153,71170],{"className":71152,"translate":43},[42],[39,71154,71156],{"className":71155},[47],[49,71157,71158],{"xmlns":51},[53,71159,71160,71168],{},[56,71161,71162,71164,71166],{},[59,71163,531],{},[133,71165,1693],{},[539,71167,546],{},[63,71169,17296],{"encoding":65},[39,71171,71173,71191],{"className":71172,"ariaHidden":70},[69],[39,71174,71176,71179,71182,71185,71188],{"className":71175},[74],[39,71177],{"className":71178,"style":561},[78],[39,71180,531],{"className":71181,"style":565},[83,84],[39,71183],{"className":71184,"style":231},[230],[39,71186,1693],{"className":71187},[235],[39,71189],{"className":71190,"style":231},[230],[39,71192,71194,71197],{"className":71193},[74],[39,71195],{"className":71196,"style":2473},[78],[39,71198,546],{"className":71199},[83]," 的必要性主要针对无限期问题。",[26,71202,71204],{"id":71203},"_73-注意要点","7.3 注意要点",[11,71206,71207,71210,71211,71380],{},[1489,71208,71209],{},"把贝尔曼方程写成没有期望的形式。"," 在确定性转移的情形下，",[39,71212,71214,71264],{"className":71213,"translate":43},[42],[39,71215,71217],{"className":71216},[47],[49,71218,71219],{"xmlns":51},[53,71220,71221,71261],{},[56,71222,71223,71225,71227,71229,71231,71233,71235,71237,71239,71241,71243,71245,71247,71249,71251,71253,71259],{},[59,71224,4230],{},[133,71226,136],{"stretchy":135},[59,71228,142],{},[133,71230,161],{"stretchy":135},[133,71232,793],{},[59,71234,401],{},[133,71236,136],{"stretchy":135},[59,71238,142],{},[133,71240,155],{"separator":70},[59,71242,158],{},[133,71244,161],{"stretchy":135},[133,71246,788],{},[59,71248,531],{},[59,71250,4230],{},[133,71252,136],{"stretchy":135},[138,71254,71255,71257],{},[59,71256,142],{},[133,71258,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,71260,161],{"stretchy":135},[63,71262,71263],{"encoding":65},"V(s)=r(s,a)+\\gamma V(s')",[39,71265,71267,71294,71330],{"className":71266,"ariaHidden":70},[69],[39,71268,71270,71273,71276,71279,71282,71285,71288,71291],{"className":71269},[74],[39,71271],{"className":71272,"style":245},[78],[39,71274,4230],{"className":71275,"style":461},[83,84],[39,71277,136],{"className":71278},[182],[39,71280,142],{"className":71281},[83,84],[39,71283,161],{"className":71284},[263],[39,71286],{"className":71287,"style":231},[230],[39,71289,793],{"className":71290},[235],[39,71292],{"className":71293,"style":231},[230],[39,71295,71297,71300,71303,71306,71309,71312,71315,71318,71321,71324,71327],{"className":71296},[74],[39,71298],{"className":71299,"style":245},[78],[39,71301,401],{"className":71302,"style":438},[83,84],[39,71304,136],{"className":71305},[182],[39,71307,142],{"className":71308},[83,84],[39,71310,155],{"className":71311},[252],[39,71313],{"className":71314,"style":256},[230],[39,71316,158],{"className":71317},[83,84],[39,71319,161],{"className":71320},[263],[39,71322],{"className":71323,"style":461},[230],[39,71325,788],{"className":71326},[465],[39,71328],{"className":71329,"style":461},[230],[39,71331,71333,71336,71339,71342,71345,71377],{"className":71332},[74],[39,71334],{"className":71335,"style":174},[78],[39,71337,531],{"className":71338,"style":565},[83,84],[39,71340,4230],{"className":71341,"style":461},[83,84],[39,71343,136],{"className":71344},[182],[39,71346,71348,71351],{"className":71347},[83],[39,71349,142],{"className":71350},[83,84],[39,71352,71354],{"className":71353},[192],[39,71355,71357],{"className":71356},[196],[39,71358,71360],{"className":71359},[200],[39,71361,71363],{"className":71362,"style":205},[204],[39,71364,71365,71368],{"style":208},[39,71366],{"className":71367,"style":213},[212],[39,71369,71371],{"className":71370},[217,218,219,220],[39,71372,71374],{"className":71373},[83,220],[39,71375,147],{"className":71376},[83,220],[39,71378,161],{"className":71379},[263]," 可以成立。但在随机转移下，必须对下一状态取期望。遗漏期望会把随机问题误写成确定性问题。",[11,71382,71383,71386,71387,71468,71469,71550],{},[1489,71384,71385],{},"把最大值放进期望内部。"," 最优方程是 ",[39,71388,71390,71413],{"className":71389,"translate":43},[42],[39,71391,71393],{"className":71392},[47],[49,71394,71395],{"xmlns":51},[53,71396,71397,71411],{},[56,71398,71399,71409],{},[613,71400,71401,71407],{},[56,71402,71403,71405],{},[59,71404,1687],{},[133,71406,1690],{},[59,71408,158],{},[59,71410,4253],{"mathvariant":421},[63,71412,45377],{"encoding":65},[39,71414,71416],{"className":71415,"ariaHidden":70},[69],[39,71417,71419,71422,71462,71465],{"className":71418},[74],[39,71420],{"className":71421,"style":45387},[78],[39,71423,71425,71428],{"className":71424},[1744],[39,71426,1687],{"className":71427},[1744],[39,71429,71431],{"className":71430},[192],[39,71432,71434,71454],{"className":71433},[196,649],[39,71435,71437,71451],{"className":71436},[200],[39,71438,71440],{"className":71439,"style":1728},[204],[39,71441,71442,71445],{"style":44021},[39,71443],{"className":71444,"style":213},[212],[39,71446,71448],{"className":71447},[217,218,219,220],[39,71449,158],{"className":71450},[83,84,220],[39,71452,673],{"className":71453},[672],[39,71455,71457],{"className":71456},[200],[39,71458,71460],{"className":71459,"style":680},[204],[39,71461],{},[39,71463],{"className":71464,"style":256},[230],[39,71466,4253],{"className":71467},[83,515],"，不是 ",[39,71470,71472,71495],{"className":71471,"translate":43},[42],[39,71473,71475],{"className":71474},[47],[49,71476,71477],{"xmlns":51},[53,71478,71479,71493],{},[56,71480,71481,71483],{},[59,71482,4253],{"mathvariant":421},[613,71484,71485,71491],{},[56,71486,71487,71489],{},[59,71488,1687],{},[133,71490,1690],{},[59,71492,158],{},[63,71494,45461],{"encoding":65},[39,71496,71498],{"className":71497,"ariaHidden":70},[69],[39,71499,71501,71504,71507,71510],{"className":71500},[74],[39,71502],{"className":71503,"style":45387},[78],[39,71505,4253],{"className":71506},[83,515],[39,71508],{"className":71509,"style":256},[230],[39,71511,71513,71516],{"className":71512},[1744],[39,71514,1687],{"className":71515},[1744],[39,71517,71519],{"className":71518},[192],[39,71520,71522,71542],{"className":71521},[196,649],[39,71523,71525,71539],{"className":71524},[200],[39,71526,71528],{"className":71527,"style":1728},[204],[39,71529,71530,71533],{"style":44021},[39,71531],{"className":71532,"style":213},[212],[39,71534,71536],{"className":71535},[217,218,219,220],[39,71537,158],{"className":71538},[83,84,220],[39,71540,673],{"className":71541},[672],[39,71543,71545],{"className":71544},[200],[39,71546,71548],{"className":71547,"style":680},[204],[39,71549],{},"。4.3 节的例子说明，二者一般相差很大。动作必须在观察到下一状态之前选定，这个时序约束正是最大值位于期望之外的原因。",[11,71552,71553,2083,71612,71686,71687,71773,71774,71926,71927,72036,72037,72081],{},[1489,71554,71555,71556,1639,71584,1607],{},"混淆 ",[39,71557,71559,71572],{"className":71558,"translate":43},[42],[39,71560,71562],{"className":71561},[47],[49,71563,71564],{"xmlns":51},[53,71565,71566,71570],{},[56,71567,71568],{},[59,71569,4230],{},[63,71571,4230],{"encoding":65},[39,71573,71575],{"className":71574,"ariaHidden":70},[69],[39,71576,71578,71581],{"className":71577},[74],[39,71579],{"className":71580,"style":79},[78],[39,71582,4230],{"className":71583,"style":461},[83,84],[39,71585,71587,71600],{"className":71586,"translate":43},[42],[39,71588,71590],{"className":71589},[47],[49,71591,71592],{"xmlns":51},[53,71593,71594,71598],{},[56,71595,71596],{},[59,71597,4295],{},[63,71599,4295],{"encoding":65},[39,71601,71603],{"className":71602,"ariaHidden":70},[69],[39,71604,71606,71609],{"className":71605},[74],[39,71607],{"className":71608,"style":6452},[78],[39,71610,4295],{"className":71611},[83,84],[39,71613,71615,71639],{"className":71614,"translate":43},[42],[39,71616,71618],{"className":71617},[47],[49,71619,71620],{"xmlns":51},[53,71621,71622,71636],{},[56,71623,71624,71630,71632,71634],{},[138,71625,71626,71628],{},[59,71627,4230],{},[59,71629,2098],{},[133,71631,136],{"stretchy":135},[59,71633,142],{},[133,71635,161],{"stretchy":135},[63,71637,71638],{"encoding":65},"V^\\pi(s)",[39,71640,71642],{"className":71641,"ariaHidden":70},[69],[39,71643,71645,71648,71677,71680,71683],{"className":71644},[74],[39,71646],{"className":71647,"style":245},[78],[39,71649,71651,71654],{"className":71650},[83],[39,71652,4230],{"className":71653,"style":461},[83,84],[39,71655,71657],{"className":71656},[192],[39,71658,71660],{"className":71659},[196],[39,71661,71663],{"className":71662},[200],[39,71664,71666],{"className":71665,"style":6410},[204],[39,71667,71668,71671],{"style":208},[39,71669],{"className":71670,"style":213},[212],[39,71672,71674],{"className":71673},[217,218,219,220],[39,71675,2098],{"className":71676,"style":2114},[83,84,220],[39,71678,136],{"className":71679},[182],[39,71681,142],{"className":71682},[83,84],[39,71684,161],{"className":71685},[263]," 已经对策略的动作分布取了平均，因此它不再区分同一状态下的不同动作。要选择动作，应使用 ",[39,71688,71690,71717],{"className":71689,"translate":43},[42],[39,71691,71693],{"className":71692},[47],[49,71694,71695],{"xmlns":51},[53,71696,71697,71715],{},[56,71698,71699,71705,71707,71709,71711,71713],{},[138,71700,71701,71703],{},[59,71702,4295],{},[133,71704,21884],{},[133,71706,136],{"stretchy":135},[59,71708,142],{},[133,71710,155],{"separator":70},[59,71712,158],{},[133,71714,161],{"stretchy":135},[63,71716,45169],{"encoding":65},[39,71718,71720],{"className":71719,"ariaHidden":70},[69],[39,71721,71723,71726,71755,71758,71761,71764,71767,71770],{"className":71722},[74],[39,71724],{"className":71725,"style":245},[78],[39,71727,71729,71732],{"className":71728},[83],[39,71730,4295],{"className":71731},[83,84],[39,71733,71735],{"className":71734},[192],[39,71736,71738],{"className":71737},[196],[39,71739,71741],{"className":71740},[200],[39,71742,71744],{"className":71743,"style":21897},[204],[39,71745,71746,71749],{"style":208},[39,71747],{"className":71748,"style":213},[212],[39,71750,71752],{"className":71751},[217,218,219,220],[39,71753,21884],{"className":71754},[465,220],[39,71756,136],{"className":71757},[182],[39,71759,142],{"className":71760},[83,84],[39,71762,155],{"className":71763},[252],[39,71765],{"className":71766,"style":256},[230],[39,71768,158],{"className":71769},[83,84],[39,71771,161],{"className":71772},[263],"，最优动作是 ",[39,71775,71777,71819],{"className":71776,"translate":43},[42],[39,71778,71780],{"className":71779},[47],[49,71781,71782],{"xmlns":51},[53,71783,71784,71816],{},[56,71785,71786,71788,71790,71800,71806,71808,71810,71812,71814],{},[59,71787,39047],{},[133,71789,1690],{},[613,71791,71792,71798],{},[56,71793,71794,71796],{},[59,71795,1687],{},[133,71797,1690],{},[59,71799,158],{},[138,71801,71802,71804],{},[59,71803,4295],{},[133,71805,21884],{},[133,71807,136],{"stretchy":135},[59,71809,142],{},[133,71811,155],{"separator":70},[59,71813,158],{},[133,71815,161],{"stretchy":135},[63,71817,71818],{"encoding":65},"\\arg\\max_a Q^*(s,a)",[39,71820,71822],{"className":71821,"ariaHidden":70},[69],[39,71823,71825,71828,71833,71836,71876,71879,71908,71911,71914,71917,71920,71923],{"className":71824},[74],[39,71826],{"className":71827,"style":245},[78],[39,71829,39206,71831],{"className":71830},[1744],[39,71832,39209],{"style":35078},[39,71834],{"className":71835,"style":256},[230],[39,71837,71839,71842],{"className":71838},[1744],[39,71840,1687],{"className":71841},[1744],[39,71843,71845],{"className":71844},[192],[39,71846,71848,71868],{"className":71847},[196,649],[39,71849,71851,71865],{"className":71850},[200],[39,71852,71854],{"className":71853,"style":1728},[204],[39,71855,71856,71859],{"style":44021},[39,71857],{"className":71858,"style":213},[212],[39,71860,71862],{"className":71861},[217,218,219,220],[39,71863,158],{"className":71864},[83,84,220],[39,71866,673],{"className":71867},[672],[39,71869,71871],{"className":71870},[200],[39,71872,71874],{"className":71873,"style":680},[204],[39,71875],{},[39,71877],{"className":71878,"style":256},[230],[39,71880,71882,71885],{"className":71881},[83],[39,71883,4295],{"className":71884},[83,84],[39,71886,71888],{"className":71887},[192],[39,71889,71891],{"className":71890},[196],[39,71892,71894],{"className":71893},[200],[39,71895,71897],{"className":71896,"style":21897},[204],[39,71898,71899,71902],{"style":208},[39,71900],{"className":71901,"style":213},[212],[39,71903,71905],{"className":71904},[217,218,219,220],[39,71906,21884],{"className":71907},[465,220],[39,71909,136],{"className":71910},[182],[39,71912,142],{"className":71913},[83,84],[39,71915,155],{"className":71916},[252],[39,71918],{"className":71919,"style":256},[230],[39,71921,158],{"className":71922},[83,84],[39,71924,161],{"className":71925},[263],"。把 ",[39,71928,71930,71964],{"className":71929,"translate":43},[42],[39,71931,71933],{"className":71932},[47],[49,71934,71935],{"xmlns":51},[53,71936,71937,71961],{},[56,71938,71939,71941,71943,71953,71955,71957,71959],{},[59,71940,39047],{},[133,71942,1690],{},[613,71944,71945,71951],{},[56,71946,71947,71949],{},[59,71948,1687],{},[133,71950,1690],{},[59,71952,158],{},[59,71954,4230],{},[133,71956,136],{"stretchy":135},[59,71958,142],{},[133,71960,161],{"stretchy":135},[63,71962,71963],{"encoding":65},"\\arg\\max_a V(s)",[39,71965,71967],{"className":71966,"ariaHidden":70},[69],[39,71968,71970,71973,71978,71981,72021,72024,72027,72030,72033],{"className":71969},[74],[39,71971],{"className":71972,"style":245},[78],[39,71974,39206,71976],{"className":71975},[1744],[39,71977,39209],{"style":35078},[39,71979],{"className":71980,"style":256},[230],[39,71982,71984,71987],{"className":71983},[1744],[39,71985,1687],{"className":71986},[1744],[39,71988,71990],{"className":71989},[192],[39,71991,71993,72013],{"className":71992},[196,649],[39,71994,71996,72010],{"className":71995},[200],[39,71997,71999],{"className":71998,"style":1728},[204],[39,72000,72001,72004],{"style":44021},[39,72002],{"className":72003,"style":213},[212],[39,72005,72007],{"className":72006},[217,218,219,220],[39,72008,158],{"className":72009},[83,84,220],[39,72011,673],{"className":72012},[672],[39,72014,72016],{"className":72015},[200],[39,72017,72019],{"className":72018,"style":680},[204],[39,72020],{},[39,72022],{"className":72023,"style":256},[230],[39,72025,4230],{"className":72026,"style":461},[83,84],[39,72028,136],{"className":72029},[182],[39,72031,142],{"className":72032},[83,84],[39,72034,161],{"className":72035},[263]," 当作最优动作是没有意义的，因为 ",[39,72038,72040,72060],{"className":72039,"translate":43},[42],[39,72041,72043],{"className":72042},[47],[49,72044,72045],{"xmlns":51},[53,72046,72047,72057],{},[56,72048,72049,72051,72053,72055],{},[59,72050,4230],{},[133,72052,136],{"stretchy":135},[59,72054,142],{},[133,72056,161],{"stretchy":135},[63,72058,72059],{"encoding":65},"V(s)",[39,72061,72063],{"className":72062,"ariaHidden":70},[69],[39,72064,72066,72069,72072,72075,72078],{"className":72065},[74],[39,72067],{"className":72068,"style":245},[78],[39,72070,4230],{"className":72071,"style":461},[83,84],[39,72073,136],{"className":72074},[182],[39,72076,142],{"className":72077},[83,84],[39,72079,161],{"className":72080},[263]," 中已经没有动作变量。",[11,72083,72084,72087,72088,72138],{},[1489,72085,72086],{},"混淆必要性、充分性与唯一性。"," 在折扣有界 MDP 中，贝尔曼方程既必要又充分，而且解唯一。但在 ",[39,72089,72091,72108],{"className":72090,"translate":43},[42],[39,72092,72094],{"className":72093},[47],[49,72095,72096],{"xmlns":51},[53,72097,72098,72106],{},[56,72099,72100,72102,72104],{},[59,72101,531],{},[133,72103,793],{},[539,72105,546],{},[63,72107,28990],{"encoding":65},[39,72109,72111,72129],{"className":72110,"ariaHidden":70},[69],[39,72112,72114,72117,72120,72123,72126],{"className":72113},[74],[39,72115],{"className":72116,"style":2654},[78],[39,72118,531],{"className":72119,"style":565},[83,84],[39,72121],{"className":72122,"style":231},[230],[39,72124,793],{"className":72125},[235],[39,72127],{"className":72128,"style":231},[230],[39,72130,72132,72135],{"className":72131},[74],[39,72133],{"className":72134,"style":2473},[78],[39,72136,546],{"className":72137},[83],"、奖励无界或状态空间不满足可测选择条件时，这些结论可能分别失效。讨论贝尔曼方程时，应明确所依赖的假设。",[11,72140,72141,72144,72145,72173],{},[1489,72142,72143],{},"认为最优策略唯一。"," 最优值函数是唯一的，但最优策略不一定唯一。例如只有一个状态、两个动作，两个动作都使状态保持不变且都获得奖励 ",[39,72146,72148,72161],{"className":72147,"translate":43},[42],[39,72149,72151],{"className":72150},[47],[49,72152,72153],{"xmlns":51},[53,72154,72155,72159],{},[56,72156,72157],{},[539,72158,546],{},[63,72160,546],{"encoding":65},[39,72162,72164],{"className":72163,"ariaHidden":70},[69],[39,72165,72167,72170],{"className":72166},[74],[39,72168],{"className":72169,"style":2473},[78],[39,72171,546],{"className":72172},[83],"。此时",[39,72175,72177],{"className":72176,"translate":43},[746],[39,72178,72180,72214],{"className":72179,"translate":43},[42],[39,72181,72183],{"className":72182},[47],[49,72184,72185],{"xmlns":51,"display":755},[53,72186,72187,72211],{},[56,72188,72189,72195,72197,72209],{},[138,72190,72191,72193],{},[59,72192,4230],{},[133,72194,21884],{},[133,72196,793],{},[5362,72198,72199,72201],{},[539,72200,546],{},[56,72202,72203,72205,72207],{},[539,72204,546],{},[133,72206,5378],{},[59,72208,531],{},[133,72210,155],{"separator":70},[63,72212,72213],{"encoding":65},"V^*=\\frac{1}{1-\\gamma},",[39,72215,72217,72261],{"className":72216,"ariaHidden":70},[69],[39,72218,72220,72223,72252,72255,72258],{"className":72219},[74],[39,72221],{"className":72222,"style":22242},[78],[39,72224,72226,72229],{"className":72225},[83],[39,72227,4230],{"className":72228,"style":461},[83,84],[39,72230,72232],{"className":72231},[192],[39,72233,72235],{"className":72234},[196],[39,72236,72238],{"className":72237},[200],[39,72239,72241],{"className":72240,"style":22242},[204],[39,72242,72243,72246],{"style":1027},[39,72244],{"className":72245,"style":213},[212],[39,72247,72249],{"className":72248},[217,218,219,220],[39,72250,21884],{"className":72251},[465,220],[39,72253],{"className":72254,"style":231},[230],[39,72256,793],{"className":72257},[235],[39,72259],{"className":72260,"style":231},[230],[39,72262,72264,72268,72342],{"className":72263},[74],[39,72265],{"className":72266,"style":72267},[78],"height:2.2019em;vertical-align:-0.8804em;",[39,72269,72271,72274,72339],{"className":72270},[83],[39,72272],{"className":72273},[182,5463],[39,72275,72277],{"className":72276},[5362],[39,72278,72280,72331],{"className":72279},[196,649],[39,72281,72283,72328],{"className":72282},[200],[39,72284,72286,72309,72317],{"className":72285,"style":18240},[204],[39,72287,72288,72291],{"style":5479},[39,72289],{"className":72290,"style":935},[212],[39,72292,72294,72297,72300,72303,72306],{"className":72293},[83],[39,72295,546],{"className":72296},[83],[39,72298],{"className":72299,"style":461},[230],[39,72301,5378],{"className":72302},[465],[39,72304],{"className":72305,"style":461},[230],[39,72307,531],{"className":72308,"style":565},[83,84],[39,72310,72311,72314],{"style":5503},[39,72312],{"className":72313,"style":935},[212],[39,72315],{"className":72316,"style":5511},[5510],[39,72318,72319,72322],{"style":5514},[39,72320],{"className":72321,"style":935},[212],[39,72323,72325],{"className":72324},[83],[39,72326,546],{"className":72327},[83],[39,72329,673],{"className":72330},[672],[39,72332,72334],{"className":72333},[200],[39,72335,72337],{"className":72336,"style":5582},[204],[39,72338],{},[39,72340],{"className":72341},[263,5463],[39,72343,155],{"className":72344},[252],[11,72346,72347],{},"两个动作在每个状态都达到最大值，因此都是最优的。唯一性属于值函数，而不属于策略。",[21,72349,72350],{"id":72350},"总结",[11,72352,72353,72354,72569,72570,72626],{},"贝尔曼方程的核心可以概括为三步。第一步，利用折扣回报的递归分解 ",[39,72355,72357,72398],{"className":72356,"translate":43},[42],[39,72358,72360],{"className":72359},[47],[49,72361,72362],{"xmlns":51},[53,72363,72364,72396],{},[56,72365,72366,72372,72374,72380,72382,72384],{},[613,72367,72368,72370],{},[59,72369,3181],{},[59,72371,621],{},[133,72373,793],{},[613,72375,72376,72378],{},[59,72377,401],{},[59,72379,621],{},[133,72381,788],{},[59,72383,531],{},[613,72385,72386,72388],{},[59,72387,3181],{},[56,72389,72390,72392,72394],{},[59,72391,621],{},[133,72393,788],{},[539,72395,546],{},[63,72397,7463],{"encoding":65},[39,72399,72401,72456,72511],{"className":72400,"ariaHidden":70},[69],[39,72402,72404,72407,72447,72450,72453],{"className":72403},[74],[39,72405],{"className":72406,"style":634},[78],[39,72408,72410,72413],{"className":72409},[83],[39,72411,3181],{"className":72412},[83,84],[39,72414,72416],{"className":72415},[192],[39,72417,72419,72439],{"className":72418},[196,649],[39,72420,72422,72436],{"className":72421},[200],[39,72423,72425],{"className":72424,"style":656},[204],[39,72426,72427,72430],{"style":966},[39,72428],{"className":72429,"style":213},[212],[39,72431,72433],{"className":72432},[217,218,219,220],[39,72434,621],{"className":72435},[83,84,220],[39,72437,673],{"className":72438},[672],[39,72440,72442],{"className":72441},[200],[39,72443,72445],{"className":72444,"style":680},[204],[39,72446],{},[39,72448],{"className":72449,"style":231},[230],[39,72451,793],{"className":72452},[235],[39,72454],{"className":72455,"style":231},[230],[39,72457,72459,72462,72502,72505,72508],{"className":72458},[74],[39,72460],{"className":72461,"style":7528},[78],[39,72463,72465,72468],{"className":72464},[83],[39,72466,401],{"className":72467,"style":438},[83,84],[39,72469,72471],{"className":72470},[192],[39,72472,72474,72494],{"className":72473},[196,649],[39,72475,72477,72491],{"className":72476},[200],[39,72478,72480],{"className":72479,"style":656},[204],[39,72481,72482,72485],{"style":2937},[39,72483],{"className":72484,"style":213},[212],[39,72486,72488],{"className":72487},[217,218,219,220],[39,72489,621],{"className":72490},[83,84,220],[39,72492,673],{"className":72493},[672],[39,72495,72497],{"className":72496},[200],[39,72498,72500],{"className":72499,"style":680},[204],[39,72501],{},[39,72503],{"className":72504,"style":461},[230],[39,72506,788],{"className":72507},[465],[39,72509],{"className":72510,"style":461},[230],[39,72512,72514,72517,72520],{"className":72513},[74],[39,72515],{"className":72516,"style":7584},[78],[39,72518,531],{"className":72519,"style":565},[83,84],[39,72521,72523,72526],{"className":72522},[83],[39,72524,3181],{"className":72525},[83,84],[39,72527,72529],{"className":72528},[192],[39,72530,72532,72561],{"className":72531},[196,649],[39,72533,72535,72558],{"className":72534},[200],[39,72536,72538],{"className":72537,"style":963},[204],[39,72539,72540,72543],{"style":966},[39,72541],{"className":72542,"style":213},[212],[39,72544,72546],{"className":72545},[217,218,219,220],[39,72547,72549,72552,72555],{"className":72548},[83,220],[39,72550,621],{"className":72551},[83,84,220],[39,72553,788],{"className":72554},[465,220],[39,72556,546],{"className":72557},[83,220],[39,72559,673],{"className":72560},[672],[39,72562,72564],{"className":72563},[200],[39,72565,72567],{"className":72566,"style":994},[204],[39,72568],{}," 与马尔可夫性，把值函数写成当前奖励与后继值函数的期望。第二步，把该递归式视为有界函数空间上的算子方程 ",[39,72571,72573,72593],{"className":72572,"translate":43},[42],[39,72574,72576],{"className":72575},[47],[49,72577,72578],{"xmlns":51},[53,72579,72580,72590],{},[56,72581,72582,72584,72586,72588],{},[59,72583,4230],{},[133,72585,793],{},[59,72587,11772],{"mathvariant":617},[59,72589,4230],{},[63,72591,72592],{"encoding":65},"V=\\mathcal{T}V",[39,72594,72596,72614],{"className":72595,"ariaHidden":70},[69],[39,72597,72599,72602,72605,72608,72611],{"className":72598},[74],[39,72600],{"className":72601,"style":79},[78],[39,72603,4230],{"className":72604,"style":461},[83,84],[39,72606],{"className":72607,"style":231},[230],[39,72609,793],{"className":72610},[235],[39,72612],{"className":72613,"style":231},[230],[39,72615,72617,72620,72623],{"className":72616},[74],[39,72618],{"className":72619,"style":79},[78],[39,72621,11772],{"className":72622,"style":11793},[83,641],[39,72624,4230],{"className":72625,"style":461},[83,84],"。第三步，利用算子的单调性、常数平移性与压缩性，证明不动点存在唯一，并由 Banach 不动点定理得到几何收敛。",[11,72628,72629,72630,72715],{},"贝尔曼期望方程刻画固定策略的值函数，其有限状态形式是一个线性方程组，系数矩阵为 ",[39,72631,72633,72656],{"className":72632,"translate":43},[42],[39,72634,72636],{"className":72635},[47],[49,72637,72638],{"xmlns":51},[53,72639,72640,72654],{},[56,72641,72642,72644,72646,72648],{},[59,72643,14373],{},[133,72645,5378],{},[59,72647,531],{},[138,72649,72650,72652],{},[59,72651,131],{},[59,72653,2098],{},[63,72655,14654],{"encoding":65},[39,72657,72659,72677],{"className":72658,"ariaHidden":70},[69],[39,72660,72662,72665,72668,72671,72674],{"className":72661},[74],[39,72663],{"className":72664,"style":454},[78],[39,72666,14373],{"className":72667,"style":14420},[83,84],[39,72669],{"className":72670,"style":461},[230],[39,72672,5378],{"className":72673},[465],[39,72675],{"className":72676,"style":461},[230],[39,72678,72680,72683,72686],{"className":72679},[74],[39,72681],{"className":72682,"style":6452},[78],[39,72684,531],{"className":72685,"style":565},[83,84],[39,72687,72689,72692],{"className":72688},[83],[39,72690,131],{"className":72691,"style":178},[83,84],[39,72693,72695],{"className":72694},[192],[39,72696,72698],{"className":72697},[196],[39,72699,72701],{"className":72700},[200],[39,72702,72704],{"className":72703,"style":6410},[204],[39,72705,72706,72709],{"style":208},[39,72707],{"className":72708,"style":213},[212],[39,72710,72712],{"className":72711},[217,218,219,220],[39,72713,2098],{"className":72714,"style":2114},[83,84,220],"。贝尔曼最优方程刻画最优值函数，其算子在动作上取最大值。两者的共同点是都把无限期问题转化为一个不动点问题。区别在于前者对策略取期望，因而在有限状态下是线性方程。后者对动作取最大值，因而含有非线性项。",[11,72717,72718],{},"最优子结构是这一整套理论的解释性核心。它说明，在可加折扣效用、马尔可夫转移与有界回报的条件下，整体最优决策可以沿着实际到达的后继状态分解为一系列子问题的最优决策。它不是一个无条件的哲学原则，而是一个由回报结构、状态表示与折扣条件共同保证的数学定理。",{"title":72720,"searchDepth":72721,"depth":72721,"links":72722},"",2,[72723,72729,72736,72743,72750,72756,72761,72766],{"id":23,"depth":72721,"text":24,"children":72724},[72725,72727,72728],{"id":28,"depth":72726,"text":29},3,{"id":2586,"depth":72726,"text":2587},{"id":6488,"depth":72726,"text":6489},{"id":10038,"depth":72721,"text":10039,"children":72730},[72731,72732,72733,72734,72735],{"id":10042,"depth":72726,"text":10043},{"id":11748,"depth":72726,"text":11749},{"id":12928,"depth":72726,"text":12929},{"id":17330,"depth":72726,"text":17331},{"id":19743,"depth":72726,"text":19744},{"id":21349,"depth":72721,"text":21350,"children":72737},[72738,72739,72740,72741,72742],{"id":21353,"depth":72726,"text":21354},{"id":23677,"depth":72726,"text":23678},{"id":26631,"depth":72726,"text":26632},{"id":29396,"depth":72726,"text":29397},{"id":34082,"depth":72726,"text":34083},{"id":36535,"depth":72721,"text":36536,"children":72744},[72745,72746,72747,72748,72749],{"id":36539,"depth":72726,"text":36540},{"id":37349,"depth":72726,"text":37350},{"id":45346,"depth":72726,"text":45347},{"id":47168,"depth":72726,"text":47169},{"id":48417,"depth":72726,"text":48418},{"id":51028,"depth":72721,"text":51029,"children":72751},[72752,72753,72754,72755],{"id":51032,"depth":72726,"text":51033},{"id":53366,"depth":72726,"text":53367},{"id":58583,"depth":72726,"text":58584},{"id":61601,"depth":72726,"text":61602},{"id":62687,"depth":72721,"text":62688,"children":72757},[72758,72759,72760],{"id":62691,"depth":72726,"text":62692},{"id":68009,"depth":72726,"text":68010},{"id":68297,"depth":72726,"text":68298},{"id":68905,"depth":72721,"text":68906,"children":72762},[72763,72764,72765],{"id":68909,"depth":72726,"text":68910},{"id":69898,"depth":72726,"text":69899},{"id":71203,"depth":72726,"text":71204},{"id":72350,"depth":72721,"text":72350},false,true,"md",null,{},"\u002Fblog\u002F2026\u002F0015-the-bellman-equation-and-its-mathematical-properties",{"title":6,"description":13},"blog\u002F2026\u002F0015-the-bellman-equation-and-its-mathematical-properties","贝尔曼方程（Bellman equation）是动态规划的核心，本质是把一个多阶段的决策问题拆成当前决策 + 剩余子问题的递归关系。它由理查德·贝尔曼在 20 世纪 50 年代提出，是强化学习、最优控制、经济学等领域的基础工具。",[72777,72778],"machine-learning","mathematics","2026-11-09T23:49:14+08:00","hpJs47-d_ZZWWr8USnYwX8xzwtvDogQjVGqetsjCV2g",{"id":72782,"title":72783,"body":72784,"description":72788,"draft":72767,"enableCodeHighlight":72767,"enableCodeHightlight":72767,"enableComment":72768,"enableD2":72767,"enableKatex":72768,"extension":72769,"image":72720,"important":72767,"location":72770,"meta":125302,"navigation":72768,"ogImage":72770,"path":125303,"seo":125304,"stem":125305,"summary":125306,"tags":125307,"time":125309,"__hash__":125310},"blog\u002Fblog\u002F2026\u002F0014-zui-you-chuan-shu-ji-he-jian-lun.md","最优传输几何概念简论",{"type":8,"value":72785,"toc":125289},[72786,72789,72792,72795,72798,72801,73140,73478,73561,73627,73836,74081,74195,74531,75133,75136,75620,75738,76104,76136,76306,76676,77374,77534,77784,77816,77919,77922,77925,77987,78199,78202,79578,79840,80612,80615,82197,82717,83334,83337,83340,83694,84127,84452,84455,84828,84831,84834,84959,85183,85377,85568,85641,86158,86336,86921,87313,87718,88432,88914,88917,89848,90959,91170,91608,91611,91984,91987,92018,92261,92398,92401,92773,92776,93192,93421,93620,94098,94652,94707,94839,95126,95315,95422,95425,95643,95676,95900,95903,95906,96299,96302,96387,97126,97501,97504,97639,98584,98587,98590,98593,98796,98968,99138,99611,100051,100154,100518,101100,101103,101920,101923,101926,101987,101990,102220,102662,103042,103534,103537,104892,105015,105173,105649,106417,106420,106424,106577,106778,106983,107106,107671,107674,107735,108021,108192,109186,109189,109675,109751,109754,110213,110446,110847,112670,112746,112749,112977,113468,113471,113919,114422,114425,114563,114793,114883,114886,115485,115889,116619,116959,117301,118081,118084,118087,118090,118093,118158,118324,118474,118915,118918,119082,119537,119540,119913,119975,120602,121551,121654,122012,122213,122284,123080,123287,123290,123293,123296,123411,123897,123929,123932,124508,124511,124514,124666,124918,124921,124924,124927,124930,124962,124965,125283,125286],[11,72787,72788],{},"我从一个很具体的困扰开始。模型对一张从没见过的图片给出的 softmax 最大值是 0.99，可我没有任何依据说它“99% 属于猫”。softmax 只做了一件事：把 logits 归一化成和为 1 的数。它回答的问题是“在已经假定的这几个类里，哪个最像”，而不是“这个样本是否属于我见过的任何一类”。",[11,72790,72791],{},"要判断一个样本有没有越界，我需要知道的不是分类器给它打了多少分，而是已知类别的数据在空间里占据了多少概率质量、这个样本离那片质量有多远。前者是密度，后者是距离，两者都是分布的几何性质，而不是分类器的性质。",[21,72793,72794],{"id":72794},"搬土",[11,72796,72797],{},"先算一道运输题。",[11,72799,72800],{},"甲矿有 3 吨煤，乙矿有 1 吨；A 厂要 2 吨，B 厂要 2 吨。运价是：甲到 A 每吨 1 块，甲到 B 每吨 2 块，乙到 A 每吨 3 块，乙到 B 每吨 1 块。怎么运最省？",[11,72802,72803,72804,72832,72833,72884,72885,72936,72937,73065,73066,73139],{},"设甲运往 A 的吨数是 ",[39,72805,72807,72820],{"className":72806,"translate":43},[42],[39,72808,72810],{"className":72809},[47],[49,72811,72812],{"xmlns":51},[53,72813,72814,72818],{},[56,72815,72816],{},[59,72817,1755],{},[63,72819,1755],{"encoding":65},[39,72821,72823],{"className":72822,"ariaHidden":70},[69],[39,72824,72826,72829],{"className":72825},[74],[39,72827],{"className":72828,"style":290},[78],[39,72830,1755],{"className":72831},[83,84],"。甲的 3 吨要分完，所以甲到 B 是 ",[39,72834,72836,72854],{"className":72835,"translate":43},[42],[39,72837,72839],{"className":72838},[47],[49,72840,72841],{"xmlns":51},[53,72842,72843,72851],{},[56,72844,72845,72847,72849],{},[539,72846,59187],{},[133,72848,5378],{},[59,72850,1755],{},[63,72852,72853],{"encoding":65},"3-x",[39,72855,72857,72875],{"className":72856,"ariaHidden":70},[69],[39,72858,72860,72863,72866,72869,72872],{"className":72859},[74],[39,72861],{"className":72862,"style":17882},[78],[39,72864,59187],{"className":72865},[83],[39,72867],{"className":72868,"style":461},[230],[39,72870,5378],{"className":72871},[465],[39,72873],{"className":72874,"style":461},[230],[39,72876,72878,72881],{"className":72877},[74],[39,72879],{"className":72880,"style":290},[78],[39,72882,1755],{"className":72883},[83,84],"；A 要 2 吨，所以乙到 A 是 ",[39,72886,72888,72906],{"className":72887,"translate":43},[42],[39,72889,72891],{"className":72890},[47],[49,72892,72893],{"xmlns":51},[53,72894,72895,72903],{},[56,72896,72897,72899,72901],{},[539,72898,17359],{},[133,72900,5378],{},[59,72902,1755],{},[63,72904,72905],{"encoding":65},"2-x",[39,72907,72909,72927],{"className":72908,"ariaHidden":70},[69],[39,72910,72912,72915,72918,72921,72924],{"className":72911},[74],[39,72913],{"className":72914,"style":17882},[78],[39,72916,17359],{"className":72917},[83],[39,72919],{"className":72920,"style":461},[230],[39,72922,5378],{"className":72923},[465],[39,72925],{"className":72926,"style":461},[230],[39,72928,72930,72933],{"className":72929},[74],[39,72931],{"className":72932,"style":290},[78],[39,72934,1755],{"className":72935},[83,84],"；乙只有 1 吨，所以乙到 B 是 ",[39,72938,72940,72974],{"className":72939,"translate":43},[42],[39,72941,72943],{"className":72942},[47],[49,72944,72945],{"xmlns":51},[53,72946,72947,72971],{},[56,72948,72949,72951,72953,72955,72957,72959,72961,72963,72965,72967,72969],{},[539,72950,546],{},[133,72952,5378],{},[133,72954,136],{"stretchy":135},[539,72956,17359],{},[133,72958,5378],{},[59,72960,1755],{},[133,72962,161],{"stretchy":135},[133,72964,793],{},[59,72966,1755],{},[133,72968,5378],{},[539,72970,546],{},[63,72972,72973],{"encoding":65},"1-(2-x)=x-1",[39,72975,72977,72995,73016,73037,73056],{"className":72976,"ariaHidden":70},[69],[39,72978,72980,72983,72986,72989,72992],{"className":72979},[74],[39,72981],{"className":72982,"style":17882},[78],[39,72984,546],{"className":72985},[83],[39,72987],{"className":72988,"style":461},[230],[39,72990,5378],{"className":72991},[465],[39,72993],{"className":72994,"style":461},[230],[39,72996,72998,73001,73004,73007,73010,73013],{"className":72997},[74],[39,72999],{"className":73000,"style":245},[78],[39,73002,136],{"className":73003},[182],[39,73005,17359],{"className":73006},[83],[39,73008],{"className":73009,"style":461},[230],[39,73011,5378],{"className":73012},[465],[39,73014],{"className":73015,"style":461},[230],[39,73017,73019,73022,73025,73028,73031,73034],{"className":73018},[74],[39,73020],{"className":73021,"style":245},[78],[39,73023,1755],{"className":73024},[83,84],[39,73026,161],{"className":73027},[263],[39,73029],{"className":73030,"style":231},[230],[39,73032,793],{"className":73033},[235],[39,73035],{"className":73036,"style":231},[230],[39,73038,73040,73044,73047,73050,73053],{"className":73039},[74],[39,73041],{"className":73042,"style":73043},[78],"height:0.6667em;vertical-align:-0.0833em;",[39,73045,1755],{"className":73046},[83,84],[39,73048],{"className":73049,"style":461},[230],[39,73051,5378],{"className":73052},[465],[39,73054],{"className":73055,"style":461},[230],[39,73057,73059,73062],{"className":73058},[74],[39,73060],{"className":73061,"style":2473},[78],[39,73063,546],{"className":73064},[83],"。四个数都得非负，于是 ",[39,73067,73069,73091],{"className":73068,"translate":43},[42],[39,73070,73072],{"className":73071},[47],[49,73073,73074],{"xmlns":51},[53,73075,73076,73088],{},[56,73077,73078,73080,73082,73084,73086],{},[539,73079,546],{},[133,73081,1824],{},[59,73083,1755],{},[133,73085,1824],{},[539,73087,17359],{},[63,73089,73090],{"encoding":65},"1\\le x\\le 2",[39,73092,73094,73112,73130],{"className":73093,"ariaHidden":70},[69],[39,73095,73097,73100,73103,73106,73109],{"className":73096},[74],[39,73098],{"className":73099,"style":14587},[78],[39,73101,546],{"className":73102},[83],[39,73104],{"className":73105,"style":231},[230],[39,73107,1824],{"className":73108},[235],[39,73110],{"className":73111,"style":231},[230],[39,73113,73115,73118,73121,73124,73127],{"className":73114},[74],[39,73116],{"className":73117,"style":59832},[78],[39,73119,1755],{"className":73120},[83,84],[39,73122],{"className":73123,"style":231},[230],[39,73125,1824],{"className":73126},[235],[39,73128],{"className":73129,"style":231},[230],[39,73131,73133,73136],{"className":73132},[74],[39,73134],{"className":73135,"style":2473},[78],[39,73137,17359],{"className":73138},[83],"。总运费",[39,73141,73143],{"className":73142,"translate":43},[746],[39,73144,73146,73225],{"className":73145,"translate":43},[42],[39,73147,73149],{"className":73148},[47],[49,73150,73151],{"xmlns":51,"display":755},[53,73152,73153,73222],{},[56,73154,73155,73157,73159,73161,73163,73165,73167,73169,73171,73173,73175,73177,73179,73181,73183,73185,73187,73189,73191,73193,73195,73197,73199,73201,73203,73205,73207,73209,73211,73214,73216,73218,73220],{},[59,73156,1755],{},[133,73158,8885],{},[539,73160,546],{},[133,73162,788],{},[133,73164,136],{"stretchy":135},[539,73166,59187],{},[133,73168,5378],{},[59,73170,1755],{},[133,73172,161],{"stretchy":135},[133,73174,8885],{},[539,73176,17359],{},[133,73178,788],{},[133,73180,136],{"stretchy":135},[539,73182,17359],{},[133,73184,5378],{},[59,73186,1755],{},[133,73188,161],{"stretchy":135},[133,73190,8885],{},[539,73192,59187],{},[133,73194,788],{},[133,73196,136],{"stretchy":135},[59,73198,1755],{},[133,73200,5378],{},[539,73202,546],{},[133,73204,161],{"stretchy":135},[133,73206,8885],{},[539,73208,546],{},[133,73210,793],{},[539,73212,73213],{},"11",[133,73215,5378],{},[539,73217,59187],{},[59,73219,1755],{},[59,73221,2903],{"mathvariant":145},[63,73223,73224],{"encoding":65},"x\\cdot1+(3-x)\\cdot2+(2-x)\\cdot3+(x-1)\\cdot1=11-3x .",[39,73226,73228,73247,73265,73286,73307,73325,73346,73367,73385,73406,73427,73445,73463],{"className":73227,"ariaHidden":70},[69],[39,73229,73231,73235,73238,73241,73244],{"className":73230},[74],[39,73232],{"className":73233,"style":73234},[78],"height:0.4445em;",[39,73236,1755],{"className":73237},[83,84],[39,73239],{"className":73240,"style":461},[230],[39,73242,8885],{"className":73243},[465],[39,73245],{"className":73246,"style":461},[230],[39,73248,73250,73253,73256,73259,73262],{"className":73249},[74],[39,73251],{"className":73252,"style":17882},[78],[39,73254,546],{"className":73255},[83],[39,73257],{"className":73258,"style":461},[230],[39,73260,788],{"className":73261},[465],[39,73263],{"className":73264,"style":461},[230],[39,73266,73268,73271,73274,73277,73280,73283],{"className":73267},[74],[39,73269],{"className":73270,"style":245},[78],[39,73272,136],{"className":73273},[182],[39,73275,59187],{"className":73276},[83],[39,73278],{"className":73279,"style":461},[230],[39,73281,5378],{"className":73282},[465],[39,73284],{"className":73285,"style":461},[230],[39,73287,73289,73292,73295,73298,73301,73304],{"className":73288},[74],[39,73290],{"className":73291,"style":245},[78],[39,73293,1755],{"className":73294},[83,84],[39,73296,161],{"className":73297},[263],[39,73299],{"className":73300,"style":461},[230],[39,73302,8885],{"className":73303},[465],[39,73305],{"className":73306,"style":461},[230],[39,73308,73310,73313,73316,73319,73322],{"className":73309},[74],[39,73311],{"className":73312,"style":17882},[78],[39,73314,17359],{"className":73315},[83],[39,73317],{"className":73318,"style":461},[230],[39,73320,788],{"className":73321},[465],[39,73323],{"className":73324,"style":461},[230],[39,73326,73328,73331,73334,73337,73340,73343],{"className":73327},[74],[39,73329],{"className":73330,"style":245},[78],[39,73332,136],{"className":73333},[182],[39,73335,17359],{"className":73336},[83],[39,73338],{"className":73339,"style":461},[230],[39,73341,5378],{"className":73342},[465],[39,73344],{"className":73345,"style":461},[230],[39,73347,73349,73352,73355,73358,73361,73364],{"className":73348},[74],[39,73350],{"className":73351,"style":245},[78],[39,73353,1755],{"className":73354},[83,84],[39,73356,161],{"className":73357},[263],[39,73359],{"className":73360,"style":461},[230],[39,73362,8885],{"className":73363},[465],[39,73365],{"className":73366,"style":461},[230],[39,73368,73370,73373,73376,73379,73382],{"className":73369},[74],[39,73371],{"className":73372,"style":17882},[78],[39,73374,59187],{"className":73375},[83],[39,73377],{"className":73378,"style":461},[230],[39,73380,788],{"className":73381},[465],[39,73383],{"className":73384,"style":461},[230],[39,73386,73388,73391,73394,73397,73400,73403],{"className":73387},[74],[39,73389],{"className":73390,"style":245},[78],[39,73392,136],{"className":73393},[182],[39,73395,1755],{"className":73396},[83,84],[39,73398],{"className":73399,"style":461},[230],[39,73401,5378],{"className":73402},[465],[39,73404],{"className":73405,"style":461},[230],[39,73407,73409,73412,73415,73418,73421,73424],{"className":73408},[74],[39,73410],{"className":73411,"style":245},[78],[39,73413,546],{"className":73414},[83],[39,73416,161],{"className":73417},[263],[39,73419],{"className":73420,"style":461},[230],[39,73422,8885],{"className":73423},[465],[39,73425],{"className":73426,"style":461},[230],[39,73428,73430,73433,73436,73439,73442],{"className":73429},[74],[39,73431],{"className":73432,"style":2473},[78],[39,73434,546],{"className":73435},[83],[39,73437],{"className":73438,"style":231},[230],[39,73440,793],{"className":73441},[235],[39,73443],{"className":73444,"style":231},[230],[39,73446,73448,73451,73454,73457,73460],{"className":73447},[74],[39,73449],{"className":73450,"style":17882},[78],[39,73452,73213],{"className":73453},[83],[39,73455],{"className":73456,"style":461},[230],[39,73458,5378],{"className":73459},[465],[39,73461],{"className":73462,"style":461},[230],[39,73464,73466,73469,73472,73475],{"className":73465},[74],[39,73467],{"className":73468,"style":2473},[78],[39,73470,59187],{"className":73471},[83],[39,73473,1755],{"className":73474},[83,84],[39,73476,2903],{"className":73477},[83],[11,73479,73480,73508,73509,73560],{},[39,73481,73483,73496],{"className":73482,"translate":43},[42],[39,73484,73486],{"className":73485},[47],[49,73487,73488],{"xmlns":51},[53,73489,73490,73494],{},[56,73491,73492],{},[59,73493,1755],{},[63,73495,1755],{"encoding":65},[39,73497,73499],{"className":73498,"ariaHidden":70},[69],[39,73500,73502,73505],{"className":73501},[74],[39,73503],{"className":73504,"style":290},[78],[39,73506,1755],{"className":73507},[83,84]," 越大越省，取 ",[39,73510,73512,73530],{"className":73511,"translate":43},[42],[39,73513,73515],{"className":73514},[47],[49,73516,73517],{"xmlns":51},[53,73518,73519,73527],{},[56,73520,73521,73523,73525],{},[59,73522,1755],{},[133,73524,793],{},[539,73526,17359],{},[63,73528,73529],{"encoding":65},"x=2",[39,73531,73533,73551],{"className":73532,"ariaHidden":70},[69],[39,73534,73536,73539,73542,73545,73548],{"className":73535},[74],[39,73537],{"className":73538,"style":290},[78],[39,73540,1755],{"className":73541},[83,84],[39,73543],{"className":73544,"style":231},[230],[39,73546,793],{"className":73547},[235],[39,73549],{"className":73550,"style":231},[230],[39,73552,73554,73557],{"className":73553},[74],[39,73555],{"className":73556,"style":2473},[78],[39,73558,17359],{"className":73559},[83],"，总运费 5 块：甲到 A 运 2 吨，甲到 B 运 1 吨，乙到 B 运 1 吨。",[11,73562,73563,73564,73594,73595,73626],{},"这张“每个产地往每个销地运多少”的表，就是运输方案。Monge 在 1781 年问的是一般情形：给定一堆土 ",[39,73565,73567,73582],{"className":73566,"translate":43},[42],[39,73568,73570],{"className":73569},[47],[49,73571,73572],{"xmlns":51},[53,73573,73574,73579],{},[56,73575,73576],{},[59,73577,73578],{},"μ",[63,73580,73581],{"encoding":65},"\\mu",[39,73583,73585],{"className":73584,"ariaHidden":70},[69],[39,73586,73588,73591],{"className":73587},[74],[39,73589],{"className":73590,"style":2654},[78],[39,73592,73578],{"className":73593},[83,84]," 和一个坑 ",[39,73596,73598,73613],{"className":73597,"translate":43},[42],[39,73599,73601],{"className":73600},[47],[49,73602,73603],{"xmlns":51},[53,73604,73605,73610],{},[56,73606,73607],{},[59,73608,73609],{},"ν",[63,73611,73612],{"encoding":65},"\\nu",[39,73614,73616],{"className":73615,"ariaHidden":70},[69],[39,73617,73619,73622],{"className":73618},[74],[39,73620],{"className":73621,"style":290},[78],[39,73623,73609],{"className":73624,"style":73625},[83,84],"margin-right:0.0637em;","，怎么搬最省。",[11,73628,73629,73630,73707,73708,73807,73808,742],{},"他的写法是找一个映射 ",[39,73631,73633,73656],{"className":73632,"translate":43},[42],[39,73634,73636],{"className":73635},[47],[49,73637,73638],{"xmlns":51},[53,73639,73640,73653],{},[56,73641,73642,73644,73646,73648,73650],{},[59,73643,11772],{},[133,73645,404],{},[59,73647,29420],{"mathvariant":617},[133,73649,418],{},[59,73651,73652],{"mathvariant":617},"Y",[63,73654,73655],{"encoding":65},"T:\\mathcal X\\to\\mathcal Y",[39,73657,73659,73677,73696],{"className":73658,"ariaHidden":70},[69],[39,73660,73662,73665,73668,73671,73674],{"className":73661},[74],[39,73663],{"className":73664,"style":79},[78],[39,73666,11772],{"className":73667,"style":178},[83,84],[39,73669],{"className":73670,"style":231},[230],[39,73672,404],{"className":73673},[235],[39,73675],{"className":73676,"style":231},[230],[39,73678,73680,73683,73687,73690,73693],{"className":73679},[74],[39,73681],{"className":73682,"style":79},[78],[39,73684,29420],{"className":73685,"style":73686},[83,641],"margin-right:0.1464em;",[39,73688],{"className":73689,"style":231},[230],[39,73691,418],{"className":73692},[235],[39,73694],{"className":73695,"style":231},[230],[39,73697,73699,73703],{"className":73698},[74],[39,73700],{"className":73701,"style":73702},[78],"height:0.7805em;vertical-align:-0.0972em;",[39,73704,73652],{"className":73705,"style":73706},[83,641],"margin-right:0.0822em;","，要求“搬完之后每个地方刚好装满”，也就是推前测度 ",[39,73709,73711,73736],{"className":73710,"translate":43},[42],[39,73712,73714],{"className":73713},[47],[49,73715,73716],{"xmlns":51},[53,73717,73718,73733],{},[56,73719,73720,73727,73729,73731],{},[613,73721,73722,73724],{},[59,73723,11772],{},[59,73725,73726],{"mathvariant":145},"#",[59,73728,73578],{},[133,73730,793],{},[59,73732,73609],{},[63,73734,73735],{"encoding":65},"T_\\#\\mu=\\nu",[39,73737,73739,73798],{"className":73738,"ariaHidden":70},[69],[39,73740,73742,73745,73786,73789,73792,73795],{"className":73741},[74],[39,73743],{"className":73744,"style":35350},[78],[39,73746,73748,73751],{"className":73747},[83],[39,73749,11772],{"className":73750,"style":178},[83,84],[39,73752,73754],{"className":73753},[192],[39,73755,73757,73778],{"className":73756},[196,649],[39,73758,73760,73775],{"className":73759},[200],[39,73761,73763],{"className":73762,"style":3419},[204],[39,73764,73766,73769],{"style":73765},"top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;",[39,73767],{"className":73768,"style":213},[212],[39,73770,73772],{"className":73771},[217,218,219,220],[39,73773,73726],{"className":73774},[83,220],[39,73776,673],{"className":73777},[672],[39,73779,73781],{"className":73780},[200],[39,73782,73784],{"className":73783,"style":35396},[204],[39,73785],{},[39,73787,73578],{"className":73788},[83,84],[39,73790],{"className":73791,"style":231},[230],[39,73793,793],{"className":73794},[235],[39,73796],{"className":73797,"style":231},[230],[39,73799,73801,73804],{"className":73800},[74],[39,73802],{"className":73803,"style":290},[78],[39,73805,73609],{"className":73806,"style":73625},[83,84],"，即对任意可测集 ",[39,73809,73811,73824],{"className":73810,"translate":43},[42],[39,73812,73814],{"className":73813},[47],[49,73815,73816],{"xmlns":51},[53,73817,73818,73822],{},[56,73819,73820],{},[59,73821,19772],{},[63,73823,19772],{"encoding":65},[39,73825,73827],{"className":73826,"ariaHidden":70},[69],[39,73828,73830,73833],{"className":73829},[74],[39,73831],{"className":73832,"style":79},[78],[39,73834,19772],{"className":73835,"style":19812},[83,84],[39,73837,73839],{"className":73838,"translate":43},[746],[39,73840,73842,73909],{"className":73841,"translate":43},[42],[39,73843,73845],{"className":73844},[47],[49,73846,73847],{"xmlns":51,"display":755},[53,73848,73849,73906],{},[56,73850,73851,73853,73859,73861,73863,73865,73867,73869,73871,73873,73876,73886,73888,73890,73892,73894,73896,73898,73900,73902,73904],{},[133,73852,136],{"stretchy":135},[613,73854,73855,73857],{},[59,73856,11772],{},[59,73858,73726],{"mathvariant":145},[59,73860,73578],{},[133,73862,161],{"stretchy":135},[133,73864,136],{"stretchy":135},[59,73866,19772],{},[133,73868,161],{"stretchy":135},[133,73870,793],{},[59,73872,73578],{},[133,73874,136],{"fence":70,"stretchy":70,"minsize":73875,"maxsize":73875},"1.2em",[138,73877,73878,73880],{},[59,73879,11772],{},[56,73881,73882,73884],{},[133,73883,5378],{},[539,73885,546],{},[133,73887,136],{"stretchy":135},[59,73889,19772],{},[133,73891,161],{"stretchy":135},[133,73893,161],{"fence":70,"stretchy":70,"minsize":73875,"maxsize":73875},[133,73895,793],{},[59,73897,73609],{},[133,73899,136],{"stretchy":135},[59,73901,19772],{},[133,73903,161],{"stretchy":135},[133,73905,155],{"separator":70},[63,73907,73908],{"encoding":65},"(T_\\#\\mu)(B)=\\mu\\bigl(T^{-1}(B)\\bigr)=\\nu(B),",[39,73910,73912,73985,74060],{"className":73911,"ariaHidden":70},[69],[39,73913,73915,73918,73921,73961,73964,73967,73970,73973,73976,73979,73982],{"className":73914},[74],[39,73916],{"className":73917,"style":41560},[78],[39,73919,136],{"className":73920},[182],[39,73922,73924,73927],{"className":73923},[83],[39,73925,11772],{"className":73926,"style":178},[83,84],[39,73928,73930],{"className":73929},[192],[39,73931,73933,73953],{"className":73932},[196,649],[39,73934,73936,73950],{"className":73935},[200],[39,73937,73939],{"className":73938,"style":3419},[204],[39,73940,73941,73944],{"style":73765},[39,73942],{"className":73943,"style":213},[212],[39,73945,73947],{"className":73946},[217,218,219,220],[39,73948,73726],{"className":73949},[83,220],[39,73951,673],{"className":73952},[672],[39,73954,73956],{"className":73955},[200],[39,73957,73959],{"className":73958,"style":35396},[204],[39,73960],{},[39,73962,73578],{"className":73963},[83,84],[39,73965,161],{"className":73966},[263],[39,73968,136],{"className":73969},[182],[39,73971,19772],{"className":73972,"style":19812},[83,84],[39,73974,161],{"className":73975},[263],[39,73977],{"className":73978,"style":231},[230],[39,73980,793],{"className":73981},[235],[39,73983],{"className":73984,"style":231},[230],[39,73986,73988,73992,73995,74001,74036,74039,74042,74045,74051,74054,74057],{"className":73987},[74],[39,73989],{"className":73990,"style":73991},[78],"height:1.2141em;vertical-align:-0.35em;",[39,73993,73578],{"className":73994},[83,84],[39,73996,73998],{"className":73997},[182],[39,73999,136],{"className":74000},[18520,4009],[39,74002,74004,74007],{"className":74003},[83],[39,74005,11772],{"className":74006,"style":178},[83,84],[39,74008,74010],{"className":74009},[192],[39,74011,74013],{"className":74012},[196],[39,74014,74016],{"className":74015},[200],[39,74017,74019],{"className":74018,"style":14940},[204],[39,74020,74021,74024],{"style":1027},[39,74022],{"className":74023,"style":213},[212],[39,74025,74027],{"className":74026},[217,218,219,220],[39,74028,74030,74033],{"className":74029},[83,220],[39,74031,5378],{"className":74032},[83,220],[39,74034,546],{"className":74035},[83,220],[39,74037,136],{"className":74038},[182],[39,74040,19772],{"className":74041,"style":19812},[83,84],[39,74043,161],{"className":74044},[263],[39,74046,74048],{"className":74047},[263],[39,74049,161],{"className":74050},[18520,4009],[39,74052],{"className":74053,"style":231},[230],[39,74055,793],{"className":74056},[235],[39,74058],{"className":74059,"style":231},[230],[39,74061,74063,74066,74069,74072,74075,74078],{"className":74062},[74],[39,74064],{"className":74065,"style":245},[78],[39,74067,73609],{"className":74068,"style":73625},[83,84],[39,74070,136],{"className":74071},[182],[39,74073,19772],{"className":74074,"style":19812},[83,84],[39,74076,161],{"className":74077},[263],[39,74079,155],{"className":74080},[252],[11,74082,74083,74084,74194],{},"代价是 ",[39,74085,74087,74131],{"className":74086,"translate":43},[42],[39,74088,74090],{"className":74089},[47],[49,74091,74092],{"xmlns":51},[53,74093,74094,74128],{},[56,74095,74096,74098,74100,74102,74104,74106,74108,74110,74112,74114,74116,74118,74120,74122,74124,74126],{},[133,74097,69087],{},[59,74099,24077],{},[133,74101,136],{"stretchy":135},[59,74103,1755],{},[133,74105,155],{"separator":70},[59,74107,11772],{},[133,74109,136],{"stretchy":135},[59,74111,1755],{},[133,74113,161],{"stretchy":135},[133,74115,161],{"stretchy":135},[35841,74117,35843],{},[59,74119,29425],{},[59,74121,73578],{},[133,74123,136],{"stretchy":135},[59,74125,1755],{},[133,74127,161],{"stretchy":135},[63,74129,74130],{"encoding":65},"\\int c(x,T(x))\\,d\\mu(x)",[39,74132,74134],{"className":74133,"ariaHidden":70},[69],[39,74135,74137,74141,74145,74148,74151,74154,74157,74160,74163,74166,74169,74172,74176,74179,74182,74185,74188,74191],{"className":74136},[74],[39,74138],{"className":74139,"style":74140},[78],"height:1.1111em;vertical-align:-0.3061em;",[39,74142,69087],{"className":74143,"style":74144},[1744,3339,3952],"margin-right:0.1945em;position:relative;top:-0.0006em;",[39,74146],{"className":74147,"style":256},[230],[39,74149,24077],{"className":74150},[83,84],[39,74152,136],{"className":74153},[182],[39,74155,1755],{"className":74156},[83,84],[39,74158,155],{"className":74159},[252],[39,74161],{"className":74162,"style":256},[230],[39,74164,11772],{"className":74165,"style":178},[83,84],[39,74167,136],{"className":74168},[182],[39,74170,1755],{"className":74171},[83,84],[39,74173,74175],{"className":74174},[263],"))",[39,74177],{"className":74178,"style":256},[230],[39,74180,29425],{"className":74181},[83,84],[39,74183,73578],{"className":74184},[83,84],[39,74186,136],{"className":74187},[182],[39,74189,1755],{"className":74190},[83,84],[39,74192,161],{"className":74193},[263],"。于是 Monge 问题就是",[39,74196,74198],{"className":74197,"translate":43},[746],[39,74199,74201,74284],{"className":74200,"translate":43},[42],[39,74202,74204],{"className":74203},[47],[49,74205,74206],{"xmlns":51,"display":755},[53,74207,74208,74281],{},[56,74209,74210,74221,74224,74230,74232,74234,74236,74238,74240,74242,74244,74246,74248,74250,74252,74254,74256,74258,74260,74262,74265,74267,74273,74275,74277,74279],{},[6556,74211,74212,74219],{},[56,74213,74214,74217],{},[59,74215,74216],{},"inf",[133,74218,1690],{},[59,74220,11772],{},[35841,74222,74223],{}," ",[613,74225,74226,74228],{},[133,74227,69087],{},[59,74229,29420],{"mathvariant":617},[59,74231,24077],{},[133,74233,136],{"fence":70,"stretchy":70,"minsize":73875,"maxsize":73875},[59,74235,1755],{},[133,74237,155],{"separator":70},[59,74239,11772],{},[133,74241,136],{"stretchy":135},[59,74243,1755],{},[133,74245,161],{"stretchy":135},[133,74247,161],{"fence":70,"stretchy":70,"minsize":73875,"maxsize":73875},[35841,74249,35843],{},[59,74251,29425],{},[59,74253,73578],{},[133,74255,136],{"stretchy":135},[59,74257,1755],{},[133,74259,161],{"stretchy":135},[230,74261],{"width":6891},[35841,74263,74264],{},"s.t.",[230,74266],{"width":6891},[613,74268,74269,74271],{},[59,74270,11772],{},[59,74272,73726],{"mathvariant":145},[59,74274,73578],{},[133,74276,793],{},[59,74278,73609],{},[59,74280,2903],{"mathvariant":145},[63,74282,74283],{"encoding":65},"\\inf_{T}\\ \\int_{\\mathcal X} c\\bigl(x,T(x)\\bigr)\\,d\\mu(x)\n\\quad\\text{s.t.}\\quad T_\\#\\mu=\\nu .",[39,74285,74287,74519],{"className":74286,"ariaHidden":70},[69],[39,74288,74290,74294,74348,74351,74354,74397,74400,74403,74409,74412,74415,74418,74421,74424,74427,74430,74436,74439,74442,74445,74448,74451,74454,74457,74464,74467,74507,74510,74513,74516],{"className":74289},[74],[39,74291],{"className":74292,"style":74293},[78],"height:2.2719em;vertical-align:-0.9119em;",[39,74295,74297],{"className":74296},[1744,3294],[39,74298,74300,74339],{"className":74299},[196,649],[39,74301,74303,74336],{"className":74302},[200],[39,74304,74306,74321],{"className":74305,"style":28824},[204],[39,74307,74309,74312],{"style":74308},"top:-2.3557em;margin-left:0em;",[39,74310],{"className":74311,"style":935},[212],[39,74313,74315],{"className":74314},[217,218,219,220],[39,74316,74318],{"className":74317},[83,220],[39,74319,11772],{"className":74320,"style":178},[83,84,220],[39,74322,74323,74326],{"style":21599},[39,74324],{"className":74325,"style":935},[212],[39,74327,74328],{},[39,74329,74331,74332],{"className":74330},[1744],"in",[39,74333,74335],{"style":74334},"margin-right:0.0778em;","f",[39,74337,673],{"className":74338},[672],[39,74340,74342],{"className":74341},[200],[39,74343,74346],{"className":74344,"style":74345},[204],"height:0.7443em;",[39,74347],{},[39,74349,74223],{"className":74350},[230],[39,74352],{"className":74353,"style":256},[230],[39,74355,74357,74360],{"className":74356},[1744],[39,74358,69087],{"className":74359,"style":69380},[1744,3339,3340],[39,74361,74363],{"className":74362},[192],[39,74364,74366,74389],{"className":74365},[196,649],[39,74367,74369,74386],{"className":74368},[200],[39,74370,74372],{"className":74371,"style":69393},[204],[39,74373,74374,74377],{"style":69396},[39,74375],{"className":74376,"style":213},[212],[39,74378,74380],{"className":74379},[217,218,219,220],[39,74381,74383],{"className":74382},[83,220],[39,74384,29420],{"className":74385,"style":73686},[83,641,220],[39,74387,673],{"className":74388},[672],[39,74390,74392],{"className":74391},[200],[39,74393,74395],{"className":74394,"style":69415},[204],[39,74396],{},[39,74398],{"className":74399,"style":256},[230],[39,74401,24077],{"className":74402},[83,84],[39,74404,74406],{"className":74405},[182],[39,74407,136],{"className":74408},[18520,4009],[39,74410,1755],{"className":74411},[83,84],[39,74413,155],{"className":74414},[252],[39,74416],{"className":74417,"style":256},[230],[39,74419,11772],{"className":74420,"style":178},[83,84],[39,74422,136],{"className":74423},[182],[39,74425,1755],{"className":74426},[83,84],[39,74428,161],{"className":74429},[263],[39,74431,74433],{"className":74432},[263],[39,74434,161],{"className":74435},[18520,4009],[39,74437],{"className":74438,"style":256},[230],[39,74440,29425],{"className":74441},[83,84],[39,74443,73578],{"className":74444},[83,84],[39,74446,136],{"className":74447},[182],[39,74449,1755],{"className":74450},[83,84],[39,74452,161],{"className":74453},[263],[39,74455],{"className":74456,"style":7240},[230],[39,74458,74461],{"className":74459},[83,74460],"text",[39,74462,74264],{"className":74463},[83],[39,74465],{"className":74466,"style":7240},[230],[39,74468,74470,74473],{"className":74469},[83],[39,74471,11772],{"className":74472,"style":178},[83,84],[39,74474,74476],{"className":74475},[192],[39,74477,74479,74499],{"className":74478},[196,649],[39,74480,74482,74496],{"className":74481},[200],[39,74483,74485],{"className":74484,"style":3419},[204],[39,74486,74487,74490],{"style":73765},[39,74488],{"className":74489,"style":213},[212],[39,74491,74493],{"className":74492},[217,218,219,220],[39,74494,73726],{"className":74495},[83,220],[39,74497,673],{"className":74498},[672],[39,74500,74502],{"className":74501},[200],[39,74503,74505],{"className":74504,"style":35396},[204],[39,74506],{},[39,74508,73578],{"className":74509},[83,84],[39,74511],{"className":74512,"style":231},[230],[39,74514,793],{"className":74515},[235],[39,74517],{"className":74518,"style":231},[230],[39,74520,74522,74525,74528],{"className":74521},[74],[39,74523],{"className":74524,"style":290},[78],[39,74526,73609],{"className":74527,"style":73625},[83,84],[39,74529,2903],{"className":74530},[83],[11,74532,74533,74534,74562,74563,74591,74592,74620,74621,74717,74718,1607,75030,75074,75075,75103,75104,75132],{},"这个提法有一个硬伤：",[39,74535,74537,74550],{"className":74536,"translate":43},[42],[39,74538,74540],{"className":74539},[47],[49,74541,74542],{"xmlns":51},[53,74543,74544,74548],{},[56,74545,74546],{},[59,74547,11772],{},[63,74549,11772],{"encoding":65},[39,74551,74553],{"className":74552,"ariaHidden":70},[69],[39,74554,74556,74559],{"className":74555},[74],[39,74557],{"className":74558,"style":79},[78],[39,74560,11772],{"className":74561,"style":178},[83,84]," 必须是确定性的，一个产地只能有一条固定去向。如果 ",[39,74564,74566,74579],{"className":74565,"translate":43},[42],[39,74567,74569],{"className":74568},[47],[49,74570,74571],{"xmlns":51},[53,74572,74573,74577],{},[56,74574,74575],{},[59,74576,73578],{},[63,74578,73581],{"encoding":65},[39,74580,74582],{"className":74581,"ariaHidden":70},[69],[39,74583,74585,74588],{"className":74584},[74],[39,74586],{"className":74587,"style":2654},[78],[39,74589,73578],{"className":74590},[83,84]," 在某个点上有原子，而 ",[39,74593,74595,74608],{"className":74594,"translate":43},[42],[39,74596,74598],{"className":74597},[47],[49,74599,74600],{"xmlns":51},[53,74601,74602,74606],{},[56,74603,74604],{},[59,74605,73609],{},[63,74607,73612],{"encoding":65},[39,74609,74611],{"className":74610,"ariaHidden":70},[69],[39,74612,74614,74617],{"className":74613},[74],[39,74615],{"className":74616,"style":290},[78],[39,74618,73609],{"className":74619,"style":73625},[83,84]," 在那里要求把质量分到两个地方，就无解。举个最小的例子：全部质量集中在甲一个点上，",[39,74622,74624,74647],{"className":74623,"translate":43},[42],[39,74625,74627],{"className":74626},[47],[49,74628,74629],{"xmlns":51},[53,74630,74631,74644],{},[56,74632,74633,74635,74637],{},[59,74634,73578],{},[133,74636,793],{},[613,74638,74639,74642],{},[59,74640,74641],{},"δ",[539,74643,541],{},[63,74645,74646],{"encoding":65},"\\mu=\\delta_0",[39,74648,74650,74668],{"className":74649,"ariaHidden":70},[69],[39,74651,74653,74656,74659,74662,74665],{"className":74652},[74],[39,74654],{"className":74655,"style":2654},[78],[39,74657,73578],{"className":74658},[83,84],[39,74660],{"className":74661,"style":231},[230],[39,74663,793],{"className":74664},[235],[39,74666],{"className":74667,"style":231},[230],[39,74669,74671,74675],{"className":74670},[74],[39,74672],{"className":74673,"style":74674},[78],"height:0.8444em;vertical-align:-0.15em;",[39,74676,74678,74682],{"className":74677},[83],[39,74679,74641],{"className":74680,"style":74681},[83,84],"margin-right:0.0379em;",[39,74683,74685],{"className":74684},[192],[39,74686,74688,74709],{"className":74687},[196,649],[39,74689,74691,74706],{"className":74690},[200],[39,74692,74694],{"className":74693,"style":963},[204],[39,74695,74697,74700],{"style":74696},"top:-2.55em;margin-left:-0.0379em;margin-right:0.05em;",[39,74698],{"className":74699,"style":213},[212],[39,74701,74703],{"className":74702},[217,218,219,220],[39,74704,541],{"className":74705},[83,220],[39,74707,673],{"className":74708},[672],[39,74710,74712],{"className":74711},[200],[39,74713,74715],{"className":74714,"style":680},[204],[39,74716],{},"；而两个厂各要一半，",[39,74719,74721,74767],{"className":74720,"translate":43},[42],[39,74722,74724],{"className":74723},[47],[49,74725,74726],{"xmlns":51},[53,74727,74728,74764],{},[56,74729,74730,74732,74734,74742,74748,74750,74758],{},[59,74731,73609],{},[133,74733,793],{},[770,74735,74736],{"scriptlevel":541,"displaystyle":135},[5362,74737,74738,74740],{},[539,74739,546],{},[539,74741,17359],{},[613,74743,74744,74746],{},[59,74745,74641],{},[539,74747,546],{},[133,74749,788],{},[770,74751,74752],{"scriptlevel":541,"displaystyle":135},[5362,74753,74754,74756],{},[539,74755,546],{},[539,74757,17359],{},[613,74759,74760,74762],{},[59,74761,74641],{},[539,74763,17359],{},[63,74765,74766],{"encoding":65},"\\nu=\\tfrac12\\delta_1+\\tfrac12\\delta_2",[39,74768,74770,74788,74916],{"className":74769,"ariaHidden":70},[69],[39,74771,74773,74776,74779,74782,74785],{"className":74772},[74],[39,74774],{"className":74775,"style":290},[78],[39,74777,73609],{"className":74778,"style":73625},[83,84],[39,74780],{"className":74781,"style":231},[230],[39,74783,793],{"className":74784},[235],[39,74786],{"className":74787,"style":231},[230],[39,74789,74791,74795,74867,74907,74910,74913],{"className":74790},[74],[39,74792],{"className":74793,"style":74794},[78],"height:1.1901em;vertical-align:-0.345em;",[39,74796,74798,74801,74864],{"className":74797},[83],[39,74799],{"className":74800},[182,5463],[39,74802,74804],{"className":74803},[5362],[39,74805,74807,74855],{"className":74806},[196,649],[39,74808,74810,74852],{"className":74809},[200],[39,74811,74814,74829,74837],{"className":74812,"style":74813},[204],"height:0.8451em;",[39,74815,74817,74820],{"style":74816},"top:-2.655em;",[39,74818],{"className":74819,"style":935},[212],[39,74821,74823],{"className":74822},[217,218,219,220],[39,74824,74826],{"className":74825},[83,220],[39,74827,17359],{"className":74828},[83,220],[39,74830,74831,74834],{"style":5503},[39,74832],{"className":74833,"style":935},[212],[39,74835],{"className":74836,"style":5511},[5510],[39,74838,74840,74843],{"style":74839},"top:-3.394em;",[39,74841],{"className":74842,"style":935},[212],[39,74844,74846],{"className":74845},[217,218,219,220],[39,74847,74849],{"className":74848},[83,220],[39,74850,546],{"className":74851},[83,220],[39,74853,673],{"className":74854},[672],[39,74856,74858],{"className":74857},[200],[39,74859,74862],{"className":74860,"style":74861},[204],"height:0.345em;",[39,74863],{},[39,74865],{"className":74866},[263,5463],[39,74868,74870,74873],{"className":74869},[83],[39,74871,74641],{"className":74872,"style":74681},[83,84],[39,74874,74876],{"className":74875},[192],[39,74877,74879,74899],{"className":74878},[196,649],[39,74880,74882,74896],{"className":74881},[200],[39,74883,74885],{"className":74884,"style":963},[204],[39,74886,74887,74890],{"style":74696},[39,74888],{"className":74889,"style":213},[212],[39,74891,74893],{"className":74892},[217,218,219,220],[39,74894,546],{"className":74895},[83,220],[39,74897,673],{"className":74898},[672],[39,74900,74902],{"className":74901},[200],[39,74903,74905],{"className":74904,"style":680},[204],[39,74906],{},[39,74908],{"className":74909,"style":461},[230],[39,74911,788],{"className":74912},[465],[39,74914],{"className":74915,"style":461},[230],[39,74917,74919,74922,74990],{"className":74918},[74],[39,74920],{"className":74921,"style":74794},[78],[39,74923,74925,74928,74987],{"className":74924},[83],[39,74926],{"className":74927},[182,5463],[39,74929,74931],{"className":74930},[5362],[39,74932,74934,74979],{"className":74933},[196,649],[39,74935,74937,74976],{"className":74936},[200],[39,74938,74940,74954,74962],{"className":74939,"style":74813},[204],[39,74941,74942,74945],{"style":74816},[39,74943],{"className":74944,"style":935},[212],[39,74946,74948],{"className":74947},[217,218,219,220],[39,74949,74951],{"className":74950},[83,220],[39,74952,17359],{"className":74953},[83,220],[39,74955,74956,74959],{"style":5503},[39,74957],{"className":74958,"style":935},[212],[39,74960],{"className":74961,"style":5511},[5510],[39,74963,74964,74967],{"style":74839},[39,74965],{"className":74966,"style":935},[212],[39,74968,74970],{"className":74969},[217,218,219,220],[39,74971,74973],{"className":74972},[83,220],[39,74974,546],{"className":74975},[83,220],[39,74977,673],{"className":74978},[672],[39,74980,74982],{"className":74981},[200],[39,74983,74985],{"className":74984,"style":74861},[204],[39,74986],{},[39,74988],{"className":74989},[263,5463],[39,74991,74993,74996],{"className":74992},[83],[39,74994,74641],{"className":74995,"style":74681},[83,84],[39,74997,74999],{"className":74998},[192],[39,75000,75002,75022],{"className":75001},[196,649],[39,75003,75005,75019],{"className":75004},[200],[39,75006,75008],{"className":75007,"style":963},[204],[39,75009,75010,75013],{"style":74696},[39,75011],{"className":75012,"style":213},[212],[39,75014,75016],{"className":75015},[217,218,219,220],[39,75017,17359],{"className":75018},[83,220],[39,75020,673],{"className":75021},[672],[39,75023,75025],{"className":75024},[200],[39,75026,75028],{"className":75027,"style":680},[204],[39,75029],{},[39,75031,75033,75053],{"className":75032,"translate":43},[42],[39,75034,75036],{"className":75035},[47],[49,75037,75038],{"xmlns":51},[53,75039,75040,75050],{},[56,75041,75042,75044,75046,75048],{},[59,75043,11772],{},[133,75045,136],{"stretchy":135},[539,75047,541],{},[133,75049,161],{"stretchy":135},[63,75051,75052],{"encoding":65},"T(0)",[39,75054,75056],{"className":75055,"ariaHidden":70},[69],[39,75057,75059,75062,75065,75068,75071],{"className":75058},[74],[39,75060],{"className":75061,"style":245},[78],[39,75063,11772],{"className":75064,"style":178},[83,84],[39,75066,136],{"className":75067},[182],[39,75069,541],{"className":75070},[83],[39,75072,161],{"className":75073},[263]," 必须同时等于 ",[39,75076,75078,75091],{"className":75077,"translate":43},[42],[39,75079,75081],{"className":75080},[47],[49,75082,75083],{"xmlns":51},[53,75084,75085,75089],{},[56,75086,75087],{},[539,75088,546],{},[63,75090,546],{"encoding":65},[39,75092,75094],{"className":75093,"ariaHidden":70},[69],[39,75095,75097,75100],{"className":75096},[74],[39,75098],{"className":75099,"style":2473},[78],[39,75101,546],{"className":75102},[83]," 和 ",[39,75105,75107,75120],{"className":75106,"translate":43},[42],[39,75108,75110],{"className":75109},[47],[49,75111,75112],{"xmlns":51},[53,75113,75114,75118],{},[56,75115,75116],{},[539,75117,17359],{},[63,75119,17359],{"encoding":65},[39,75121,75123],{"className":75122,"ariaHidden":70},[69],[39,75124,75126,75129],{"className":75125},[74],[39,75127],{"className":75128,"style":2473},[78],[39,75130,17359],{"className":75131},[83],"，函数做不到。",[11,75134,75135],{},"可这件事在现实中天天发生：一辆车分两个厂送货而已。Kantorovich 在 1940 年代的办法是把映射换成联合分布。令",[39,75137,75139],{"className":75138,"translate":43},[746],[39,75140,75142,75257],{"className":75141,"translate":43},[42],[39,75143,75145],{"className":75144},[47],[49,75146,75147],{"xmlns":51,"display":755},[53,75148,75149,75254],{},[56,75150,75151,75154,75156,75158,75160,75162,75164,75166,75168,75170,75172,75174,75176,75178,75180,75182,75184,75186,75188,75190,75192,75206,75212,75214,75216,75218,75220,75222,75224,75238,75244,75246,75248,75250,75252],{},[59,75152,75153],{"mathvariant":145},"Π",[133,75155,136],{"stretchy":135},[59,75157,73578],{},[133,75159,155],{"separator":70},[59,75161,73609],{},[133,75163,161],{"stretchy":135},[133,75165,793],{},[133,75167,12951],{"fence":70,"stretchy":70,"minsize":65017,"maxsize":65017},[59,75169,2098],{},[133,75171,534],{},[59,75173,131],{"mathvariant":617},[133,75175,136],{"stretchy":135},[59,75177,29420],{"mathvariant":617},[133,75179,409],{},[59,75181,73652],{"mathvariant":617},[133,75183,161],{"stretchy":135},[35841,75185,74223],{},[133,75187,404],{},[35841,75189,74223],{},[133,75191,136],{"stretchy":135},[613,75193,75194,75204],{},[56,75195,75196,75198,75200,75202],{},[59,75197,11],{"mathvariant":145},[59,75199,401],{"mathvariant":145},[59,75201,35010],{"mathvariant":145},[59,75203,13446],{"mathvariant":145},[59,75205,29420],{"mathvariant":617},[613,75207,75208,75210],{},[133,75209,161],{"stretchy":135},[59,75211,73726],{"mathvariant":145},[59,75213,2098],{},[133,75215,793],{},[59,75217,73578],{},[133,75219,155],{"separator":70},[35841,75221,74223],{},[133,75223,136],{"stretchy":135},[613,75225,75226,75236],{},[56,75227,75228,75230,75232,75234],{},[59,75229,11],{"mathvariant":145},[59,75231,401],{"mathvariant":145},[59,75233,35010],{"mathvariant":145},[59,75235,13446],{"mathvariant":145},[59,75237,73652],{"mathvariant":617},[613,75239,75240,75242],{},[133,75241,161],{"stretchy":135},[59,75243,73726],{"mathvariant":145},[59,75245,2098],{},[133,75247,793],{},[59,75249,73609],{},[133,75251,12965],{"fence":70,"stretchy":70,"minsize":65017,"maxsize":65017},[133,75253,155],{"separator":70},[63,75255,75256],{"encoding":65},"\\Pi(\\mu,\\nu)=\\Bigl\\{\\pi\\in\\mathcal P(\\mathcal X\\times\\mathcal Y)\\ :\\ (\\mathrm{proj}_{\\mathcal X})_\\#\\pi=\\mu,\\ (\\mathrm{proj}_{\\mathcal 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读作“从 ",[39,75681,75683,75696],{"className":75682,"translate":43},[42],[39,75684,75686],{"className":75685},[47],[49,75687,75688],{"xmlns":51},[53,75689,75690,75694],{},[56,75691,75692],{},[59,75693,1755],{},[63,75695,1755],{"encoding":65},[39,75697,75699],{"className":75698,"ariaHidden":70},[69],[39,75700,75702,75705],{"className":75701},[74],[39,75703],{"className":75704,"style":290},[78],[39,75706,1755],{"className":75707},[83,84]," 搬到 ",[39,75710,75712,75725],{"className":75711,"translate":43},[42],[39,75713,75715],{"className":75714},[47],[49,75716,75717],{"xmlns":51},[53,75718,75719,75723],{},[56,75720,75721],{},[59,75722,29625],{},[63,75724,29625],{"encoding":65},[39,75726,75728],{"className":75727,"ariaHidden":70},[69],[39,75729,75731,75734],{"className":75730},[74],[39,75732],{"className":75733,"style":2654},[78],[39,75735,29625],{"className":75736,"style":2114},[83,84]," 的那部分质量”，于是",[39,75739,75741],{"className":75740,"translate":43},[746],[39,75742,75744,75842],{"className":75743,"translate":43},[42],[39,75745,75747],{"className":75746},[47],[49,75748,75749],{"xmlns":51,"display":755},[53,75750,75751,75839],{},[56,75752,75753,75759,75761,75763,75765,75767,75769,75771,75797,75809,75811,75813,75815,75817,75819,75821,75823,75825,75827,75829,75831,75833,75835,75837],{},[613,75754,75755,75757],{},[59,75756,23703],{},[59,75758,24077],{},[133,75760,136],{"stretchy":135},[59,75762,73578],{},[133,75764,155],{"separator":70},[59,75766,73609],{},[133,75768,161],{"stretchy":135},[133,75770,793],{},[6556,75772,75773,75779],{},[56,75774,75775,75777],{},[59,75776,74216],{},[133,75778,1690],{},[56,75780,75781,75783,75785,75787,75789,75791,75793,75795],{},[59,75782,2098],{},[133,75784,534],{},[59,75786,75153],{"mathvariant":145},[133,75788,136],{"stretchy":135},[59,75790,73578],{},[133,75792,155],{"separator":70},[59,75794,73609],{},[133,75796,161],{"stretchy":135},[613,75798,75799,75801],{},[133,75800,69087],{},[56,75802,75803,75805,75807],{},[59,75804,29420],{"mathvariant":617},[133,75806,409],{},[59,75808,73652],{"mathvariant":617},[59,75810,24077],{},[133,75812,136],{"stretchy":135},[59,75814,1755],{},[133,75816,155],{"separator":70},[59,75818,29625],{},[133,75820,161],{"stretchy":135},[35841,75822,35843],{},[59,75824,29425],{},[59,75826,2098],{},[133,75828,136],{"stretchy":135},[59,75830,1755],{},[133,75832,155],{"separator":70},[59,75834,29625],{},[133,75836,161],{"stretchy":135},[59,75838,2903],{"mathvariant":145},[63,75840,75841],{"encoding":65},"W_c(\\mu,\\nu)=\\inf_{\\pi\\in\\Pi(\\mu,\\nu)}\\int_{\\mathcal 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就是那张运输表。",[11,76137,76138,76139,76276,76277,76305],{},"取 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是度量），则",[39,76307,76309],{"className":76308,"translate":43},[746],[39,76310,76312,76408],{"className":76311,"translate":43},[42],[39,76313,76315],{"className":76314},[47],[49,76316,76317],{"xmlns":51,"display":755},[53,76318,76319,76405],{},[56,76320,76321,76327,76329,76331,76333,76335,76337,76339],{},[613,76322,76323,76325],{},[59,76324,23703],{},[59,76326,11],{},[133,76328,136],{"stretchy":135},[59,76330,73578],{},[133,76332,155],{"separator":70},[59,76334,73609],{},[133,76336,161],{"stretchy":135},[133,76338,793],{},[138,76340,76341,76397],{},[56,76342,76343,76345,76371,76373,76375,76377,76379,76381,76383,76389,76391,76393,76395],{},[133,76344,136],{"fence":70},[6556,76346,76347,76353],{},[56,76348,76349,76351],{},[59,76350,74216],{},[133,76352,1690],{},[56,76354,76355,76357,76359,76361,76363,76365,76367,76369],{},[59,76356,2098],{},[133,76358,534],{},[59,76360,75153],{"mathvariant":145},[133,76362,136],{"stretchy":135},[59,76364,73578],{},[133,76366,155],{"separator":70},[59,76368,73609],{},[133,76370,161],{"stretchy":135},[133,76372,69087],{},[59,76374,29425],{},[133,76376,136],{"stretchy":135},[59,76378,1755],{},[133,76380,155],{"separator":70},[59,76382,29625],{},[138,76384,76385,76387],{},[133,76386,161],{"stretchy":135},[59,76388,11],{},[35841,76390,35843],{},[59,76392,29425],{},[59,76394,2098],{},[133,76396,161],{"fence":70},[56,76398,76399,76401,76403],{},[539,76400,546],{},[59,76402,45691],{"mathvariant":145},[59,76404,11],{},[63,76406,76407],{"encoding":65},"W_p(\\mu,\\nu)=\\left(\\inf_{\\pi\\in\\Pi(\\mu,\\nu)}\\int 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是个有限的数，而 ",[39,77172,77174,77218],{"className":77173,"translate":43},[42],[39,77175,77177],{"className":77176},[47],[49,77178,77179],{"xmlns":51},[53,77180,77181,77215],{},[56,77182,77183,77191,77193,77199,77201,77207,77209,77211,77213],{},[56,77184,77185,77188],{},[59,77186,77187],{"mathvariant":145},"K",[59,77189,77190],{"mathvariant":145},"L",[133,77192,136],{"stretchy":135},[613,77194,77195,77197],{},[59,77196,74641],{},[59,77198,1755],{},[59,77200,15389],{"mathvariant":145},[613,77202,77203,77205],{},[59,77204,74641],{},[59,77206,29625],{},[133,77208,161],{"stretchy":135},[133,77210,793],{},[133,77212,788],{},[59,77214,1696],{"mathvariant":145},[63,77216,77217],{"encoding":65},"\\mathrm{KL}(\\delta_x\\|\\delta_y)=+\\infty",[39,77219,77221,77332],{"className":77220,"ariaHidden":70},[69],[39,77222,77224,77227,77234,77237,77277,77280,77320,77323,77326,77329],{"className":77223},[74],[39,77225],{"className":77226,"style":41560},[78],[39,77228,77230],{"className":77229},[83],[39,77231,77233],{"className":77232},[83,35077],"KL",[39,77235,136],{"className":77236},[182],[39,77238,77240,77243],{"className":77239},[83],[39,77241,74641],{"className":77242,"style":74681},[83,84],[39,77244,77246],{"className":77245},[192],[39,77247,77249,77269],{"className":77248},[196,649],[39,77250,77252,77266],{"className":77251},[200],[39,77253,77255],{"className":77254,"style":1728},[204],[39,77256,77257,77260],{"style":74696},[39,77258],{"className":77259,"style":213},[212],[39,77261,77263],{"className":77262},[217,218,219,220],[39,77264,1755],{"className":77265},[83,84,220],[39,77267,673],{"className":77268},[672],[39,77270,77272],{"className":77271},[200],[39,77273,77275],{"className":77274,"style":680},[204],[39,77276],{},[39,77278,15389],{"className":77279},[83],[39,77281,77283,77286],{"className":77282},[83],[39,77284,74641],{"className":77285,"style":74681},[83,84],[39,77287,77289],{"className":77288},[192],[39,77290,77292,77312],{"className":77291},[196,649],[39,77293,77295,77309],{"className":77294},[200],[39,77296,77298],{"className":77297,"style":1728},[204],[39,77299,77300,77303],{"style":74696},[39,77301],{"className":77302,"style":213},[212],[39,77304,77306],{"className":77305},[217,218,219,220],[39,77307,29625],{"className":77308,"style":2114},[83,84,220],[39,77310,673],{"className":77311},[672],[39,77313,77315],{"className":77314},[200],[39,77316,77318],{"className":77317,"style":35396},[204],[39,77319],{},[39,77321,161],{"className":77322},[263],[39,77324],{"className":77325,"style":231},[230],[39,77327,793],{"className":77328},[235],[39,77330],{"className":77331,"style":231},[230],[39,77333,77335,77338,77341],{"className":77334},[74],[39,77336],{"className":77337,"style":73043},[78],[39,77339,788],{"className":77340},[83],[39,77342,1696],{"className":77343},[83],"。支撑不交的时候，KL 要么发散要么给出零梯度，",[39,77346,77348,77361],{"className":77347,"translate":43},[42],[39,77349,77351],{"className":77350},[47],[49,77352,77353],{"xmlns":51},[53,77354,77355,77359],{},[56,77356,77357],{},[59,77358,23703],{},[63,77360,23703],{"encoding":65},[39,77362,77364],{"className":77363,"ariaHidden":70},[69],[39,77365,77367,77370],{"className":77366},[74],[39,77368],{"className":77369,"style":79},[78],[39,77371,23703],{"className":77372,"style":178},[83,84]," 却仍然告诉你“隔了多远”。",[11,77375,77376,77377,77405,77406,13349,77435,38249,77505,77533],{},"平移的例子更直白。把 ",[39,77378,77380,77393],{"className":77379,"translate":43},[42],[39,77381,77383],{"className":77382},[47],[49,77384,77385],{"xmlns":51},[53,77386,77387,77391],{},[56,77388,77389],{},[59,77390,73578],{},[63,77392,73581],{"encoding":65},[39,77394,77396],{"className":77395,"ariaHidden":70},[69],[39,77397,77399,77402],{"className":77398},[74],[39,77400],{"className":77401,"style":2654},[78],[39,77403,73578],{"className":77404},[83,84]," 整体挪 ",[39,77407,77409,77423],{"className":77408,"translate":43},[42],[39,77410,77412],{"className":77411},[47],[49,77413,77414],{"xmlns":51},[53,77415,77416,77420],{},[56,77417,77418],{},[59,77419,58009],{},[63,77421,77422],{"encoding":65},"\\varepsilon",[39,77424,77426],{"className":77425,"ariaHidden":70},[69],[39,77427,77429,77432],{"className":77428},[74],[39,77430],{"className":77431,"style":290},[78],[39,77433,58009],{"className":77434},[83,84],[39,77436,77438,77456],{"className":77437,"translate":43},[42],[39,77439,77441],{"className":77440},[47],[49,77442,77443],{"xmlns":51},[53,77444,77445,77453],{},[56,77446,77447],{},[613,77448,77449,77451],{},[59,77450,23703],{},[59,77452,11],{},[63,77454,77455],{"encoding":65},"W_p",[39,77457,77459],{"className":77458,"ariaHidden":70},[69],[39,77460,77462,77465],{"className":77461},[74],[39,77463],{"className":77464,"style":35350},[78],[39,77466,77468,77471],{"className":77467},[83],[39,77469,23703],{"className":77470,"style":178},[83,84],[39,77472,77474],{"className":77473},[192],[39,77475,77477,77497],{"className":77476},[196,649],[39,77478,77480,77494],{"className":77479},[200],[39,77481,77483],{"className":77482,"style":1728},[204],[39,77484,77485,77488],{"style":73765},[39,77486],{"className":77487,"style":213},[212],[39,77489,77491],{"className":77490},[217,218,219,220],[39,77492,11],{"className":77493},[83,84,220],[39,77495,673],{"className":77496},[672],[39,77498,77500],{"className":77499},[200],[39,77501,77503],{"className":77502,"style":35396},[204],[39,77504],{},[39,77506,77508,77521],{"className":77507,"translate":43},[42],[39,77509,77511],{"className":77510},[47],[49,77512,77513],{"xmlns":51},[53,77514,77515,77519],{},[56,77516,77517],{},[59,77518,58009],{},[63,77520,77422],{"encoding":65},[39,77522,77524],{"className":77523,"ariaHidden":70},[69],[39,77525,77527,77530],{"className":77526},[74],[39,77528],{"className":77529,"style":290},[78],[39,77531,58009],{"className":77532},[83,84]," 量级，而",[39,77535,77537],{"className":77536,"translate":43},[746],[39,77538,77540,77603],{"className":77539,"translate":43},[42],[39,77541,77543],{"className":77542},[47],[49,77544,77545],{"xmlns":51,"display":755},[53,77546,77547,77600],{},[56,77548,77549,77555,77557,77560,77562,77564,77566,77568,77570,77572,77574,77576,77578,77580,77582,77584,77586,77588,77598],{},[56,77550,77551,77553],{},[59,77552,77187],{"mathvariant":145},[59,77554,77190],{"mathvariant":145},[133,77556,136],{"fence":70,"stretchy":70,"minsize":73875,"maxsize":73875},[59,77558,77559],{"mathvariant":617},"N",[133,77561,136],{"stretchy":135},[539,77563,541],{},[133,77565,155],{"separator":70},[539,77567,546],{},[133,77569,161],{"stretchy":135},[59,77571,15389],{"mathvariant":145},[59,77573,77559],{"mathvariant":617},[133,77575,136],{"stretchy":135},[59,77577,58009],{},[133,77579,155],{"separator":70},[539,77581,546],{},[133,77583,161],{"stretchy":135},[133,77585,161],{"fence":70,"stretchy":70,"minsize":73875,"maxsize":73875},[133,77587,793],{},[5362,77589,77590,77596],{},[138,77591,77592,77594],{},[59,77593,58009],{},[539,77595,17359],{},[539,77597,17359],{},[133,77599,155],{"separator":70},[63,77601,77602],{"encoding":65},"\\mathrm{KL}\\bigl(\\mathcal N(0,1)\\|\\mathcal N(\\varepsilon,1)\\bigr)=\\frac{\\varepsilon^2}{2},",[39,77604,77606,77685],{"className":77605,"ariaHidden":70},[69],[39,77607,77609,77612,77618,77624,77628,77631,77634,77637,77640,77643,77646,77649,77652,77655,77658,77661,77664,77667,77670,77676,77679,77682],{"className":77608},[74],[39,77610],{"className":77611,"style":908},[78],[39,77613,77615],{"className":77614},[83],[39,77616,77233],{"className":77617},[83,35077],[39,77619,77621],{"className":77620},[182],[39,77622,136],{"className":77623},[18520,4009],[39,77625,77559],{"className":77626,"style":77627},[83,641],"margin-right:0.1474em;",[39,77629,136],{"className":77630},[182],[39,77632,541],{"className":77633},[83],[39,77635,155],{"className":77636},[252],[39,77638],{"className":77639,"style":256},[230],[39,77641,546],{"className":77642},[83],[39,77644,161],{"className":77645},[263],[39,77647,15389],{"className":77648},[83],[39,77650,77559],{"className":77651,"style":77627},[83,641],[39,77653,136],{"className":77654},[182],[39,77656,58009],{"className":77657},[83,84],[39,77659,155],{"className":77660},[252],[39,77662],{"className":77663,"style":256},[230],[39,77665,546],{"className":77666},[83],[39,77668,161],{"className":77669},[263],[39,77671,77673],{"className":77672},[263],[39,77674,161],{"className":77675},[18520,4009],[39,77677],{"className":77678,"style":231},[230],[39,77680,793],{"className":77681},[235],[39,77683],{"className":77684,"style":231},[230],[39,77686,77688,77692,77781],{"className":77687},[74],[39,77689],{"className":77690,"style":77691},[78],"height:2.1771em;vertical-align:-0.686em;",[39,77693,77695,77698,77778],{"className":77694},[83],[39,77696],{"className":77697},[182,5463],[39,77699,77701],{"className":77700},[5362],[39,77702,77704,77770],{"className":77703},[196,649],[39,77705,77707,77767],{"className":77706},[200],[39,77708,77711,77722,77730],{"className":77709,"style":77710},[204],"height:1.4911em;",[39,77712,77713,77716],{"style":5479},[39,77714],{"className":77715,"style":935},[212],[39,77717,77719],{"className":77718},[83],[39,77720,17359],{"className":77721},[83],[39,77723,77724,77727],{"style":5503},[39,77725],{"className":77726,"style":935},[212],[39,77728],{"className":77729,"style":5511},[5510],[39,77731,77732,77735],{"style":5514},[39,77733],{"className":77734,"style":935},[212],[39,77736,77738],{"className":77737},[83],[39,77739,77741,77744],{"className":77740},[83],[39,77742,58009],{"className":77743},[83,84],[39,77745,77747],{"className":77746},[192],[39,77748,77750],{"className":77749},[196],[39,77751,77753],{"className":77752},[200],[39,77754,77756],{"className":77755,"style":19710},[204],[39,77757,77758,77761],{"style":208},[39,77759],{"className":77760,"style":213},[212],[39,77762,77764],{"className":77763},[217,218,219,220],[39,77765,17359],{"className":77766},[83,220],[39,77768,673],{"className":77769},[672],[39,77771,77773],{"className":77772},[200],[39,77774,77776],{"className":77775,"style":19487},[204],[39,77777],{},[39,77779],{"className":77780},[263,5463],[39,77782,155],{"className":77783},[252],[11,77785,77786,77787,77815],{},"是二阶的。所以小位移时 KL 的梯度趋近于零，",[39,77788,77790,77803],{"className":77789,"translate":43},[42],[39,77791,77793],{"className":77792},[47],[49,77794,77795],{"xmlns":51},[53,77796,77797,77801],{},[56,77798,77799],{},[59,77800,23703],{},[63,77802,23703],{"encoding":65},[39,77804,77806],{"className":77805,"ariaHidden":70},[69],[39,77807,77809,77812],{"className":77808},[74],[39,77810],{"className":77811,"style":79},[78],[39,77813,23703],{"className":77814,"style":178},[83,84]," 的梯度是常数。生成模型里那句“Wasserstein 提供不饱和梯度”，准确的数学内容就是这一行。",[11,77817,77818,77819,77889,77890,77918],{},"顺带说一句，计算机视觉里管它叫 Earth Mover's Distance，数学里叫 Wasserstein 距离，是同一个东西；EMD 通常特指 ",[39,77820,77822,77840],{"className":77821,"translate":43},[42],[39,77823,77825],{"className":77824},[47],[49,77826,77827],{"xmlns":51},[53,77828,77829,77837],{},[56,77830,77831],{},[613,77832,77833,77835],{},[59,77834,23703],{},[539,77836,546],{},[63,77838,77839],{"encoding":65},"W_1",[39,77841,77843],{"className":77842,"ariaHidden":70},[69],[39,77844,77846,77849],{"className":77845},[74],[39,77847],{"className":77848,"style":634},[78],[39,77850,77852,77855],{"className":77851},[83],[39,77853,23703],{"className":77854,"style":178},[83,84],[39,77856,77858],{"className":77857},[192],[39,77859,77861,77881],{"className":77860},[196,649],[39,77862,77864,77878],{"className":77863},[200],[39,77865,77867],{"className":77866,"style":963},[204],[39,77868,77869,77872],{"style":73765},[39,77870],{"className":77871,"style":213},[212],[39,77873,77875],{"className":77874},[217,218,219,220],[39,77876,546],{"className":77877},[83,220],[39,77879,673],{"className":77880},[672],[39,77882,77884],{"className":77883},[200],[39,77885,77887],{"className":77886,"style":680},[204],[39,77888],{},"，或者指带整数质量的离散版本。把一张 28×28 的手写数字的灰度值当成质量，",[39,77891,77893,77906],{"className":77892,"translate":43},[42],[39,77894,77896],{"className":77895},[47],[49,77897,77898],{"xmlns":51},[53,77899,77900,77904],{},[56,77901,77902],{},[59,77903,23703],{},[63,77905,23703],{"encoding":65},[39,77907,77909],{"className":77908,"ariaHidden":70},[69],[39,77910,77912,77915],{"className":77911},[74],[39,77913],{"className":77914,"style":79},[78],[39,77916,23703],{"className":77917,"style":178},[83,84]," 就是“把这张画改写成那张画的最小工作量”，我最初就是靠这个画面理解它的。",[21,77920,77921],{"id":77921},"对偶",[11,77923,77924],{},"回到那道运输题。我算出最低运费是 5 块。如果有人说他能做到 4 块，我怎么证明他做不到？",[11,77926,77927,77928,77956,77957,77986],{},"不去枚举所有运输表。我给每个矿定一个“提货加价” ",[39,77929,77931,77944],{"className":77930,"translate":43},[42],[39,77932,77934],{"className":77933},[47],[49,77935,77936],{"xmlns":51},[53,77937,77938,77942],{},[56,77939,77940],{},[59,77941,35321],{},[63,77943,35321],{"encoding":65},[39,77945,77947],{"className":77946,"ariaHidden":70},[69],[39,77948,77950,77953],{"className":77949},[74],[39,77951],{"className":77952,"style":290},[78],[39,77954,35321],{"className":77955},[83,84],"，每个厂定一个“到货减价” ",[39,77958,77960,77974],{"className":77959,"translate":43},[42],[39,77961,77963],{"className":77962},[47],[49,77964,77965],{"xmlns":51},[53,77966,77967,77972],{},[56,77968,77969],{},[59,77970,77971],{},"v",[63,77973,77971],{"encoding":65},[39,77975,77977],{"className":77976,"ariaHidden":70},[69],[39,77978,77980,77983],{"className":77979},[74],[39,77981],{"className":77982,"style":290},[78],[39,77984,77971],{"className":77985,"style":2114},[83,84],"，只要求一条规矩：对任意一条路线，两个价加起来不超过运费，",[39,77988,77990],{"className":77989,"translate":43},[746],[39,77991,77993,78033],{"className":77992,"translate":43},[42],[39,77994,77996],{"className":77995},[47],[49,77997,77998],{"xmlns":51,"display":755},[53,77999,78000,78030],{},[56,78001,78002,78008,78010,78016,78018,78028],{},[613,78003,78004,78006],{},[59,78005,35321],{},[59,78007,13377],{},[133,78009,788],{},[613,78011,78012,78014],{},[59,78013,77971],{},[59,78015,13446],{},[133,78017,1824],{},[613,78019,78020,78022],{},[59,78021,24077],{},[56,78023,78024,78026],{},[59,78025,13377],{},[59,78027,13446],{},[59,78029,2903],{"mathvariant":145},[63,78031,78032],{"encoding":65},"u_i+v_j\\le c_{ij}.",[39,78034,78036,78091,78147],{"className":78035,"ariaHidden":70},[69],[39,78037,78039,78042,78082,78085,78088],{"className":78038},[74],[39,78040],{"className":78041,"style":7528},[78],[39,78043,78045,78048],{"className":78044},[83],[39,78046,35321],{"className":78047},[83,84],[39,78049,78051],{"className":78050},[192],[39,78052,78054,78074],{"className":78053},[196,649],[39,78055,78057,78071],{"className":78056},[200],[39,78058,78060],{"className":78059,"style":13604},[204],[39,78061,78062,78065],{"style":966},[39,78063],{"className":78064,"style":213},[212],[39,78066,78068],{"className":78067},[217,218,219,220],[39,78069,13377],{"className":78070},[83,84,220],[39,78072,673],{"className":78073},[672],[39,78075,78077],{"className":78076},[200],[39,78078,78080],{"className":78079,"style":680},[204],[39,78081],{},[39,78083],{"className":78084,"style":461},[230],[39,78086,788],{"className":78087},[465],[39,78089],{"className":78090,"style":461},[230],[39,78092,78094,78098,78138,78141,78144],{"className":78093},[74],[39,78095],{"className":78096,"style":78097},[78],"height:0.9221em;vertical-align:-0.2861em;",[39,78099,78101,78104],{"className":78100},[83],[39,78102,77971],{"className":78103,"style":2114},[83,84],[39,78105,78107],{"className":78106},[192],[39,78108,78110,78130],{"className":78109},[196,649],[39,78111,78113,78127],{"className":78112},[200],[39,78114,78116],{"className":78115,"style":13604},[204],[39,78117,78118,78121],{"style":41316},[39,78119],{"className":78120,"style":213},[212],[39,78122,78124],{"className":78123},[217,218,219,220],[39,78125,13446],{"className":78126,"style":13731},[83,84,220],[39,78128,673],{"className":78129},[672],[39,78131,78133],{"className":78132},[200],[39,78134,78136],{"className":78135,"style":35396},[204],[39,78137],{},[39,78139],{"className":78140,"style":231},[230],[39,78142,1824],{"className":78143},[235],[39,78145],{"className":78146,"style":231},[230],[39,78148,78150,78153,78196],{"className":78149},[74],[39,78151],{"className":78152,"style":41295},[78],[39,78154,78156,78159],{"className":78155},[83],[39,78157,24077],{"className":78158},[83,84],[39,78160,78162],{"className":78161},[192],[39,78163,78165,78188],{"className":78164},[196,649],[39,78166,78168,78185],{"className":78167},[200],[39,78169,78171],{"className":78170,"style":13604},[204],[39,78172,78173,78176],{"style":966},[39,78174],{"className":78175,"style":213},[212],[39,78177,78179],{"className":78178},[217,218,219,220],[39,78180,78182],{"className":78181},[83,220],[39,78183,13732],{"className":78184,"style":13731},[83,84,220],[39,78186,673],{"className":78187},[672],[39,78189,78191],{"className":78190},[200],[39,78192,78194],{"className":78193,"style":35396},[204],[39,78195],{},[39,78197,2903],{"className":78198},[83],[11,78200,78201],{},"那么任何一张运单的总价都满足",[39,78203,78205],{"className":78204,"translate":43},[746],[39,78206,78208,78449],{"className":78207,"translate":43},[42],[39,78209,78211],{"className":78210},[47],[49,78212,78213],{"xmlns":51,"display":755},[53,78214,78215,78446],{},[759,78216,78217,78303,78387],{"rowspacing":761,"columnalign":762,"columnspacing":146},[764,78218,78219,78255],{},[767,78220,78221],{},[770,78222,78223],{"scriptlevel":541,"displaystyle":70},[56,78224,78225,78235,78245],{},[6556,78226,78227,78229],{},[133,78228,3191],{},[56,78230,78231,78233],{},[59,78232,13377],{},[59,78234,13446],{},[613,78236,78237,78239],{},[59,78238,2098],{},[56,78240,78241,78243],{},[59,78242,13377],{},[59,78244,13446],{},[613,78246,78247,78249],{},[59,78248,24077],{},[56,78250,78251,78253],{},[59,78252,13377],{},[59,78254,13446],{},[767,78256,78257],{},[770,78258,78259],{"scriptlevel":541,"displaystyle":70},[56,78260,78261,78263,78265,78275,78285,78287,78293,78295,78301],{},[56,78262],{},[133,78264,25173],{},[6556,78266,78267,78269],{},[133,78268,3191],{},[56,78270,78271,78273],{},[59,78272,13377],{},[59,78274,13446],{},[613,78276,78277,78279],{},[59,78278,2098],{},[56,78280,78281,78283],{},[59,78282,13377],{},[59,78284,13446],{},[133,78286,136],{"stretchy":135},[613,78288,78289,78291],{},[59,78290,35321],{},[59,78292,13377],{},[133,78294,788],{},[613,78296,78297,78299],{},[59,78298,77971],{},[59,78300,13446],{},[133,78302,161],{"stretchy":135},[764,78304,78305,78311],{},[767,78306,78307],{},[770,78308,78309],{"scriptlevel":541,"displaystyle":70},[56,78310],{},[767,78312,78313],{},[770,78314,78315],{"scriptlevel":541,"displaystyle":70},[56,78316,78317,78319,78321,78327,78333,78335,78341,78351,78353,78355,78361,78367,78369,78375,78385],{},[56,78318],{},[133,78320,793],{},[6556,78322,78323,78325],{},[133,78324,3191],{},[59,78326,13377],{},[613,78328,78329,78331],{},[59,78330,35321],{},[59,78332,13377],{},[133,78334,136],{"fence":70,"stretchy":70,"minsize":65017,"maxsize":65017},[6556,78336,78337,78339],{},[133,78338,3191],{},[59,78340,13446],{},[613,78342,78343,78345],{},[59,78344,2098],{},[56,78346,78347,78349],{},[59,78348,13377],{},[59,78350,13446],{},[133,78352,161],{"fence":70,"stretch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u_i\\Bigl(\\sum_j\\pi_{ij}\\Bigr)+\\sum_j v_j\\Bigl(\\sum_i\\pi_{ij}\\Bigr)\\\\\n&=\\sum_i a_iu_i+\\sum_j b_jv_j 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",[39,79582,79584,79602],{"className":79583,"translate":43},[42],[39,79585,79587],{"className":79586},[47],[49,79588,79589],{"xmlns":51},[53,79590,79591,79599],{},[56,79592,79593,79595,79597],{},[59,79594,35321],{},[133,79596,155],{"separator":70},[59,79598,77971],{},[63,79600,79601],{"encoding":65},"u,v",[39,79603,79605],{"className":79604,"ariaHidden":70},[69],[39,79606,79608,79611,79614,79617,79620],{"className":79607},[74],[39,79609],{"className":79610,"style":2654},[78],[39,79612,35321],{"className":79613},[83,84],[39,79615,155],{"className":79616},[252],[39,79618],{"className":79619,"style":256},[230],[39,79621,77971],{"className":79622,"style":2114},[83,84]," 有关，跟运单无关；而中间那步把 ",[39,79625,79627,79640],{"className":79626,"translate":43},[42],[39,79628,79630],{"className":79629},[47],[49,79631,79632],{"xmlns":51},[53,79633,79634,79638],{},[56,79635,79636],{},[59,79637,2098],{},[63,79639,2101],{"encoding":65},[39,79641,79643],{"className":79642,"ariaHidden":70},[69],[39,79644,79646,79649],{"className":79645},[74],[39,79647],{"className":79648,"style":290},[78],[39,79650,2098],{"className":79651,"style":2114},[83,84]," 的边缘和换成了产量 ",[39,79654,79656,79674],{"className":79655,"translate":43},[42],[39,79657,79659],{"className":79658},[47],[49,79660,79661],{"xmlns":51},[53,79662,79663,79671],{},[56,79664,79665],{},[613,79666,79667,79669],{},[59,79668,158],{},[59,79670,13377],{},[63,79672,79673],{"encoding":65},"a_i",[39,79675,79677],{"className":79676,"ariaHidden":70},[69],[39,79678,79680,79683],{"className":79679},[74],[39,79681],{"className":79682,"style":2916},[78],[39,79684,79686,79689],{"className":79685},[83],[39,79687,158],{"className":79688},[83,84],[39,79690,79692],{"className":79691},[192],[39,79693,79695,79715],{"className":79694},[196,649],[39,79696,79698,79712],{"className":79697},[200],[39,79699,79701],{"className":79700,"style":13604},[204],[39,79702,79703,79706],{"style":966},[39,79704],{"className":79705,"style":213},[212],[39,79707,79709],{"className":79708},[217,218,219,220],[39,79710,13377],{"className":79711},[83,84,220],[39,79713,673],{"className":79714},[672],[39,79716,79718],{"className":79717},[200],[39,79719,79721],{"className":79720,"style":680},[204],[39,79722],{},"、需求 ",[39,79725,79727,79745],{"className":79726,"translate":43},[42],[39,79728,79730],{"className":79729},[47],[49,79731,79732],{"xmlns":51},[53,79733,79734,79742],{},[56,79735,79736],{},[613,79737,79738,79740],{},[59,79739,78435],{},[59,79741,13446],{},[63,79743,79744],{"encoding":65},"b_j",[39,79746,79748],{"className":79747,"ariaHidden":70},[69],[39,79749,79751,79754],{"className":79750},[74],[39,79752],{"className":79753,"style":76899},[78],[39,79755,79757,79760],{"className":79756},[83],[39,79758,78435],{"className":79759},[83,84],[39,79761,79763],{"className":79762},[192],[39,79764,79766,79786],{"className":79765},[196,649],[39,79767,79769,79783],{"className":79768},[200],[39,79770,79772],{"className":79771,"style":13604},[204],[39,79773,79774,79777],{"style":966},[39,79775],{"className":79776,"style":213},[212],[39,79778,79780],{"className":79779},[217,218,219,220],[39,79781,13446],{"className":79782,"style":13731},[83,84,220],[39,79784,673],{"className":79785},[672],[39,79787,79789],{"className":79788},[200],[39,79790,79792],{"className":79791,"style":35396},[204],[39,79793],{},"。所以只要我找到一组 ",[39,79796,79798,79815],{"className":79797,"translate":43},[42],[39,79799,79801],{"className":79800},[47],[49,79802,79803],{"xmlns":51},[53,79804,79805,79813],{},[56,79806,79807,79809,79811],{},[59,79808,35321],{},[133,79810,155],{"separator":70},[59,79812,77971],{},[63,79814,79601],{"encoding":65},[39,79816,79818],{"className":79817,"ariaHidden":70},[69],[39,79819,79821,79824,79827,79830,79833],{"className":79820},[74],[39,79822],{"className":79823,"style":2654},[78],[39,79825,35321],{"className":79826},[83,84],[39,79828,155],{"className":79829},[252],[39,79831],{"className":79832,"style":256},[230],[39,79834,77971],{"className":79835,"style":2114},[83,84],"，使这个和等于 5，我就",[1489,79838,5595],{},"了运费不可能低于 5 块。",[11,79841,79842,79843,742,79922,79996,79997,80092,80093,80188,80189,80289,80290,80390,80391,80611],{},"试一试：取 ",[39,79844,79846,79874],{"className":79845,"translate":43},[42],[39,79847,79849],{"className":79848},[47],[49,79850,79851],{"xmlns":51},[53,79852,79853,79871],{},[56,79854,79855,79857,79859,79861,79863,79865,79867,79869],{},[59,79856,35321],{},[133,79858,793],{},[133,79860,136],{"stretchy":135},[539,79862,541],{},[133,79864,155],{"separator":70},[133,79866,5378],{},[539,79868,546],{},[133,79870,161],{"stretchy":135},[63,79872,79873],{"encoding":65},"u=(0,-1)",[39,79875,79877,79895],{"className":79876,"ariaHidden":70},[69],[39,79878,79880,79883,79886,79889,79892],{"className":79879},[74],[39,79881],{"className":79882,"style":290},[78],[39,79884,35321],{"className":79885},[83,84],[39,79887],{"className":79888,"style":231},[230],[39,79890,793],{"className":79891},[235],[39,79893],{"className":79894,"style":231},[230],[39,79896,79898,79901,79904,79907,79910,79913,79916,79919],{"className":79897},[74],[39,79899],{"className":79900,"style":245},[78],[39,79902,136],{"className":79903},[182],[39,79905,541],{"className":79906},[83],[39,79908,155],{"className":79909},[252],[39,79911],{"className":79912,"style":256},[230],[39,79914,5378],{"className":79915},[83],[39,79917,546],{"className":79918},[83],[39,79920,161],{"className":79921},[263],[39,79923,79925,79951],{"className":79924,"translate":43},[42],[39,79926,79928],{"className":79927},[47],[49,79929,79930],{"xmlns":51},[53,79931,79932,79948],{},[56,79933,79934,79936,79938,79940,79942,79944,79946],{},[59,79935,77971],{},[133,79937,793],{},[133,79939,136],{"stretchy":135},[539,79941,546],{},[133,79943,155],{"separator":70},[539,79945,17359],{},[133,79947,161],{"stretchy":135},[63,79949,79950],{"encoding":65},"v=(1,2)",[39,79952,79954,79972],{"className":79953,"ariaHidden":70},[69],[39,79955,79957,79960,79963,79966,79969],{"className":79956},[74],[39,79958],{"className":79959,"style":290},[78],[39,79961,77971],{"className":79962,"style":2114},[83,84],[39,79964],{"className":79965,"style":231},[230],[39,79967,793],{"className":79968},[235],[39,79970],{"className":79971,"style":231},[230],[39,79973,79975,79978,79981,79984,79987,79990,79993],{"className":79974},[74],[39,79976],{"className":79977,"style":245},[78],[39,79979,136],{"className":79980},[182],[39,79982,546],{"className":79983},[83],[39,79985,155],{"className":79986},[252],[39,79988],{"className":79989,"style":256},[230],[39,79991,17359],{"className":79992},[83],[39,79994,161],{"className":79995},[263],"。四条路线逐一检查：甲到 A 是 ",[39,79998,80000,80026],{"className":79999,"translate":43},[42],[39,80001,80003],{"className":80002},[47],[49,80004,80005],{"xmlns":51},[53,80006,80007,80023],{},[56,80008,80009,80011,80013,80015,80017,80019,80021],{},[539,80010,541],{},[133,80012,788],{},[539,80014,546],{},[133,80016,793],{},[539,80018,546],{},[133,80020,1824],{},[539,80022,546],{},[63,80024,80025],{"encoding":65},"0+1=1\\le 1",[39,80027,80029,80047,80065,80083],{"className":80028,"ariaHidden":70},[69],[39,80030,80032,80035,80038,80041,80044],{"className":80031},[74],[39,80033],{"className":80034,"style":17882},[78],[39,80036,541],{"className":80037},[83],[39,80039],{"className":80040,"style":461},[230],[39,80042,788],{"className":80043},[465],[39,80045],{"className":80046,"style":461},[230],[39,80048,80050,80053,80056,80059,80062],{"className":80049},[74],[39,80051],{"className":80052,"style":2473},[78],[39,80054,546],{"className":80055},[83],[39,80057],{"className":80058,"style":231},[230],[39,80060,793],{"className":80061},[235],[39,80063],{"className":80064,"style":231},[230],[39,80066,80068,80071,80074,80077,80080],{"className":80067},[74],[39,80069],{"className":80070,"style":14587},[78],[39,80072,546],{"className":80073},[83],[39,80075],{"className":80076,"style":231},[230],[39,80078,1824],{"className":80079},[235],[39,80081],{"className":80082,"style":231},[230],[39,80084,80086,80089],{"className":80085},[74],[39,80087],{"className":80088,"style":2473},[78],[39,80090,546],{"className":80091},[83],"，甲到 B 是 ",[39,80094,80096,80122],{"className":80095,"translate":43},[42],[39,80097,80099],{"className":80098},[47],[49,80100,80101],{"xmlns":51},[53,80102,80103,80119],{},[56,80104,80105,80107,80109,80111,80113,80115,80117],{},[539,80106,541],{},[133,80108,788],{},[539,80110,17359],{},[133,80112,793],{},[539,80114,17359],{},[133,80116,1824],{},[539,80118,17359],{},[63,80120,80121],{"encoding":65},"0+2=2\\le 2",[39,80123,80125,80143,80161,80179],{"className":80124,"ariaHidden":70},[69],[39,80126,80128,80131,80134,80137,80140],{"className":80127},[74],[39,80129],{"className":80130,"style":17882},[78],[39,80132,541],{"className":80133},[83],[39,80135],{"className":80136,"style":461},[230],[39,80138,788],{"className":80139},[465],[39,80141],{"className":80142,"style":461},[230],[39,80144,80146,80149,80152,80155,80158],{"className":80145},[74],[39,80147],{"className":80148,"style":2473},[78],[39,80150,17359],{"className":80151},[83],[39,80153],{"className":80154,"style":231},[230],[39,80156,793],{"className":80157},[235],[39,80159],{"className":80160,"style":231},[230],[39,80162,80164,80167,80170,80173,80176],{"className":80163},[74],[39,80165],{"className":80166,"style":14587},[78],[39,80168,17359],{"className":80169},[83],[39,80171],{"className":80172,"style":231},[230],[39,80174,1824],{"className":80175},[235],[39,80177],{"className":80178,"style":231},[230],[39,80180,80182,80185],{"className":80181},[74],[39,80183],{"className":80184,"style":2473},[78],[39,80186,17359],{"className":80187},[83],"，乙到 A 是 ",[39,80190,80192,80220],{"className":80191,"translate":43},[42],[39,80193,80195],{"className":80194},[47],[49,80196,80197],{"xmlns":51},[53,80198,80199,80217],{},[56,80200,80201,80203,80205,80207,80209,80211,80213,80215],{},[133,80202,5378],{},[539,80204,546],{},[133,80206,788],{},[539,80208,546],{},[133,80210,793],{},[539,80212,541],{},[133,80214,1824],{},[539,80216,59187],{},[63,80218,80219],{"encoding":65},"-1+1=0\\le 3",[39,80221,80223,80244,80262,80280],{"className":80222,"ariaHidden":70},[69],[39,80224,80226,80229,80232,80235,80238,80241],{"className":80225},[74],[39,80227],{"className":80228,"style":17882},[78],[39,80230,5378],{"className":80231},[83],[39,80233,546],{"className":80234},[83],[39,80236],{"className":80237,"style":461},[230],[39,80239,788],{"className":80240},[465],[39,80242],{"className":80243,"style":461},[230],[39,80245,80247,80250,80253,80256,80259],{"className":80246},[74],[39,80248],{"className":80249,"style":2473},[78],[39,80251,546],{"className":80252},[83],[39,80254],{"className":80255,"style":231},[230],[39,80257,793],{"className":80258},[235],[39,80260],{"className":80261,"style":231},[230],[39,80263,80265,80268,80271,80274,80277],{"className":80264},[74],[39,80266],{"className":80267,"style":14587},[78],[39,80269,541],{"className":80270},[83],[39,80272],{"className":80273,"style":231},[230],[39,80275,1824],{"className":80276},[235],[39,80278],{"className":80279,"style":231},[230],[39,80281,80283,80286],{"className":80282},[74],[39,80284],{"className":80285,"style":2473},[78],[39,80287,59187],{"className":80288},[83],"，乙到 B 是 ",[39,80291,80293,80321],{"className":80292,"translate":43},[42],[39,80294,80296],{"className":80295},[47],[49,80297,80298],{"xmlns":51},[53,80299,80300,80318],{},[56,80301,80302,80304,80306,80308,80310,80312,80314,80316],{},[133,80303,5378],{},[539,80305,546],{},[133,80307,788],{},[539,80309,17359],{},[133,80311,793],{},[539,80313,546],{},[133,80315,1824],{},[539,80317,546],{},[63,80319,80320],{"encoding":65},"-1+2=1\\le 1",[39,80322,80324,80345,80363,80381],{"className":80323,"ariaHidden":70},[69],[39,80325,80327,80330,80333,80336,80339,80342],{"className":80326},[74],[39,80328],{"className":80329,"style":17882},[78],[39,80331,5378],{"className":80332},[83],[39,80334,546],{"className":80335},[83],[39,80337],{"className":80338,"style":461},[230],[39,80340,788],{"className":80341},[465],[39,80343],{"className":80344,"style":461},[230],[39,80346,80348,80351,80354,80357,80360],{"className":80347},[74],[39,80349],{"className":80350,"style":2473},[78],[39,80352,17359],{"className":80353},[83],[39,80355],{"className":80356,"style":231},[230],[39,80358,793],{"className":80359},[235],[39,80361],{"className":80362,"style":231},[230],[39,80364,80366,80369,80372,80375,80378],{"className":80365},[74],[39,80367],{"className":80368,"style":14587},[78],[39,80370,546],{"className":80371},[83],[39,80373],{"className":80374,"style":231},[230],[39,80376,1824],{"className":80377},[235],[39,80379],{"className":80380,"style":231},[230],[39,80382,80384,80387],{"className":80383},[74],[39,80385],{"className":80386,"style":2473},[78],[39,80388,546],{"className":80389},[83],"，全部成立。而 ",[39,80392,80394,80446],{"className":80393,"translate":43},[42],[39,80395,80397],{"className":80396},[47],[49,80398,80399],{"xmlns":51},[53,80400,80401,80443],{},[56,80402,80403,80405,80407,80409,80411,80413,80415,80417,80419,80421,80423,80425,80427,80429,80431,80433,80435,80437,80439,80441],{},[539,80404,59187],{},[133,80406,8885],{},[539,80408,541],{},[133,80410,788],{},[539,80412,546],{},[133,80414,8885],{},[133,80416,136],{"stretchy":135},[133,80418,5378],{},[539,80420,546],{},[133,80422,161],{"stretchy":135},[133,80424,788],{},[539,80426,17359],{},[133,80428,8885],{},[539,80430,546],{},[133,80432,788],{},[539,80434,17359],{},[133,80436,8885],{},[539,80438,17359],{},[133,80440,793],{},[539,80442,59259],{},[63,80444,80445],{"encoding":65},"3\\cdot0+1\\cdot(-1)+2\\cdot1+2\\cdot2=5",[39,80447,80449,80467,80485,80503,80530,80548,80566,80584,80602],{"className":80448,"ariaHidden":70},[69],[39,80450,80452,80455,80458,80461,80464],{"className":80451},[74],[39,80453],{"className":80454,"style":2473},[78],[39,80456,59187],{"className":80457},[83],[39,80459],{"className":80460,"style":461},[230],[39,80462,8885],{"className":80463},[465],[39,80465],{"className":80466,"style":461},[230],[39,80468,80470,80473,80476,80479,80482],{"className":80469},[74],[39,80471],{"className":80472,"style":17882},[78],[39,80474,541],{"className":80475},[83],[39,80477],{"className":80478,"style":461},[230],[39,80480,788],{"className":80481},[465],[39,80483],{"className":80484,"style":461},[230],[39,80486,80488,80491,80494,80497,80500],{"className":80487},[74],[39,80489],{"className":80490,"style":2473},[78],[39,80492,546],{"className":80493},[83],[39,80495],{"className":80496,"style":461},[230],[39,80498,8885],{"className":80499},[465],[39,80501],{"className":80502,"style":461},[230],[39,80504,80506,80509,80512,80515,80518,80521,80524,80527],{"className":80505},[74],[39,80507],{"className":80508,"style":245},[78],[39,80510,136],{"className":80511},[182],[39,80513,5378],{"className":80514},[83],[39,80516,546],{"className":80517},[83],[39,80519,161],{"className":80520},[263],[39,80522],{"className":80523,"style":461},[230],[39,80525,788],{"className":80526},[465],[39,80528],{"className":80529,"style":461},[230],[39,80531,80533,80536,80539,80542,80545],{"className":80532},[74],[39,80534],{"className":80535,"style":2473},[78],[39,80537,17359],{"className":80538},[83],[39,80540],{"className":80541,"style":461},[230],[39,80543,8885],{"className":80544},[465],[39,80546],{"className":80547,"style":461},[230],[39,80549,80551,80554,80557,80560,80563],{"className":80550},[74],[39,80552],{"className":80553,"style":17882},[78],[39,80555,546],{"className":80556},[83],[39,80558],{"className":80559,"style":461},[230],[39,80561,788],{"className":80562},[465],[39,80564],{"className":80565,"style":461},[230],[39,80567,80569,80572,80575,80578,80581],{"className":80568},[74],[39,80570],{"className":80571,"style":2473},[78],[39,80573,17359],{"className":80574},[83],[39,80576],{"className":80577,"style":461},[230],[39,80579,8885],{"className":80580},[465],[39,80582],{"className":80583,"style":461},[230],[39,80585,80587,80590,80593,80596,80599],{"className":80586},[74],[39,80588],{"className":80589,"style":2473},[78],[39,80591,17359],{"className":80592},[83],[39,80594],{"className":80595,"style":231},[230],[39,80597,793],{"className":80598},[235],[39,80600],{"className":80601,"style":231},[230],[39,80603,80605,80608],{"className":80604},[74],[39,80606],{"className":80607,"style":2473},[78],[39,80609,59259],{"className":80610},[83],"。证书找到了。",[11,80613,80614],{},"这就是对偶的思想：不去找运单，而是找一张“价格表”，让价格表卡住运费的下界。最优的价格表能卡到最紧，那个值就等于最低运费。离散情形下，这件事可以写成拉格朗日形式，把两组等式约束配乘子：",[39,80616,80618],{"className":80617,"translate":43},[746],[39,80619,80621,80893],{"className":80620,"translate":43},[42],[39,80622,80624],{"className":80623},[47],[49,80625,80626],{"xmlns":51,"display":755},[53,80627,80628,80890],{},[759,80629,80630,80780],{"rowspacing":761,"columnalign":762,"columnspacing":146},[764,80631,80632,80656],{},[767,80633,80634],{},[770,80635,80636],{"scriptlevel":541,"displaystyle":70},[56,80637,80638,80640,80642,80644,80646,80649,80651,80654],{},[59,80639,77190],{},[133,80641,136],{"stretchy":135},[59,80643,2098],{},[133,80645,155],{"separator":70},[59,80647,80648],{},"φ",[133,80650,155],{"separator":70},[59,80652,80653],{},"ψ",[133,80655,161],{"stretchy":135},[767,80657,80658],{},[770,80659,80660],{"scriptlevel":541,"displaystyle":70},[56,80661,80662,80664,80666,80676,80686,80696,80698,80704,80710,80712,80718,80720,80726,80736,80738,80740,80746,80752,80754,80760,80762,80768,80778],{},[56,80663],{},[133,80665,793],{},[6556,80667,80668,80670],{},[133,80669,3191],{},[56,80671,80672,80674],{},[59,80673,13377],{},[59,80675,13446],{},[613,80677,80678,80680],{},[59,80679,2098],{},[56,80681,80682,80684],{},[59,80683,13377],{},[59,80685,13446],{},[613,80687,80688,80690],{},[59,80689,48444],{},[56,80691,80692,80694],{},[59,80693,13377],{},[59,80695,13446],{},[133,80697,788],{},[6556,80699,80700,80702],{},[133,80701,3191],{},[59,80703,13377],{},[613,80705,80706,80708],{},[59,80707,80648],{},[59,80709,13377],{},[133,80711,136],{"fence":70,"stretchy":70,"minsize":65017,"maxsize":65017},[613,80713,80714,80716],{},[59,80715,158],{},[59,80717,13377],{},[133,80719,5378],{},[6556,80721,80722,80724],{},[133,80723,3191],{},[59,80725,13446],{},[613,80727,80728,80730],{},[59,80729,2098],{},[56,80731,80732,80734],{},[59,80733,13377],{},[59,80735,13446],{},[133,80737,161],{"fence":70,"stretchy":70,"minsize":65017,"maxsize":65017},[133,80739,788],{},[6556,80741,80742,80744],{},[133,80743,3191],{},[59,80745,13446],{},[613,80747,80748,80750],{},[59,80749,80653],{},[59,80751,13446],{},[133,80753,136],{"fence":70,"stretchy":70,"minsize":65017,"maxsize":65017},[613,80755,80756,80758],{},[59,80757,78435],{},[59,80759,13446],{},[133,80761,5378],{},[6556,80763,80764,80766],{},[133,80765,3191],{},[59,80767,13377],{},[613,80769,80770,80772],{},[59,80771,2098],{},[56,80773,80774,80776],{},[59,80775,13377],{},[59,80777,13446],{},[133,80779,161],{"fence":70,"stretchy":70,"minsize":65017,"maxsize":65017},[764,80781,80782,80788],{},[767,80783,80784],{},[770,80785,80786],{"scriptlevel":541,"displaystyle":70},[56,80787],{},[767,80789,80790],{},[770,80791,80792],{"scriptlevel":541,"displaystyle":70},[56,80793,80794,80796,80798,80804,80810,80816,80818,80824,80830,80836,80838,80848,80858,80860,80870,80872,80878,80880,80886,80888],{},[56,80795],{},[133,80797,793],{},[6556,80799,80800,80802],{},[133,80801,3191],{},[59,80803,13377],{},[613,80805,80806,80808],{},[59,80807,158],{},[59,80809,13377],{},[613,80811,80812,80814],{},[59,80813,80648],{},[59,80815,13377],{},[133,80817,788],{},[6556,80819,80820,80822],{},[133,80821,3191],{},[59,80823,13446],{},[613,80825,80826,80828],{},[59,80827,78435],{},[59,80829,13446],{},[613,80831,80832,80834],{},[59,80833,80653],{},[59,80835,13446],{},[133,80837,788],{},[6556,80839,80840,80842],{},[133,80841,3191],{},[56,80843,80844,80846],{},[59,80845,13377],{},[59,80847,13446],{},[613,80849,80850,80852],{},[59,80851,2098],{},[56,80853,80854,80856],{},[59,80855,13377],{},[59,80857,13446],{},[133,80859,136],{"fence":70,"stretchy":70,"minsize":73875,"maxsize":73875},[613,80861,80862,80864],{},[59,80863,48444],{},[56,80865,80866,80868],{},[59,80867,13377],{},[59,80869,13446],{},[133,80871,5378],{},[613,80873,80874,80876],{},[59,80875,80648],{},[59,80877,13377],{},[133,80879,5378],{},[613,80881,80882,80884],{},[59,80883,80653],{},[59,80885,13446],{},[133,80887,161],{"fence":70,"stretchy":70,"minsize":73875,"maxsize":73875},[59,80889,2903],{"mathvariant":145},[63,80891,80892],{"encoding":65},"\\begin{aligned}\nL(\\pi,\\varphi,\\psi)\n&=\\sum_{ij}\\pi_{ij}C_{ij}+\\sum_i\\varphi_i\\Bigl(a_i-\\sum_j\\pi_{ij}\\Bigr)+\\sum_j\\psi_j\\Bigl(b_j-\\sum_i\\pi_{ij}\\Bigr)\\\\\n&=\\sum_i a_i\\varphi_i+\\sum_j b_j\\psi_j+\\sum_{ij}\\pi_{ij}\\bigl(C_{ij}-\\varphi_i-\\psi_j\\bigr).\n\\end{aligned}",[39,80894,80896],{"className":80895,"ariaHidden":70},[69],[39,80897,80899,80903],{"className":80898},[74],[39,80900],{"className":80901,"style":80902},[78],"height:5.3276em;vertical-align:-2.4138em;",[39,80904,80906],{"className":80905},[83],[39,80907,80909,80982],{"className":80908},[759],[39,80910,80912],{"className":80911},[918],[39,80913,80915,80973],{"className":80914},[196,649],[39,80916,80918,80970],{"className":80917},[200],[39,80919,80922,80961],{"className":80920,"style":80921},[204],"height:2.9138em;",[39,80923,80925,80928],{"style":80924},"top:-4.9138em;",[39,80926],{"className":80927,"style":33100},[212],[39,80929,80931,80934,80937,80940,80943,80946,80949,80952,80955,80958],{"className":80930},[83],[39,80932,77190],{"className":80933},[83,84],[39,80935,136],{"className":80936},[182],[39,80938,2098],{"className":80939,"style":2114},[83,84],[39,80941,155],{"className":80942},[252],[39,80944],{"className":80945,"style":256},[230],[39,80947,80648],{"className":80948},[83,84],[39,80950,155],{"className":80951},[252],[39,80953],{"className":80954,"style":256},[230],[39,80956,80653],{"className":80957,"style":2114},[83,84],[39,80959,161],{"className":80960},[263],[39,80962,80964,80967],{"style":80963},"top:-2.15em;",[39,80965],{"className":80966,"style":33100},[212],[39,80968],{"className":80969},[83],[39,80971,673],{"className":80972},[672],[39,80974,80976],{"className":80975},[200],[39,80977,80980],{"className":80978,"style":80979},[204],"height:2.4138em;",[39,80981],{},[39,80983,80985],{"className":80984},[1223],[39,80986,80988,82189],{"className":80987},[196,649],[39,80989,80991,82186],{"className":80990},[200],[39,80992,80994,81645],{"className":80993,"style":80921},[204],[39,80995,80996,80999],{"style":80924},[39,80997],{"className":80998,"style":33100},[212],[39,81000,81002,81005,81008,81011,81014,81061,81064,81107,81151,81154,81157,81160,81204,81207,81247,81253,81293,81296,81299,81302,81346,81349,81392,81398,81401,81404,81407,81451,81454,81494,81500,81540,81543,81546,81549,81593,81596,81639],{"className":81001},[83],[39,81003],{"className":81004},[83],[39,81006],{"className":81007,"style":231},[230],[39,81009,793],{"className":81010},[235],[39,81012],{"className":81013,"style":231},[230],[39,81015,81017],{"className":81016},[1744,3294],[39,81018,81020,81053],{"className":81019},[196,649],[39,81021,81023,81050],{"className":81022},[200],[39,81024,81026,81040],{"className":81025,"style":6670},[204],[39,81027,81028,81031],{"style":13926},[39,81029],{"className":81030,"style":3311},[212],[39,81032,81034],{"className":81033},[217,218,219,220],[39,81035,81037],{"className":81036},[83,220],[39,81038,13732],{"className":81039,"style":13731},[83,84,220],[39,81041,81042,81045],{"style":3330},[39,81043],{"className":81044,"style":3311},[212],[39,81046,81047],{},[39,81048,3191],{"className":81049},[1744,3339,3340],[39,81051,673],{"className":81052},[672],[39,81054,81056],{"className":81055},[200],[39,81057,81059],{"className":81058,"style":13958},[204],[39,81060],{},[39,81062],{"className":81063,"style":256},[230],[39,81065,81067,81070],{"className":81066},[83],[39,81068,2098],{"className":81069,"style":2114},[83,84],[39,81071,81073],{"className":81072},[192],[39,81074,81076,81099],{"className":81075},[196,649],[39,81077,81079,81096],{"className":81078},[200],[39,81080,81082],{"className":81081,"style":13604},[204],[39,81083,81084,81087],{"style":41316},[39,81085],{"className":81086,"style":213},[212],[39,81088,81090],{"className":81089},[217,218,219,220],[39,81091,81093],{"className":81092},[83,220],[39,81094,13732],{"className":81095,"style":13731},[83,84,220],[39,81097,673],{"className":81098},[672],[39,81100,81102],{"className":81101},[200],[39,81103,81105],{"className":81104,"style":35396},[204],[39,81106],{},[39,81108,81110,81113],{"className":81109},[83],[39,81111,48444],{"className":81112,"style":48487},[83,84],[39,81114,81116],{"className":81115},[192],[39,81117,81119,81143],{"className":81118},[196,649],[39,81120,81122,81140],{"className":81121},[200],[39,81123,81125],{"className":81124,"style":13604},[204],[39,81126,81128,81131],{"style":81127},"top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;",[39,81129],{"className":81130,"style":213},[212],[39,81132,81134],{"className":81133},[217,218,219,220],[39,81135,81137],{"className":81136},[83,220],[39,81138,13732],{"className":81139,"style":13731},[83,84,220],[39,81141,673],{"className":81142},[672],[39,81144,81146],{"className":81145},[200],[39,81147,81149],{"className":81148,"style":35396},[204],[39,81150],{},[39,81152],{"className":81153,"style":461},[230],[39,81155,788],{"className":81156},[465],[39,81158],{"className":81159,"style":461},[230],[39,81161,81163],{"className":81162},[1744,3294],[39,81164,81166,81196],{"className":81165},[196,649],[39,81167,81169,81193],{"className":81168},[200],[39,81170,81172,81183],{"className":81171,"style":6670},[204],[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取下确界：只要有一对 ",[39,82252,82254,82276],{"className":82253,"translate":43},[42],[39,82255,82257],{"className":82256},[47],[49,82258,82259],{"xmlns":51},[53,82260,82261,82273],{},[56,82262,82263,82265,82267,82269,82271],{},[133,82264,136],{"stretchy":135},[59,82266,13377],{},[133,82268,155],{"separator":70},[59,82270,13446],{},[133,82272,161],{"stretchy":135},[63,82274,82275],{"encoding":65},"(i,j)",[39,82277,82279],{"className":82278,"ariaHidden":70},[69],[39,82280,82282,82285,82288,82291,82294,82297,82300],{"className":82281},[74],[39,82283],{"className":82284,"style":245},[78],[39,82286,136],{"className":82287},[182],[39,82289,13377],{"className":82290},[83,84],[39,82292,155],{"className":82293},[252],[39,82295],{"className":82296,"style":256},[230],[39,82298,13446],{"className":82299,"style":13731},[83,84],[39,82301,161],{"className":82302},[263]," 使 ",[39,82305,82307,82349],{"className":82306,"translate":43},[42],[39,82308,82310],{"className":82309},[47],[49,82311,82312],{"xmlns":51},[53,82313,82314,82346],{},[56,82315,82316,82326,82328,82334,82336,82342,82344],{},[613,82317,82318,82320],{},[59,82319,48444],{},[56,82321,82322,82324],{},[59,82323,13377],{},[59,82325,13446],{},[133,82327,5378],{},[613,82329,82330,82332],{},[59,82331,80648],{},[59,82333,13377],{},[133,82335,5378],{},[613,82337,82338,82340],{},[59,82339,80653],{},[59,82341,13446],{},[133,82343,1693],{},[539,82345,541],{},[63,82347,82348],{"encoding":65},"C_{ij}-\\varphi_i-\\psi_j\u003C0",[39,82350,82352,82410,82466,82521],{"className":82351,"ariaHidden":70},[69],[39,82353,82355,82358,82401,82404,82407],{"className":82354},[74],[39,82356],{"className":82357,"style":35350},[78],[39,82359,82361,82364],{"className":82360},[83],[39,82362,48444],{"className":82363,"style":48487},[83,84],[39,82365,82367],{"className":82366},[192],[39,82368,82370,82393],{"className":82369},[196,649],[39,82371,82373,82390],{"className":82372},[200],[39,82374,82376],{"className":82375,"style":13604},[204],[39,82377,82378,82381],{"style":81127},[39,82379],{"className":82380,"style":213},[212],[39,82382,82384],{"className":82383},[217,218,219,220],[39,82385,82387],{"className":82386},[83,220],[39,82388,13732],{"className":82389,"style":13731},[83,84,220],[39,82391,673],{"className":82392},[672],[39,82394,82396],{"className":82395},[200],[39,82397,82399],{"className":82398,"style":35396},[204],[39,82400],{},[39,82402],{"className":82403,"style":461},[230],[39,82405,5378],{"className":82406},[465],[39,82408],{"className":82409,"style":461},[230],[39,82411,82413,82417,82457,82460,82463],{"className":82412},[74],[39,82414],{"className":82415,"style":82416},[78],"height:0.7778em;vertical-align:-0.1944em;",[39,82418,82420,82423],{"className":82419},[83],[39,82421,80648],{"className":82422},[83,84],[39,82424,82426],{"className":82425},[192],[39,82427,82429,82449],{"className":82428},[196,649],[39,82430,82432,82446],{"className":82431},[200],[39,82433,82435],{"className":82434,"style":13604},[204],[39,82436,82437,82440],{"style":966},[39,82438],{"className":82439,"style":213},[212],[39,82441,82443],{"className":82442},[217,218,219,220],[39,82444,13377],{"className":82445},[83,84,220],[39,82447,673],{"className":82448},[672],[39,82450,82452],{"className":82451},[200],[39,82453,82455],{"className":82454,"style":680},[204],[39,82456],{},[39,82458],{"className":82459,"style":461},[230],[39,82461,5378],{"className":82462},[465],[39,82464],{"className":82465,"style":461},[230],[39,82467,82469,82472,82512,82515,82518],{"className":82468},[74],[39,82470],{"className":82471,"style":76899},[78],[39,82473,82475,82478],{"className":82474},[83],[39,82476,80653],{"className":82477,"style":2114},[83,84],[39,82479,82481],{"className":82480},[192],[39,82482,82484,82504],{"className":82483},[196,649],[39,82485,82487,82501],{"className":82486},[200],[39,82488,82490],{"className":82489,"style":13604},[204],[39,82491,82492,82495],{"style":41316},[39,82493],{"className":82494,"style":213},[212],[39,82496,82498],{"className":82497},[217,218,219,220],[39,82499,13446],{"className":82500,"style":13731},[83,84,220],[39,82502,673],{"className":82503},[672],[39,82505,82507],{"className":82506},[200],[39,82508,82510],{"className":82509,"style":35396},[204],[39,82511],{},[39,82513],{"className":82514,"style":231},[230],[39,82516,1693],{"className":82517},[235],[39,82519],{"className":82520,"style":231},[230],[39,82522,82524,82527],{"className":82523},[74],[39,82525],{"className":82526,"style":2473},[78],[39,82528,541],{"className":82529},[83],[39,82531,82533,82559],{"className":82532,"translate":43},[42],[39,82534,82536],{"className":82535},[47],[49,82537,82538],{"xmlns":51},[53,82539,82540,82556],{},[56,82541,82542,82552,82554],{},[613,82543,82544,82546],{},[59,82545,2098],{},[56,82547,82548,82550],{},[59,82549,13377],{},[59,82551,13446],{},[133,82553,418],{},[59,82555,1696],{"mathvariant":145},[63,82557,82558],{"encoding":65},"\\pi_{ij}\\to\\infty",[39,82560,82562,82620],{"className":82561,"ariaHidden":70},[69],[39,82563,82565,82568,82611,82614,82617],{"className":82564},[74],[39,82566],{"className":82567,"style":41295},[78],[39,82569,82571,82574],{"className":82570},[83],[39,82572,2098],{"className":82573,"style":2114},[83,84],[39,82575,82577],{"className":82576},[192],[39,82578,82580,82603],{"className":82579},[196,649],[39,82581,82583,82600],{"className":82582},[200],[39,82584,82586],{"className":82585,"style":13604},[204],[39,82587,82588,82591],{"style":41316},[39,82589],{"className":82590,"style":213},[212],[39,82592,82594],{"className":82593},[217,218,219,220],[39,82595,82597],{"className":82596},[83,220],[39,82598,13732],{"className":82599,"style":13731},[83,84,220],[39,82601,673],{"className":82602},[672],[39,82604,82606],{"className":82605},[200],[39,82607,82609],{"className":82608,"style":35396},[204],[39,82610],{},[39,82612],{"className":82613,"style":231},[230],[39,82615,418],{"className":82616},[235],[39,82618],{"className":82619,"style":231},[230],[39,82621,82623,82626],{"className":82622},[74],[39,82624],{"className":82625,"style":290},[78],[39,82627,1696],{"className":82628},[83]," 就得到 ",[39,82631,82633,82649],{"className":82632,"translate":43},[42],[39,82634,82636],{"className":82635},[47],[49,82637,82638],{"xmlns":51},[53,82639,82640,82646],{},[56,82641,82642,82644],{},[133,82643,5378],{},[59,82645,1696],{"mathvariant":145},[63,82647,82648],{"encoding":65},"-\\infty",[39,82650,82652],{"className":82651,"ariaHidden":70},[69],[39,82653,82655,82658,82661],{"className":82654},[74],[39,82656],{"className":82657,"style":73043},[78],[39,82659,5378],{"className":82660},[83],[39,82662,1696],{"className":82663},[83],"；否则下确界在 ",[39,82666,82668,82686],{"className":82667,"translate":43},[42],[39,82669,82671],{"className":82670},[47],[49,82672,82673],{"xmlns":51},[53,82674,82675,82683],{},[56,82676,82677,82679,82681],{},[59,82678,2098],{},[133,82680,793],{},[539,82682,541],{},[63,82684,82685],{"encoding":65},"\\pi=0",[39,82687,82689,82707],{"className":82688,"ariaHidden":70},[69],[39,82690,82692,82695,82698,82701,82704],{"className":82691},[74],[39,82693],{"className":82694,"style":290},[78],[39,82696,2098],{"className":82697,"style":2114},[83,84],[39,82699],{"className":82700,"style":231},[230],[39,82702,793],{"className":82703},[235],[39,82705],{"className":82706,"style":231},[230],[39,82708,82710,82713],{"className":82709},[74],[39,82711],{"className":82712,"style":2473},[78],[39,82714,541],{"className":82715},[83]," 处取到。于是对偶问题就是",[39,82718,82720],{"className":82719,"translate":43},[746],[39,82721,82723,82825],{"className":82722,"translate":43},[42],[39,82724,82726],{"className":82725},[47],[49,82727,82728],{"xmlns":51,"display":755},[53,82729,82730,82822],{},[56,82731,82732,82748,82750,82756,82762,82768,82770,82776,82782,82788,82790,82792,82794,82800,82802,82808,82810,82820],{},[6556,82733,82734,82740],{},[56,82735,82736,82738],{},[59,82737,1687],{},[133,82739,1690],{},[56,82741,82742,82744,82746],{},[59,82743,80648],{},[133,82745,155],{"separator":70},[59,82747,80653],{},[35841,82749,74223],{},[6556,82751,82752,82754],{},[133,82753,3191],{},[59,82755,13377],{},[613,82757,82758,82760],{},[59,82759,158],{},[59,82761,13377],{},[613,82763,82764,82766],{},[59,82765,80648],{},[59,82767,13377],{},[133,82769,788],{},[6556,82771,82772,82774],{},[133,82773,3191],{},[59,82775,13446],{},[613,82777,82778,82780],{},[59,82779,78435],{},[59,82781,13446],{},[613,82783,82784,82786],{},[59,82785,80653],{},[59,82787,13446],{},[230,82789],{"width":6891},[35841,82791,74264],{},[230,82793],{"width":6891},[613,82795,82796,82798],{},[59,82797,80648],{},[59,82799,13377],{},[133,82801,788],{},[613,82803,82804,82806],{},[59,82805,80653],{},[59,82807,13446],{},[133,82809,1824],{},[613,82811,82812,82814],{},[59,82813,48444],{},[56,82815,82816,82818],{},[59,82817,13377],{},[59,82819,13446],{},[133,82821,155],{"separator":70},[63,82823,82824],{"encoding":65},"\\max_{\\varphi,\\psi}\\ \\sum_i a_i\\varphi_i+\\sum_j b_j\\psi_j\n\\quad\\text{s.t.}\\quad \\varphi_i+\\psi_j\\le 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",[39,83749,83751,83795],{"className":83750,"translate":43},[42],[39,83752,83754],{"className":83753},[47],[49,83755,83756],{"xmlns":51},[53,83757,83758,83792],{},[56,83759,83760,83762,83764,83766,83768,83770,83772,83774,83776,83778,83780,83782,83784,83786,83788,83790],{},[59,83761,80648],{},[133,83763,136],{"stretchy":135},[59,83765,1755],{},[133,83767,161],{"stretchy":135},[133,83769,788],{},[59,83771,80653],{},[133,83773,136],{"stretchy":135},[59,83775,29625],{},[133,83777,161],{"stretchy":135},[133,83779,1824],{},[59,83781,29425],{},[133,83783,136],{"stretchy":135},[59,83785,1755],{},[133,83787,155],{"separator":70},[59,83789,29625],{},[133,83791,161],{"stretchy":135},[63,83793,83794],{"encoding":65},"\\varphi(x)+\\psi(y)\\le 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允许我取 ",[39,83881,83883,83903],{"className":83882,"translate":43},[42],[39,83884,83886],{"className":83885},[47],[49,83887,83888],{"xmlns":51},[53,83889,83890,83900],{},[56,83891,83892,83894,83896,83898],{},[59,83893,80653],{},[133,83895,793],{},[133,83897,5378],{},[59,83899,80648],{},[63,83901,83902],{"encoding":65},"\\psi=-\\varphi",[39,83904,83906,83924],{"className":83905,"ariaHidden":70},[69],[39,83907,83909,83912,83915,83918,83921],{"className":83908},[74],[39,83910],{"className":83911,"style":51280},[78],[39,83913,80653],{"className":83914,"style":2114},[83,84],[39,83916],{"className":83917,"style":231},[230],[39,83919,793],{"className":83920},[235],[39,83922],{"className":83923,"style":231},[230],[39,83925,83927,83930,83933],{"className":83926},[74],[39,83928],{"className":83929,"style":82416},[78],[39,83931,5378],{"className":83932},[83],[39,83934,80648],{"className":83935},[83,84],"，此时它变成 ",[39,83938,83940,83984],{"className":83939,"translate":43},[42],[39,83941,83943],{"className":83942},[47],[49,83944,83945],{"xmlns":51},[53,83946,83947,83981],{},[56,83948,83949,83951,83953,83955,83957,83959,83961,83963,83965,83967,83969,83971,83973,83975,83977,83979],{},[59,83950,80648],{},[133,83952,136],{"stretchy":135},[59,83954,1755],{},[133,83956,161],{"stretchy":135},[133,83958,5378],{},[59,83960,80648],{},[133,83962,136],{"stretchy":135},[59,83964,29625],{},[133,83966,161],{"stretchy":135},[133,83968,1824],{},[59,83970,29425],{},[133,83972,136],{"stretchy":135},[59,83974,1755],{},[133,83976,155],{"separator":70},[59,83978,29625],{},[133,83980,161],{"stretchy":135},[63,83982,83983],{"encoding":65},"\\varphi(x)-\\varphi(y)\\le d(x,y)",[39,83985,83987,84014,84041],{"className":83986,"ariaHidden":70},[69],[39,83988,83990,83993,83996,83999,84002,84005,84008,84011],{"className":83989},[74],[39,83991],{"className":83992,"style":245},[78],[39,83994,80648],{"className":83995},[83,84],[39,83997,136],{"className":83998},[182],[39,84000,1755],{"className":84001},[83,84],[39,84003,161],{"className":84004},[263],[39,84006],{"className":84007,"style":461},[230],[39,84009,5378],{"className":84010},[465],[39,84012],{"className":84013,"style":461},[230],[39,84015,84017,84020,84023,84026,84029,84032,84035,84038],{"className":84016},[74],[39,84018],{"className":84019,"style":245},[78],[39,84021,80648],{"className":84022},[83,84],[39,84024,136],{"className":84025},[182],[39,84027,29625],{"className":84028,"style":2114},[83,84],[39,84030,161],{"className":84031},[263],[39,84033],{"className":84034,"style":231},[230],[39,84036,1824],{"className":84037},[235],[39,84039],{"className":84040,"style":231},[230],[39,84042,84044,84047,84050,84053,84056,84059,84062,84065],{"className":84043},[74],[39,84045],{"className":84046,"style":245},[78],[39,84048,29425],{"className":84049},[83,84],[39,84051,136],{"className":84052},[182],[39,84054,1755],{"className":84055},[83,84],[39,84057,155],{"className":84058},[252],[39,84060],{"className":84061,"style":256},[230],[39,84063,29625],{"className":84064,"style":2114},[83,84],[39,84066,161],{"className":84067},[263],"，也就是 ",[39,84070,84072,84086],{"className":84071,"translate":43},[42],[39,84073,84075],{"className":84074},[47],[49,84076,84077],{"xmlns":51},[53,84078,84079,84083],{},[56,84080,84081],{},[59,84082,80648],{},[63,84084,84085],{"encoding":65},"\\varphi",[39,84087,84089],{"className":84088,"ariaHidden":70},[69],[39,84090,84092,84095],{"className":84091},[74],[39,84093],{"className":84094,"style":2654},[78],[39,84096,80648],{"className":84097},[83,84],[39,84099,84101,84114],{"className":84100,"translate":43},[42],[39,84102,84104],{"className":84103},[47],[49,84105,84106],{"xmlns":51},[53,84107,84108,84112],{},[56,84109,84110],{},[539,84111,546],{},[63,84113,546],{"encoding":65},[39,84115,84117],{"className":84116,"ariaHidden":70},[69],[39,84118,84120,84123],{"className":84119},[74],[39,84121],{"className":84122,"style":2473},[78],[39,84124,546],{"className":84125},[83],"-Lipschitz 的。所以",[39,84128,84130],{"className":84129,"translate":43},[746],[39,84131,84133,84213],{"className":84132,"translate":43},[42],[39,84134,84136],{"className":84135},[47],[49,84137,84138],{"xmlns":51,"display":755},[53,84139,84140,84210],{},[56,84141,84142,84148,84150,84152,84154,84156,84158,84160,84190,84192,84194,84196,84198,84200,84202,84204,84206,84208],{},[613,84143,84144,84146],{},[59,84145,23703],{},[539,84147,546],{},[133,84149,136],{"stretchy":135},[59,84151,73578],{},[133,84153,155],{"separator":70},[59,84155,73609],{},[133,84157,161],{"stretchy":135},[133,84159,793],{},[6556,84161,84162,84168],{},[56,84163,84164,84166],{},[59,84165,21465],{},[133,84167,1690],{},[56,84169,84170,84172,84174,84186,84188],{},[59,84171,15389],{"mathvariant":145},[59,84173,80648],{},[613,84175,84176,84178],{},[59,84177,15389],{"mathvariant":145},[56,84179,84180,84182,84184],{},[59,84181,77190],{"mathvariant":145},[59,84183,13377],{"mathvariant":145},[59,84185,11],{"mathvariant":145},[133,84187,1824],{},[539,84189,546],{},[133,84191,69087],{},[59,84193,80648],{},[35841,84195,35843],{},[59,84197,29425],{},[133,84199,136],{"stretchy":135},[59,84201,73578],{},[133,84203,5378],{},[59,84205,73609],{},[133,84207,161],{"stretchy":135},[59,84209,2903],{"mathvariant":145},[63,84211,84212],{"encoding":65},"W_1(\\mu,\\nu)=\\sup_{\\|\\varphi\\|_{\\mathrm{Lip}}\\le 1}\\int\\varphi\\,d(\\mu-\\nu).",[39,84214,84216,84289,84437],{"className":84215,"ariaHidden":70},[69],[39,84217,84219,84222,84262,84265,84268,84271,84274,84277,84280,84283,84286],{"className":84218},[74],[39,84220],{"className":84221,"style":245},[78],[39,84223,84225,84228],{"className":84224},[83],[39,84226,23703],{"className":84227,"style":178},[83,84],[39,84229,84231],{"className":84230},[192],[39,84232,84234,84254],{"className":84233},[196,649],[39,84235,84237,84251],{"className":84236},[200],[39,84238,84240],{"className":84239,"style":963},[204],[39,84241,84242,84245],{"style":73765},[39,84243],{"className":84244,"style":213},[212],[39,84246,84248],{"className":84247},[217,218,219,220],[39,84249,546],{"className":84250},[83,220],[39,84252,673],{"className":84253},[672],[39,84255,84257],{"className":84256},[200],[39,84258,84260],{"className":84259,"style":680},[204],[39,84261],{},[39,84263,136],{"className":84264},[182],[39,84266,73578],{"className":84267},[83,84],[39,84269,155],{"className":84270},[252],[39,84272],{"className":84273,"style":256},[230],[39,84275,73609],{"className":84276,"style":73625},[83,84],[39,84278,161],{"className":84279},[263],[39,84281],{"className":84282,"style":231},[230],[39,84284,793],{"className":84285},[235],[39,84287],{"className":84288,"style":231},[230],[39,84290,84292,84296,84404,84407,84410,84413,84416,84419,84422,84425,84428,84431,84434],{"className":84291},[74],[39,84293],{"className":84294,"style":84295},[78],"height:2.543em;vertical-align:-1.183em;",[39,84297,84299],{"className":84298},[1744,3294],[39,84300,84302,84395],{"className":84301},[196,649],[39,84303,84305,84392],{"className":84304},[200],[39,84306,84308,84382],{"className":84307,"style":290},[204],[39,84309,84311,84314],{"style":84310},"top:-2.1146em;margin-left:0em;",[39,84312],{"className":84313,"style":935},[212],[39,84315,84317],{"className":84316},[217,218,219,220],[39,84318,84320,84323,84326,84376,84379],{"className":84319},[83,220],[39,84321,15389],{"className":84322},[83,220],[39,84324,80648],{"className":84325},[83,84,220],[39,84327,84329,84332],{"className":84328},[83,220],[39,84330,15389],{"className":84331},[83,220],[39,84333,84335],{"className":84334},[192],[39,84336,84338,84367],{"className":84337},[196,649],[39,84339,84341,84364],{"className":84340},[200],[39,84342,84345],{"className":84343,"style":84344},[204],"height:0.3448em;",[39,84346,84348,84351],{"style":84347},"top:-2.3567em;margin-left:0em;margin-right:0.0714em;",[39,84349],{"className":84350,"style":4004},[212],[39,84352,84354],{"className":84353},[217,4008,4009,220],[39,84355,84357],{"className":84356},[83,220],[39,84358,84360],{"className":84359},[83,220],[39,84361,84363],{"className":84362},[83,35077,220],"Lip",[39,84365,673],{"className":84366},[672],[39,84368,84370],{"className":84369},[200],[39,84371,84374],{"className":84372,"style":84373},[204],"height:0.2822em;",[39,84375],{},[39,84377,1824],{"className":84378},[235,220],[39,84380,546],{"className":84381},[83,220],[39,84383,84384,84387],{"style":21599},[39,84385],{"className":84386,"style":935},[212],[39,84388,84389],{},[39,84390,21465],{"className":84391},[1744],[39,84393,673],{"className":84394},[672],[39,84396,84398],{"className":84397},[200],[39,84399,84402],{"className":84400,"style":84401},[204],"height:1.183em;",[39,84403],{},[39,84405],{"className":84406,"style":256},[230],[39,84408,69087],{"className":84409,"style":69380},[1744,3339,3340],[39,84411],{"className":84412,"style":256},[230],[39,84414,80648],{"className":84415},[83,84],[39,84417],{"className":84418,"style":256},[230],[39,84420,29425],{"className":84421},[83,84],[39,84423,136],{"className":84424},[182],[39,84426,73578],{"className":84427},[83,84],[39,84429],{"className":84430,"style":461},[230],[39,84432,5378],{"className":84433},[465],[39,84435],{"className":84436,"style":461},[230],[39,84438,84440,84443,84446,84449],{"className":84439},[74],[39,84441],{"className":84442,"style":245},[78],[39,84444,73609],{"className":84445,"style":73625},[83,84],[39,84447,161],{"className":84448},[263],[39,84450,2903],{"className":84451},[83],[11,84453,84454],{},"这个式子的分量很重：左边是关于分布的优化问题，右边是关于函数的优化问题，而函数可以用神经网络参数化。WGAN 干的就是这件事，只差把 Lipschitz 约束用 weight clipping 或 gradient penalty 近似实现。",[11,84456,84457,84458,84648,84649,84677,84678,84706,84707,84764,84765,84793,84794,1607],{},"二次代价 ",[39,84459,84461,84499],{"className":84460,"translate":43},[42],[39,84462,84464],{"className":84463},[47],[49,84465,84466],{"xmlns":51},[53,84467,84468,84496],{},[56,84469,84470,84472,84474,84482,84484,84486,84488,84490],{},[59,84471,24077],{},[133,84473,793],{},[770,84475,84476],{"scriptlevel":541,"displaystyle":135},[5362,84477,84478,84480],{},[539,84479,546],{},[539,84481,17359],{},[59,84483,15389],{"mathvariant":145},[59,84485,1755],{},[133,84487,5378],{},[59,84489,29625],{},[138,84491,84492,84494],{},[59,84493,15389],{"mathvariant":145},[539,84495,17359],{},[63,84497,84498],{"encoding":65},"c=\\tfrac12\\|x-y\\|^2",[39,84500,84502,84520,84609],{"className":84501,"ariaHidden":70},[69],[39,84503,84505,84508,84511,84514,84517],{"className":84504},[74],[39,84506],{"className":84507,"style":290},[78],[39,84509,24077],{"className":84510},[83,84],[39,84512],{"className":84513,"style":231},[230],[39,84515,793],{"className":84516},[235],[39,84518],{"className":84519,"style":231},[230],[39,84521,84523,84526,84594,84597,84600,84603,84606],{"className":84522},[74],[39,84524],{"className":84525,"style":74794},[78],[39,84527,84529,84532,84591],{"className":84528},[83],[39,84530],{"className":84531},[182,5463],[39,84533,84535],{"className":84534},[5362],[39,84536,84538,84583],{"className":84537},[196,649],[39,84539,84541,84580],{"className":84540},[200],[39,84542,84544,84558,84566],{"className":84543,"style":74813},[204],[39,84545,84546,84549],{"style":74816},[39,84547],{"className":84548,"style":935},[212],[39,84550,84552],{"className":84551},[217,218,219,220],[39,84553,84555],{"className":84554},[83,220],[39,84556,17359],{"className":84557},[83,220],[39,84559,84560,84563],{"style":5503},[39,84561],{"className":84562,"style":935},[212],[39,84564],{"className":84565,"style":5511},[5510],[39,84567,84568,84571],{"style":74839},[39,84569],{"className":84570,"style":935},[212],[39,84572,84574],{"className":84573},[217,218,219,220],[39,84575,84577],{"className":84576},[83,220],[39,84578,546],{"className":84579},[83,220],[39,84581,673],{"className":84582},[672],[39,84584,84586],{"className":84585},[200],[39,84587,84589],{"className":84588,"style":74861},[204],[39,84590],{},[39,84592],{"className":84593},[263,5463],[39,84595,15389],{"className":84596},[83],[39,84598,1755],{"className":84599},[83,84],[39,84601],{"className":84602,"style":461},[230],[39,84604,5378],{"className":84605},[465],[39,84607],{"className":84608,"style":461},[230],[39,84610,84612,84616,84619],{"className":84611},[74],[39,84613],{"className":84614,"style":84615},[78],"height:1.0641em;vertical-align:-0.25em;",[39,84617,29625],{"className":84618,"style":2114},[83,84],[39,84620,84622,84625],{"className":84621},[83],[39,84623,15389],{"className":84624},[83],[39,84626,84628],{"className":84627},[192],[39,84629,84631],{"className":84630},[196],[39,84632,84634],{"className":84633},[200],[39,84635,84637],{"className":84636,"style":19710},[204],[39,84638,84639,84642],{"style":208},[39,84640],{"className":84641,"style":213},[212],[39,84643,84645],{"className":84644},[217,218,219,220],[39,84646,17359],{"className":84647},[83,220]," 时还有一个 Brenier 定理：若 ",[39,84650,84652,84665],{"className":84651,"translate":43},[42],[39,84653,84655],{"className":84654},[47],[49,84656,84657],{"xmlns":51},[53,84658,84659,84663],{},[56,84660,84661],{},[59,84662,73578],{},[63,84664,73581],{"encoding":65},[39,84666,84668],{"className":84667,"ariaHidden":70},[69],[39,84669,84671,84674],{"className":84670},[74],[39,84672],{"className":84673,"style":2654},[78],[39,84675,73578],{"className":84676},[83,84]," 关于 Lebesgue 测度绝对连续，最优的 ",[39,84679,84681,84694],{"className":84680,"translate":43},[42],[39,84682,84684],{"className":84683},[47],[49,84685,84686],{"xmlns":51},[53,84687,84688,84692],{},[56,84689,84690],{},[59,84691,2098],{},[63,84693,2101],{"encoding":65},[39,84695,84697],{"className":84696,"ariaHidden":70},[69],[39,84698,84700,84703],{"className":84699},[74],[39,84701],{"className":84702,"style":290},[78],[39,84704,2098],{"className":84705,"style":2114},[83,84]," 由映射 ",[39,84708,84710,84731],{"className":84709,"translate":43},[42],[39,84711,84713],{"className":84712},[47],[49,84714,84715],{"xmlns":51},[53,84716,84717,84728],{},[56,84718,84719,84721,84723,84726],{},[59,84720,11772],{},[133,84722,793],{},[59,84724,84725],{"mathvariant":145},"∇",[59,84727,35321],{},[63,84729,84730],{"encoding":65},"T=\\nabla u",[39,84732,84734,84752],{"className":84733,"ariaHidden":70},[69],[39,84735,84737,84740,84743,84746,84749],{"className":84736},[74],[39,84738],{"className":84739,"style":79},[78],[39,84741,11772],{"className":84742,"style":178},[83,84],[39,84744],{"className":84745,"style":231},[230],[39,84747,793],{"className":84748},[235],[39,84750],{"className":84751,"style":231},[230],[39,84753,84755,84758,84761],{"className":84754},[74],[39,84756],{"className":84757,"style":79},[78],[39,84759,84725],{"className":84760},[83],[39,84762,35321],{"className":84763},[83,84]," 给出，其中 ",[39,84766,84768,84781],{"className":84767,"translate":43},[42],[39,84769,84771],{"className":84770},[47],[49,84772,84773],{"xmlns":51},[53,84774,84775,84779],{},[56,84776,84777],{},[59,84778,35321],{},[63,84780,35321],{"encoding":65},[39,84782,84784],{"className":84783,"ariaHidden":70},[69],[39,84785,84787,84790],{"className":84786},[74],[39,84788],{"className":84789,"style":290},[78],[39,84791,35321],{"className":84792},[83,84]," 是凸函数。换句话说，最优搬运方案是某个凸势函数的梯度，是一个“单调”的变换。这句话是后半段所有流模型的伏笔：如果我要把一个分布搬到另一个分布，最优的那个可逆映射长得像 ",[39,84795,84797,84813],{"className":84796,"translate":43},[42],[39,84798,84800],{"className":84799},[47],[49,84801,84802],{"xmlns":51},[53,84803,84804,84810],{},[56,84805,84806,84808],{},[59,84807,84725],{"mathvariant":145},[59,84809,35321],{},[63,84811,84812],{"encoding":65},"\\nabla u",[39,84814,84816],{"className":84815,"ariaHidden":70},[69],[39,84817,84819,84822,84825],{"className":84818},[74],[39,84820],{"className":84821,"style":79},[78],[39,84823,84725],{"className":84824},[83],[39,84826,35321],{"className":84827},[83,84],[21,84829,84830],{"id":84830},"一维与高斯",[11,84832,84833],{},"A 公司的薪资在 0 到 1 万之间均匀分布，B 公司在 0 到 2 万之间均匀分布。一个人从 A 跳到 B，平均涨多少？",[11,84835,84836,84837,84865,84866,84894,84895,84929,84930,84958],{},"“最省力”的搬法是按百分位对齐：A 的第 30 百分位是 0.3 万，B 的第 30 百分位是 0.6 万，所以 A 里处在 30 分位的那个人涨 0.3 万。一般地，分位 ",[39,84838,84840,84853],{"className":84839,"translate":43},[42],[39,84841,84843],{"className":84842},[47],[49,84844,84845],{"xmlns":51},[53,84846,84847,84851],{},[56,84848,84849],{},[59,84850,621],{},[63,84852,621],{"encoding":65},[39,84854,84856],{"className":84855,"ariaHidden":70},[69],[39,84857,84859,84862],{"className":84858},[74],[39,84860],{"className":84861,"style":709},[78],[39,84863,621],{"className":84864},[83,84]," 的人从 ",[39,84867,84869,84882],{"className":84868,"translate":43},[42],[39,84870,84872],{"className":84871},[47],[49,84873,84874],{"xmlns":51},[53,84875,84876,84880],{},[56,84877,84878],{},[59,84879,621],{},[63,84881,621],{"encoding":65},[39,84883,84885],{"className":84884,"ariaHidden":70},[69],[39,84886,84888,84891],{"className":84887},[74],[39,84889],{"className":84890,"style":709},[78],[39,84892,621],{"className":84893},[83,84]," 万变成 ",[39,84896,84898,84914],{"className":84897,"translate":43},[42],[39,84899,84901],{"className":84900},[47],[49,84902,84903],{"xmlns":51},[53,84904,84905,84911],{},[56,84906,84907,84909],{},[539,84908,17359],{},[59,84910,621],{},[63,84912,84913],{"encoding":65},"2t",[39,84915,84917],{"className":84916,"ariaHidden":70},[69],[39,84918,84920,84923,84926],{"className":84919},[74],[39,84921],{"className":84922,"style":2473},[78],[39,84924,17359],{"className":84925},[83],[39,84927,621],{"className":84928},[83,84]," 万，涨幅是 ",[39,84931,84933,84946],{"className":84932,"translate":43},[42],[39,84934,84936],{"className":84935},[47],[49,84937,84938],{"xmlns":51},[53,84939,84940,84944],{},[56,84941,84942],{},[59,84943,621],{},[63,84945,621],{"encoding":65},[39,84947,84949],{"className":84948,"ariaHidden":70},[69],[39,84950,84952,84955],{"className":84951},[74],[39,84953],{"className":84954,"style":709},[78],[39,84956,621],{"className":84957},[83,84]," 万，平均",[39,84960,84962],{"className":84961,"translate":43},[746],[39,84963,84965,85008],{"className":84964,"translate":43},[42],[39,84966,84968],{"className":84967},[47],[49,84969,84970],{"xmlns":51,"display":755},[53,84971,84972,85005],{},[56,84973,84974,84982,84984,84986,84988,84990,84992,85000,85003],{},[9932,84975,84976,84978,84980],{},[133,84977,69087],{},[539,84979,541],{},[539,84981,546],{},[59,84983,621],{},[35841,84985,35843],{},[59,84987,29425],{},[59,84989,621],{},[133,84991,793],{},[770,84993,84994],{"scriptlevel":541,"displaystyle":135},[5362,84995,84996,84998],{},[539,84997,546],{},[539,84999,17359],{},[35841,85001,85002],{}," 万",[59,85004,2903],{"mathvariant":145},[63,85006,85007],{"encoding":65},"\\int_0^1 t\\,dt=\\tfrac12\\ \\text{万}.",[39,85009,85011,85095],{"className":85010,"ariaHidden":70},[69],[39,85012,85014,85018,85071,85074,85077,85080,85083,85086,85089,85092],{"className":85013},[74],[39,85015],{"className":85016,"style":85017},[78],"height:2.476em;vertical-align:-0.9119em;",[39,85019,85021,85024],{"className":85020},[1744],[39,85022,69087],{"className":85023,"style":69380},[1744,3339,3340],[39,85025,85027],{"className":85026},[192],[39,85028,85030,85063],{"className":85029},[196,649],[39,85031,85033,85060],{"className":85032},[200],[39,85034,85037,85048],{"className":85035,"style":85036},[204],"height:1.564em;",[39,85038,85039,85042],{"style":69396},[39,85040],{"className":85041,"style":213},[212],[39,85043,85045],{"className":85044},[217,218,219,220],[39,85046,541],{"className":85047},[83,220],[39,85049,85051,85054],{"style":85050},"top:-3.8129em;margin-right:0.05em;",[39,85052],{"className":85053,"style":213},[212],[39,85055,85057],{"className":85056},[217,218,219,220],[39,85058,546],{"className":85059},[83,220],[39,85061,673],{"className":85062},[672],[39,85064,85066],{"className":85065},[200],[39,85067,85069],{"className":85068,"style":69415},[204],[39,85070],{},[39,85072],{"className":85073,"style":256},[230],[39,85075,621],{"className":85076},[83,84],[39,85078],{"className":85079,"style":256},[230],[39,85081,29425],{"className":85082},[83,84],[39,85084,621],{"className":85085},[83,84],[39,85087],{"className":85088,"style":231},[230],[39,85090,793],{"className":85091},[235],[39,85093],{"className":85094,"style":231},[230],[39,85096,85098,85101,85169,85172,85180],{"className":85097},[74],[39,85099],{"className":85100,"style":74794},[78],[39,85102,85104,85107,85166],{"className":85103},[83],[39,85105],{"className":85106},[182,5463],[39,85108,85110],{"className":85109},[5362],[39,85111,85113,85158],{"className":85112},[196,649],[39,85114,85116,85155],{"className":85115},[200],[39,85117,85119,85133,85141],{"className":85118,"style":74813},[204],[39,85120,85121,85124],{"style":74816},[39,85122],{"className":85123,"style":935},[212],[39,85125,85127],{"className":85126},[217,218,219,220],[39,85128,85130],{"className":85129},[83,220],[39,85131,17359],{"className":85132},[83,220],[39,85134,85135,85138],{"style":5503},[39,85136],{"className":85137,"style":935},[212],[39,85139],{"className":85140,"style":5511},[5510],[39,85142,85143,85146],{"style":74839},[39,85144],{"className":85145,"style":935},[212],[39,85147,85149],{"className":85148},[217,218,219,220],[39,85150,85152],{"className":85151},[83,220],[39,85153,546],{"className":85154},[83,220],[39,85156,673],{"className":85157},[672],[39,85159,85161],{"className":85160},[200],[39,85162,85164],{"className":85163,"style":74861},[204],[39,85165],{},[39,85167],{"className":85168},[263,5463],[39,85170,74223],{"className":85171},[230],[39,85173,85175],{"className":85174},[83,74460],[39,85176,85179],{"className":85177},[83,85178],"cjk_fallback","万",[39,85181,2903],{"className":85182},[83],[11,85184,85185,85186,85242,85243,85307,85308,1607],{},"这里可以直接用 ",[39,85187,85189,85209],{"className":85188,"translate":43},[42],[39,85190,85192],{"className":85191},[47],[49,85193,85194],{"xmlns":51},[53,85195,85196,85206],{},[56,85197,85198,85200,85202,85204],{},[539,85199,17359],{},[59,85201,621],{},[133,85203,5378],{},[59,85205,621],{},[63,85207,85208],{"encoding":65},"2t-t",[39,85210,85212,85233],{"className":85211,"ariaHidden":70},[69],[39,85213,85215,85218,85221,85224,85227,85230],{"className":85214},[74],[39,85216],{"className":85217,"style":17882},[78],[39,85219,17359],{"className":85220},[83],[39,85222,621],{"className":85223},[83,84],[39,85225],{"className":85226,"style":461},[230],[39,85228,5378],{"className":85229},[465],[39,85231],{"className":85232,"style":461},[230],[39,85234,85236,85239],{"className":85235},[74],[39,85237],{"className":85238,"style":709},[78],[39,85240,621],{"className":85241},[83,84]," 而不是 ",[39,85244,85246,85270],{"className":85245,"translate":43},[42],[39,85247,85249],{"className":85248},[47],[49,85250,85251],{"xmlns":51},[53,85252,85253,85267],{},[56,85254,85255,85257,85259,85261,85263,85265],{},[59,85256,150],{"mathvariant":145},[539,85258,17359],{},[59,85260,621],{},[133,85262,5378],{},[59,85264,621],{},[59,85266,150],{"mathvariant":145},[63,85268,85269],{"encoding":65},"|2t-t|",[39,85271,85273,85295],{"className":85272,"ariaHidden":70},[69],[39,85274,85276,85279,85283,85286,85289,85292],{"className":85275},[74],[39,85277],{"className":85278,"style":245},[78],[39,85280,85282],{"className":85281},[83],"∣2",[39,85284,621],{"className":85285},[83,84],[39,85287],{"className":85288,"style":461},[230],[39,85290,5378],{"className":85291},[465],[39,85293],{"className":85294,"style":461},[230],[39,85296,85298,85301,85304],{"className":85297},[74],[39,85299],{"className":85300,"style":245},[78],[39,85302,621],{"className":85303},[83,84],[39,85305,150],{"className":85306},[83],"，因为所有人都在往同一个方向涨。而这个“平均涨幅”正是 ",[39,85309,85311,85328],{"className":85310,"translate":43},[42],[39,85312,85314],{"className":85313},[47],[49,85315,85316],{"xmlns":51},[53,85317,85318,85326],{},[56,85319,85320],{},[613,85321,85322,85324],{},[59,85323,23703],{},[539,85325,546],{},[63,85327,77839],{"encoding":65},[39,85329,85331],{"className":85330,"ariaHidden":70},[69],[39,85332,85334,85337],{"className":85333},[74],[39,85335],{"className":85336,"style":634},[78],[39,85338,85340,85343],{"className":85339},[83],[39,85341,23703],{"className":85342,"style":178},[83,84],[39,85344,85346],{"className":85345},[192],[39,85347,85349,85369],{"className":85348},[196,649],[39,85350,85352,85366],{"className":85351},[200],[39,85353,85355],{"className":85354,"style":963},[204],[39,85356,85357,85360],{"style":73765},[39,85358],{"className":85359,"style":213},[212],[39,85361,85363],{"className":85362},[217,218,219,220],[39,85364,546],{"className":85365},[83,220],[39,85367,673],{"className":85368},[672],[39,85370,85372],{"className":85371},[200],[39,85373,85375],{"className":85374,"style":680},[204],[39,85376],{},[11,85378,85379,85380,85439,85440,85497,85498,85567],{},"如果反过来，A 是 ",[39,85381,85383,85408],{"className":85382,"translate":43},[42],[39,85384,85386],{"className":85385},[47],[49,85387,85388],{"xmlns":51},[53,85389,85390,85405],{},[56,85391,85392,85395,85397,85399,85401,85403],{},[59,85393,85394],{},"U",[133,85396,136],{"stretchy":135},[539,85398,541],{},[133,85400,155],{"separator":70},[539,85402,17359],{},[133,85404,161],{"stretchy":135},[63,85406,85407],{"encoding":65},"U(0,2)",[39,85409,85411],{"className":85410,"ariaHidden":70},[69],[39,85412,85414,85417,85421,85424,85427,85430,85433,85436],{"className":85413},[74],[39,85415],{"className":85416,"style":245},[78],[39,85418,85394],{"className":85419,"style":85420},[83,84],"margin-right:0.109em;",[39,85422,136],{"className":85423},[182],[39,85425,541],{"className":85426},[83],[39,85428,155],{"className":85429},[252],[39,85431],{"className":85432,"style":256},[230],[39,85434,17359],{"className":85435},[83],[39,85437,161],{"className":85438},[263],"、B 是 ",[39,85441,85443,85467],{"className":85442,"translate":43},[42],[39,85444,85446],{"className":85445},[47],[49,85447,85448],{"xmlns":51},[53,85449,85450,85464],{},[56,85451,85452,85454,85456,85458,85460,85462],{},[59,85453,85394],{},[133,85455,136],{"stretchy":135},[539,85457,541],{},[133,85459,155],{"separator":70},[539,85461,546],{},[133,85463,161],{"stretchy":135},[63,85465,85466],{"encoding":65},"U(0,1)",[39,85468,85470],{"className":85469,"ariaHidden":70},[69],[39,85471,85473,85476,85479,85482,85485,85488,85491,85494],{"className":85472},[74],[39,85474],{"className":85475,"style":245},[78],[39,85477,85394],{"className":85478,"style":85420},[83,84],[39,85480,136],{"className":85481},[182],[39,85483,541],{"className":85484},[83],[39,85486,155],{"className":85487},[252],[39,85489],{"className":85490,"style":256},[230],[39,85492,546],{"className":85493},[83],[39,85495,161],{"className":85496},[263],"，那就是平均降 0.5 万，",[39,85499,85501,85518],{"className":85500,"translate":43},[42],[39,85502,85504],{"className":85503},[47],[49,85505,85506],{"xmlns":51},[53,85507,85508,85516],{},[56,85509,85510],{},[613,85511,85512,85514],{},[59,85513,23703],{},[539,85515,546],{},[63,85517,77839],{"encoding":65},[39,85519,85521],{"className":85520,"ariaHidden":70},[69],[39,85522,85524,85527],{"className":85523},[74],[39,85525],{"className":85526,"style":634},[78],[39,85528,85530,85533],{"className":85529},[83],[39,85531,23703],{"className":85532,"style":178},[83,84],[39,85534,85536],{"className":85535},[192],[39,85537,85539,85559],{"className":85538},[196,649],[39,85540,85542,85556],{"className":85541},[200],[39,85543,85545],{"className":85544,"style":963},[204],[39,85546,85547,85550],{"style":73765},[39,85548],{"className":85549,"style":213},[212],[39,85551,85553],{"className":85552},[217,218,219,220],[39,85554,546],{"className":85555},[83,220],[39,85557,673],{"className":85558},[672],[39,85560,85562],{"className":85561},[200],[39,85563,85565],{"className":85564,"style":680},[204],[39,85566],{}," 仍然是 0.5——它量的是“搬了多少”，不管涨还是跌。",[11,85569,85570,85571,85640],{},"一般地，一维的 ",[39,85572,85574,85591],{"className":85573,"translate":43},[42],[39,85575,85577],{"className":85576},[47],[49,85578,85579],{"xmlns":51},[53,85580,85581,85589],{},[56,85582,85583],{},[613,85584,85585,85587],{},[59,85586,23703],{},[59,85588,11],{},[63,85590,77455],{"encoding":65},[39,85592,85594],{"className":85593,"ariaHidden":70},[69],[39,85595,85597,85600],{"className":85596},[74],[39,85598],{"className":85599,"style":35350},[78],[39,85601,85603,85606],{"className":85602},[83],[39,85604,23703],{"className":85605,"style":178},[83,84],[39,85607,85609],{"className":85608},[192],[39,85610,85612,85632],{"className":85611},[196,649],[39,85613,85615,85629],{"className":85614},[200],[39,85616,85618],{"className":85617,"style":1728},[204],[39,85619,85620,85623],{"style":73765},[39,85621],{"className":85622,"style":213},[212],[39,85624,85626],{"className":85625},[217,218,219,220],[39,85627,11],{"className":85628},[83,84,220],[39,85630,673],{"className":85631},[672],[39,85633,85635],{"className":85634},[200],[39,85636,85638],{"className":85637,"style":35396},[204],[39,85639],{}," 有闭式解：",[39,85642,85644],{"className":85643,"translate":43},[746],[39,85645,85647,85741],{"className":85646,"translate":43},[42],[39,85648,85650],{"className":85649},[47],[49,85651,85652],{"xmlns":51,"display":755},[53,85653,85654,85738],{},[56,85655,85656,85664,85666,85668,85670,85672,85674,85676,85684,85686,85698,85700,85702,85704,85706,85718,85720,85722,85724,85730,85732,85734,85736],{},[9932,85657,85658,85660,85662],{},[59,85659,23703],{},[59,85661,11],{},[59,85663,11],{},[133,85665,136],{"stretchy":135},[59,85667,73578],{},[133,85669,155],{"separator":70},[59,85671,73609],{},[133,85673,161],{"stretchy":135},[133,85675,793],{},[9932,85677,85678,85680,85682],{},[133,85679,69087],{},[539,85681,541],{},[539,85683,546],{},[133,85685,150],{"fence":70,"stretchy":70,"minsize":73875,"maxsize":73875},[9932,85687,85688,85690,85692],{},[59,85689,618],{},[59,85691,73578],{},[56,85693,85694,85696],{},[133,85695,5378],{},[539,85697,546],{},[133,85699,136],{"stretchy":135},[59,85701,621],{},[133,85703,161],{"stretchy":135},[133,85705,5378],{},[9932,85707,85708,85710,85712],{},[59,85709,618],{},[59,85711,73609],{},[56,85713,85714,85716],{},[133,85715,5378],{},[539,85717,546],{},[133,85719,136],{"stretchy":135},[59,85721,621],{},[133,85723,161],{"stretchy":135},[138,85725,85726,85728],{},[133,85727,150],{"fence":70,"stretchy":70,"minsize":73875,"maxsize":73875},[59,85729,11],{},[35841,85731,35843],{},[59,85733,29425],{},[59,85735,621],{},[133,85737,155],{"separator":70},[63,85739,85740],{"encoding":65},"W_p^p(\\mu,\\nu)=\\int_0^1\\bigl|F_\\mu^{-1}(t)-F_\\nu^{-1}(t)\\bigr|^p\\,dt 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时有更漂亮的形式。用上一节的对偶，对绝对连续的 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时等号成立。于是",[39,87314,87316],{"className":87315,"translate":43},[746],[39,87317,87319,87393],{"className":87318,"translate":43},[42],[39,87320,87322],{"className":87321},[47],[49,87323,87324],{"xmlns":51,"display":755},[53,87325,87326,87390],{},[56,87327,87328,87334,87336,87338,87340,87342,87344,87346,87352,87354,87360,87362,87364,87366,87368,87374,87376,87378,87380,87382,87384,87386,87388],{},[613,87329,87330,87332],{},[59,87331,23703],{},[539,87333,546],{},[133,87335,136],{"stretchy":135},[59,87337,73578],{},[133,87339,155],{"separator":70},[59,87341,73609],{},[133,87343,161],{"stretchy":135},[133,87345,793],{},[613,87347,87348,87350],{},[133,87349,69087],{},[59,87351,422],{"mathvariant":421},[133,87353,150],{"fence":70,"stretchy":70,"minsize":73875,"maxsize":73875},[613,87355,87356,87358],{},[59,87357,618],{},[59,87359,73578],{},[133,87361,136],{"stretchy":135},[59,87363,1755],{},[133,87365,161],{"stretchy":135},[133,87367,5378],{},[613,87369,87370,87372],{},[59,87371,618],{},[59,87373,73609],{},[133,87375,136],{"stretchy":135},[59,87377,1755],{},[133,87379,161],{"stretchy":135},[133,87381,150],{"fence":70,"stretchy":70,"minsize":73875,"maxsize":73875},[35841,87383,35843],{},[59,87385,29425],{},[59,87387,1755],{},[59,87389,2903],{"mathvariant":145},[63,87391,87392],{"encoding":65},"W_1(\\mu,\\nu)=\\int_{\\mathbb 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.",[39,87394,87396,87469,87614],{"className":87395,"ariaHidden":70},[69],[39,87397,87399,87402,87442,87445,87448,87451,87454,87457,87460,87463,87466],{"className":87398},[74],[39,87400],{"className":87401,"style":245},[78],[39,87403,87405,87408],{"className":87404},[83],[39,87406,23703],{"className":87407,"style":178},[83,84],[39,87409,87411],{"className":87410},[192],[39,87412,87414,87434],{"className":87413},[196,649],[39,87415,87417,87431],{"className":87416},[200],[39,87418,87420],{"className":87419,"style":963},[204],[39,87421,87422,87425],{"style":73765},[39,87423],{"className":87424,"style":213},[212],[39,87426,87428],{"className":87427},[217,218,219,220],[39,87429,546],{"className":87430},[83,220],[39,87432,673],{"className":87433},[672],[39,87435,87437],{"className":87436},[200],[39,87438,87440],{"className":87439,"style":680},[204],[39,87441],{},[39,87443,136],{"className":87444},[182],[39,87446,73578],{"className":87447},[83,84],[39,87449,155],{"className":87450},[252],[39,87452],{"className":87453,"style":256},[230],[39,87455,73609],{"className":87456,"style":73625},[83,84],[39,87458,161],{"className":87459},[263],[39,87461],{"className":87462,"style":231},[230],[39,87464,793],{"className":87465},[235],[39,87467],{"className":87468,"style":231},[230],[39,87470,87472,87475,87519,87556,87596,87599,87602,87605,87608,87611],{"className":87471},[74],[39,87473],{"className":87474,"style":74293},[78],[39,87476,87478,87481],{"className":87477},[1744],[39,87479,69087],{"className":87480,"style":69380},[1744,3339,3340],[39,87482,87484],{"className":87483},[192],[39,87485,87487,87511],{"className":87486},[196,649],[39,87488,87490,87508],{"className":87489},[200],[39,87491,87494],{"className":87492,"style":87493},[204],"height:-0.4297em;",[39,87495,87496,87499],{"style":69396},[39,87497],{"className":87498,"style":213},[212],[39,87500,87502],{"className":87501},[217,218,219,220],[39,87503,87505],{"className":87504},[83,220],[39,87506,422],{"className":87507},[83,515,220],[39,87509,673],{"className":87510},[672],[39,87512,87514],{"className":87513},[200],[39,87515,87517],{"className":87516,"style":69415},[204],[39,87518],{},[39,87520,87522],{"className":87521},[182],[39,87523,87525],{"className":87524},[18520,85892],[39,87526,87528,87548],{"className":87527},[196,649],[39,87529,87531,87545],{"className":87530},[200],[39,87532,87534],{"className":87533,"style":928},[204],[39,87535,87536,87539],{"style":85904},[39,87537],{"className":87538,"style":85908},[212],[39,87540,87541],{"style":85911},[85913,87542,87543],{"xmlns":85915,"width":85916,"height":73875,"viewBox":85917},[85919,87544],{"d":85921},[39,87546,673],{"className":87547},[672],[39,87549,87551],{"className":87550},[200],[39,87552,87554],{"className":87553,"style":1217},[204],[39,87555],{},[39,87557,87559,87562],{"className":87558},[83],[39,87560,618],{"className":87561,"style":178},[83,84],[39,87563,87565],{"className":87564},[192],[39,87566,87568,87588],{"className":87567},[196,649],[39,87569,87571,87585],{"className":87570},[200],[39,87572,87574],{"className":87573,"style":1728},[204],[39,87575,87576,87579],{"style":73765},[39,87577],{"className":87578,"style":213},[212],[39,87580,87582],{"className":87581},[217,218,219,220],[39,87583,73578],{"className":87584},[83,84,220],[39,87586,673],{"className":87587},[672],[39,87589,87591],{"className":87590},[200],[39,87592,87594],{"className":87593,"style":35396},[204],[39,87595],{},[39,87597,136],{"className":87598},[182],[39,87600,1755],{"className":87601},[83,84],[39,87603,161],{"className":87604},[263],[39,87606],{"className":87607,"style":461},[230],[39,87609,5378],{"className":87610},[465],[39,87612],{"className":87613,"style":461},[230],[39,87615,87617,87620,87660,87663,87666,87669,87706,87709,87712,87715],{"className":87616},[74],[39,87618],{"className":87619,"style":908},[78],[39,87621,87623,87626],{"className":87622},[83],[39,87624,618],{"className":87625,"style":178},[83,84],[39,87627,87629],{"className":87628},[192],[39,87630,87632,87652],{"className":87631},[196,649],[39,87633,87635,87649],{"className":87634},[200],[39,87636,87638],{"className":87637,"style":1728},[204],[39,87639,87640,87643],{"style":73765},[39,87641],{"className":87642,"style":213},[212],[39,87644,87646],{"className":87645},[217,218,219,220],[39,87647,73609],{"className":87648,"style":73625},[83,84,220],[39,87650,673],{"className":87651},[672],[39,87653,87655],{"className":87654},[200],[39,87656,87658],{"className":87657,"style":680},[204],[39,87659],{},[39,87661,136],{"className":87662},[182],[39,87664,1755],{"className":87665},[83,84],[39,87667,161],{"className":87668},[263],[39,87670,87672],{"className":87671},[263],[39,87673,87675],{"className":87674},[18520,85892],[39,87676,87678,87698],{"className":87677},[196,649],[39,87679,87681,87695],{"className":87680},[200],[39,87682,87684],{"className":87683,"style":928},[204],[39,87685,87686,87689],{"style":85904},[39,87687],{"className":87688,"style":85908},[212],[39,87690,876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CDF 曲线之间的面积，就是搬运的总量。拿刚才的薪资例验一下：在 ",[39,87722,87724,87746],{"className":87723,"translate":43},[42],[39,87725,87727],{"className":87726},[47],[49,87728,87729],{"xmlns":51},[53,87730,87731,87743],{},[56,87732,87733,87735,87737,87739,87741],{},[133,87734,537],{"stretchy":135},[539,87736,541],{},[133,87738,155],{"separator":70},[539,87740,546],{},[133,87742,4280],{"stretchy":135},[63,87744,87745],{"encoding":65},"[0,1]",[39,87747,87749],{"className":87748,"ariaHidden":70},[69],[39,87750,87752,87755,87758,87761,87764,87767,87770],{"className":87751},[74],[39,87753],{"className":87754,"style":245},[78],[39,87756,537],{"className":87757},[182],[39,87759,541],{"className":87760},[83],[39,87762,155],{"className":87763},[252],[39,87765],{"className":87766,"style":256},[230],[39,87768,546],{"className":87769},[83],[39,87771,4280],{"className":87772},[263]," 上 ",[39,87775,87777,87801],{"className":87776,"translate":43},[42],[39,87778,87780],{"className":87779},[47],[49,87781,87782],{"xmlns":51},[53,87783,87784,87798],{},[56,87785,87786,87788,87790,87792,87794,87796],{},[59,87787,618],{},[133,87789,136],{"stretchy":135},[59,87791,1755],{},[133,87793,161],{"stretchy":135},[133,87795,793],{},[59,87797,1755],{},[63,87799,87800],{"encoding":65},"F(x)=x",[39,87802,87804,87831],{"className":87803,"ariaHidden":70},[69],[39,87805,87807,87810,87813,87816,87819,87822,87825,87828],{"className":87806},[74],[39,87808],{"className":87809,"style":245},[78],[39,87811,618],{"className":87812,"style":178},[83,84],[39,87814,136],{"className":87815},[182],[39,87817,1755],{"className":87818},[83,84],[39,87820,161],{"className":87821},[263],[39,87823],{"className":87824,"style":231},[230],[39,87826,793],{"className":87827},[235],[39,87829],{"className":87830,"style":231},[230],[39,87832,87834,87837],{"className":87833},[74],[39,87835],{"className":87836,"style":290},[78],[39,87838,1755],{"className":87839},[83,84],"、",[39,87842,87844,87872],{"className":87843,"translate":43},[42],[39,87845,87847],{"className":87846},[47],[49,87848,87849],{"xmlns":51},[53,87850,87851,87869],{},[56,87852,87853,87855,87857,87859,87861,87863,87865,87867],{},[59,87854,3181],{},[133,87856,136],{"stretchy":135},[59,87858,1755],{},[133,87860,161],{"stretchy":135},[133,87862,793],{},[59,87864,1755],{},[59,87866,45691],{"mathvariant":145},[539,87868,17359],{},[63,87870,87871],{"encoding":65},"G(x)=x\u002F2",[39,87873,87875,87902],{"className":87874,"ariaHidden":70},[69],[39,87876,87878,87881,87884,87887,87890,87893,87896,87899],{"className":87877},[74],[39,87879],{"className":87880,"style":245},[78],[39,87882,3181],{"className":87883},[83,84],[39,87885,136],{"className":87886},[182],[39,87888,1755],{"className":87889},[83,84],[39,87891,161],{"className":87892},[263],[39,87894],{"className":87895,"style":231},[230],[39,87897,793],{"className":87898},[235],[39,87900],{"className":87901,"style":231},[230],[39,87903,87905,87908,87911],{"className":87904},[74],[39,87906],{"className":87907,"style":245},[78],[39,87909,1755],{"className":87910},[83,84],[39,87912,87914],{"className":87913},[83],"\u002F2","，差是 ",[39,87917,87919,87937],{"className":87918,"translate":43},[42],[39,87920,87922],{"className":87921},[47],[49,87923,87924],{"xmlns":51},[53,87925,87926,87934],{},[56,87927,87928,87930,87932],{},[59,87929,1755],{},[59,87931,45691],{"mathvariant":145},[539,87933,17359],{},[63,87935,87936],{"encoding":65},"x\u002F2",[39,87938,87940],{"className":87939,"ariaHidden":70},[69],[39,87941,87943,87946,87949],{"className":87942},[74],[39,87944],{"className":87945,"style":245},[78],[39,87947,1755],{"className":87948},[83,84],[39,87950,87914],{"className":87951},[83],"，积出 ",[39,87954,87956,87976],{"className":87955,"translate":43},[42],[39,87957,87959],{"className":87958},[47],[49,87960,87961],{"xmlns":51},[53,87962,87963,87973],{},[56,87964,87965],{},[770,87966,87967],{"scriptlevel":541,"displaystyle":135},[5362,87968,87969,87971],{},[539,87970,546],{},[539,87972,59223],{},[63,87974,87975],{"encoding":65},"\\tfrac14",[39,87977,87979],{"className":87978,"ariaHidden":70},[69],[39,87980,87982,87985],{"className":87981},[74],[39,87983],{"className":87984,"style":74794},[78],[39,87986,87988,87991,88050],{"className":87987},[83],[39,87989],{"className":87990},[182,5463],[39,87992,87994],{"className":87993},[5362],[39,87995,87997,88042],{"className":87996},[196,649],[39,87998,88000,88039],{"className":87999},[200],[39,88001,88003,88017,88025],{"className":88002,"style":74813},[204],[39,88004,88005,88008],{"style":74816},[39,88006],{"className":88007,"style":935},[212],[39,88009,88011],{"className":88010},[217,218,219,220],[39,88012,88014],{"className":88013},[83,220],[39,88015,59223],{"className":88016},[83,220],[39,88018,88019,88022],{"style":5503},[39,88020],{"className":88021,"style":935},[212],[39,88023],{"className":88024,"style":5511},[5510],[39,88026,88027,88030],{"style":74839},[39,88028],{"className":88029,"style":935},[212],[39,88031,88033],{"className":88032},[217,218,219,220],[39,88034,88036],{"className":88035},[83,220],[39,88037,546],{"className":88038},[83,220],[39,88040,673],{"className":88041},[672],[39,88043,88045],{"className":88044},[200],[39,88046,88048],{"className":88047,"style":74861},[204],[39,88049],{},[39,88051],{"className":88052},[263,5463],"；在 ",[39,88055,88057,88079],{"className":88056,"translate":43},[42],[39,88058,88060],{"className":88059},[47],[49,88061,88062],{"xmlns":51},[53,88063,88064,88076],{},[56,88065,88066,88068,88070,88072,88074],{},[133,88067,537],{"stretchy":135},[539,88069,546],{},[133,88071,155],{"separator":70},[539,88073,17359],{},[133,88075,4280],{"stretchy":135},[63,88077,88078],{"encoding":65},"[1,2]",[39,88080,88082],{"className":88081,"ariaHidden":70},[69],[39,88083,88085,88088,88091,88094,88097,88100,88103],{"className":88084},[74],[39,88086],{"className":88087,"style":245},[78],[39,88089,537],{"className":88090},[182],[39,88092,546],{"className":88093},[83],[39,88095,155],{"className":88096},[252],[39,88098],{"className":88099,"style":256},[230],[39,88101,17359],{"className":88102},[83],[39,88104,4280],{"className":88105},[263],[39,88107,88109,88133],{"className":88108,"translate":43},[42],[39,88110,88112],{"className":88111},[47],[49,88113,88114],{"xmlns":51},[53,88115,88116,88130],{},[56,88117,88118,88120,88122,88124,88126,88128],{},[59,88119,618],{},[133,88121,136],{"stretchy":135},[59,88123,1755],{},[133,88125,161],{"stretchy":135},[133,88127,793],{},[539,88129,546],{},[63,88131,88132],{"encoding":65},"F(x)=1",[39,88134,88136,88163],{"className":88135,"ariaHidden":70},[69],[39,88137,88139,88142,88145,88148,88151,88154,88157,88160],{"className":88138},[74],[39,88140],{"className":88141,"style":245},[78],[39,88143,618],{"className":88144,"style":178},[83,84],[39,88146,136],{"className":88147},[182],[39,88149,1755],{"className":88150},[83,84],[39,88152,161],{"className":88153},[263],[39,88155],{"className":88156,"style":231},[230],[39,88158,793],{"className":88159},[235],[39,88161],{"className":88162,"style":231},[230],[39,88164,88166,88169],{"className":88165},[74],[39,88167],{"className":88168,"style":2473},[78],[39,88170,546],{"className":88171},[83],[39,88173,88175,88197],{"className":88174,"translate":43},[42],[39,88176,88178],{"className":88177},[47],[49,88179,88180],{"xmlns":51},[53,88181,88182,88194],{},[56,88183,88184,88186,88188,88190,88192],{},[539,88185,546],{},[133,88187,5378],{},[59,88189,1755],{},[59,88191,45691],{"mathvariant":145},[539,88193,17359],{},[63,88195,88196],{"encoding":65},"1-x\u002F2",[39,88198,88200,88218],{"className":88199,"ariaHidden":70},[69],[39,88201,88203,88206,88209,88212,88215],{"className":88202},[74],[39,88204],{"className":88205,"style":17882},[78],[39,88207,546],{"className":88208},[83],[39,88210],{"className":88211,"style":461},[230],[39,88213,5378],{"className":88214},[465],[39,88216],{"className":88217,"style":461},[230],[39,88219,88221,88224,88227],{"className":88220},[74],[39,88222],{"className":88223,"style":245},[78],[39,88225,1755],{"className":88226},[83,84],[39,88228,87914],{"className":88229},[83],"，也积出 ",[39,88232,88234,88253],{"className":88233,"translate":43},[42],[39,88235,88237],{"className":88236},[47],[49,88238,88239],{"xmlns":51},[53,88240,88241,88251],{},[56,88242,88243],{},[770,88244,88245],{"scriptlevel":541,"displaystyle":135},[5362,88246,88247,88249],{},[539,88248,546],{},[539,88250,59223],{},[63,88252,87975],{"encoding":65},[39,88254,88256],{"className":88255,"ariaHidden":70},[69],[39,88257,88259,88262],{"className":88258},[74],[39,88260],{"className":88261,"style":74794},[78],[39,88263,88265,88268,88327],{"className":88264},[83],[39,88266],{"className":88267},[182,5463],[39,88269,88271],{"className":88270},[5362],[39,88272,88274,88319],{"className":88273},[196,649],[39,88275,88277,88316],{"className":88276},[200],[39,88278,88280,88294,88302],{"className":88279,"style":74813},[204],[39,88281,88282,88285],{"style":74816},[39,88283],{"className":88284,"style":935},[212],[39,88286,88288],{"className":88287},[217,218,219,220],[39,88289,88291],{"className":88290},[83,220],[39,88292,59223],{"className":88293},[83,220],[39,88295,88296,88299],{"style":5503},[39,88297],{"className":88298,"style":935},[212],[39,88300],{"className":88301,"style":5511},[5510],[39,88303,88304,88307],{"style":74839},[39,88305],{"className":88306,"style":935},[212],[39,88308,88310],{"className":88309},[217,218,219,220],[39,88311,88313],{"className":88312},[83,220],[39,88314,546],{"className":88315},[83,220],[39,88317,673],{"className":88318},[672],[39,88320,88322],{"className":88321},[200],[39,88323,88325],{"className":88324,"style":74861},[204],[39,88326],{},[39,88328],{"className":88329},[263,5463],"。合计 ",[39,88332,88334,88354],{"className":88333,"translate":43},[42],[39,88335,88337],{"className":88336},[47],[49,88338,88339],{"xmlns":51},[53,88340,88341,88351],{},[56,88342,88343],{},[770,88344,88345],{"scriptlevel":541,"displaystyle":135},[5362,88346,88347,88349],{},[539,88348,546],{},[539,88350,17359],{},[63,88352,88353],{"encoding":65},"\\tfrac12",[39,88355,88357],{"className":88356,"ariaHidden":70},[69],[39,88358,88360,88363],{"className":88359},[74],[39,88361],{"className":88362,"style":74794},[78],[39,88364,88366,88369,88428],{"className":88365},[83],[39,88367],{"className":88368},[182,5463],[39,88370,88372],{"className":88371},[5362],[39,88373,88375,88420],{"className":88374},[196,649],[39,88376,88378,88417],{"className":88377},[200],[39,88379,88381,88395,88403],{"className":88380,"style":74813},[204],[39,88382,88383,88386],{"style":74816},[39,88384],{"className":88385,"style":935},[212],[39,88387,88389],{"className":88388},[217,218,219,220],[39,88390,88392],{"className":88391},[83,220],[39,88393,17359],{"className":88394},[83,220],[39,88396,88397,88400],{"style":5503},[39,88398],{"className":88399,"style":935},[212],[39,88401],{"className":88402,"style":5511},[5510],[39,88404,88405,88408],{"style":74839},[39,88406],{"className":88407,"style":935},[212],[39,88409,88411],{"className":88410},[217,218,219,220],[39,88412,88414],{"className":88413},[83,220],[39,88415,546],{"className":88416},[83,220],[39,88418,673],{"className":88419},[672],[39,88421,88423],{"className":88422},[200],[39,88424,88426],{"className":88425,"style":74861},[204],[39,88427],{},[39,88429],{"className":88430},[263,5463],"，和分位数算法的结果一致。",[11,88433,88434,88435,88568,88569,88732,88733,88913],{},"一维之所以这么简单，是因为凸代价下最优耦合是“单调”的：不会出现 ",[39,88436,88438,88464],{"className":88437,"translate":43},[42],[39,88439,88441],{"className":88440},[47],[49,88442,88443],{"xmlns":51},[53,88444,88445,88461],{},[56,88446,88447,88453,88455],{},[613,88448,88449,88451],{},[59,88450,1755],{},[539,88452,546],{},[133,88454,1693],{},[613,88456,88457,88459],{},[59,88458,1755],{},[539,88460,17359],{},[63,88462,88463],{"encoding":65},"x_1\u003Cx_2",[39,88465,88467,88522],{"className":88466,"ariaHidden":70},[69],[39,88468,88470,88473,88513,88516,88519],{"className":88469},[74],[39,88471],{"className":88472,"style":29956},[78],[39,88474,88476,88479],{"className":88475},[83],[39,88477,1755],{"className":88478},[83,84],[39,88480,88482],{"className":88481},[192],[39,88483,88485,88505],{"className":88484},[196,649],[39,88486,88488,88502],{"className":88487},[200],[39,88489,88491],{"className":88490,"style":963},[204],[39,88492,88493,88496],{"style":966},[39,88494],{"className":88495,"style":213},[212],[39,88497,88499],{"className":88498},[217,218,219,220],[39,88500,546],{"className":88501},[83,220],[39,88503,673],{"className":88504},[672],[39,88506,88508],{"className":88507},[200],[39,88509,88511],{"className":88510,"style":680},[204],[39,88512],{},[39,88514],{"className":88515,"style":231},[230],[39,88517,1693],{"className":88518},[235],[39,88520],{"className":88521,"style":231},[230],[39,88523,88525,88528],{"className":88524},[74],[39,88526],{"className":88527,"style":2916},[78],[39,88529,88531,88534],{"className":88530},[83],[39,88532,1755],{"className":88533},[83,84],[39,88535,88537],{"className":88536},[192],[39,88538,88540,88560],{"className":88539},[196,649],[39,88541,88543,88557],{"className":88542},[200],[39,88544,88546],{"className":88545,"style":963},[204],[39,88547,88548,88551],{"style":966},[39,88549],{"className":88550,"style":213},[212],[39,88552,88554],{"className":88553},[217,218,219,220],[39,88555,17359],{"className":88556},[83,220],[39,88558,673],{"className":88559},[672],[39,88561,88563],{"className":88562},[200],[39,88564,88566],{"className":88565,"style":680},[204],[39,88567],{}," 却 ",[39,88570,88572,88610],{"className":88571,"translate":43},[42],[39,88573,88575],{"className":88574},[47],[49,88576,88577],{"xmlns":51},[53,88578,88579,88607],{},[56,88580,88581,88583,88585,88591,88593,88595,88597,88599,88605],{},[59,88582,11772],{},[133,88584,136],{"stretchy":135},[613,88586,88587,88589],{},[59,88588,1755],{},[539,88590,546],{},[133,88592,161],{"stretchy":135},[133,88594,31514],{},[59,88596,11772],{},[133,88598,136],{"stretchy":135},[613,88600,88601,88603],{},[59,88602,1755],{},[539,88604,17359],{},[133,88606,161],{"stretchy":135},[63,88608,88609],{"encoding":65},"T(x_1)>T(x_2)",[39,88611,88613,88677],{"className":88612,"ariaHidden":70},[69],[39,88614,88616,88619,88622,88625,88665,88668,88671,88674],{"className":88615},[74],[39,88617],{"className":88618,"style":245},[78],[39,88620,11772],{"className":88621,"style":178},[83,84],[39,88623,136],{"className":88624},[182],[39,88626,88628,88631],{"className":88627},[83],[39,88629,1755],{"className":88630},[83,84],[39,88632,88634],{"className":88633},[192],[39,88635,88637,88657],{"className":88636},[196,649],[39,88638,88640,88654],{"className":88639},[200],[39,88641,88643],{"className":88642,"style":963},[204],[39,88644,88645,88648],{"style":966},[39,88646],{"className":88647,"style":213},[212],[39,88649,88651],{"className":88650},[217,218,219,220],[39,88652,546],{"className":88653},[83,220],[39,88655,673],{"className":88656},[672],[39,88658,88660],{"className":88659},[200],[39,88661,88663],{"className":88662,"style":680},[204],[39,88664],{},[39,88666,161],{"className":88667},[263],[39,88669],{"className":88670,"style":231},[230],[39,88672,31514],{"className":88673},[235],[39,88675],{"className":88676,"style":231},[230],[39,88678,88680,88683,88686,88689,88729],{"className":88679},[74],[39,88681],{"className":88682,"style":245},[78],[39,88684,11772],{"className":88685,"style":178},[83,84],[39,88687,136],{"className":88688},[182],[39,88690,88692,88695],{"className":88691},[83],[39,88693,1755],{"className":88694},[83,84],[39,88696,88698],{"className":88697},[192],[39,88699,88701,88721],{"className":88700},[196,649],[39,88702,88704,88718],{"className":88703},[200],[39,88705,88707],{"className":88706,"style":963},[204],[39,88708,88709,88712],{"style":966},[39,88710],{"className":88711,"style":213},[212],[39,88713,88715],{"className":88714},[217,218,219,220],[39,88716,17359],{"className":88717},[83,220],[39,88719,673],{"className":88720},[672],[39,88722,88724],{"className":88723},[200],[39,88725,88727],{"className":88726,"style":680},[204],[39,88728],{},[39,88730,161],{"className":88731},[263],"，否则交换一下就更省。所以 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F_\\mu",[39,88774,88776,88794,88867],{"className":88775,"ariaHidden":70},[69],[39,88777,88779,88782,88785,88788,88791],{"className":88778},[74],[39,88780],{"className":88781,"style":79},[78],[39,88783,11772],{"className":88784,"style":178},[83,84],[39,88786],{"className":88787,"style":231},[230],[39,88789,793],{"className":88790},[235],[39,88792],{"className":88793,"style":231},[230],[39,88795,88797,88801,88858,88861,88864],{"className":88796},[74],[39,88798],{"className":88799,"style":88800},[78],"height:1.0611em;vertical-align:-0.247em;",[39,88802,88804,88807],{"className":88803},[83],[39,88805,618],{"className":88806,"style":178},[83,84],[39,88808,88810],{"className":88809},[192],[39,88811,88813,88850],{"className":88812},[196,649],[39,88814,88816,88847],{"className":88815},[200],[39,88817,88819,88830],{"className":88818,"style":19710},[204],[39,88820,88821,88824],{"style":13718},[39,88822],{"className":88823,"style":213},[212],[39,88825,88827],{"className":88826},[217,218,219,220],[39,88828,73609],{"className":88829,"style":73625},[83,84,220],[39,88831,88832,88835],{"style":208},[39,88833],{"className":88834,"style":213},[212],[39,88836,88838],{"className":88837},[217,218,219,220],[39,88839,88841,88844],{"className":88840},[83,220],[39,88842,5378],{"className":88843},[83,220],[39,88845,546],{"className":88846},[83,220],[39,88848,673],{"className":88849},[672],[39,88851,88853],{"className":88852},[200],[39,88854,88856],{"className":88855,"style":10003},[204],[39,88857],{},[39,88859],{"className":88860,"style":461},[230],[39,88862,88763],{"className":88863},[465],[39,88865],{"className":88866,"style":461},[230],[39,88868,88870,88873],{"className":88869},[74],[39,88871],{"className":88872,"style":35350},[78],[39,88874,88876,88879],{"className":88875},[83],[39,88877,618],{"className":88878,"style":178},[83,84],[39,88880,88882],{"className":88881},[192],[39,88883,88885,88905],{"className":88884},[196,649],[39,88886,88888,88902],{"className":88887},[200],[39,88889,88891],{"className":88890,"style":1728},[204],[39,88892,88893,88896],{"style":73765},[39,88894],{"className":88895,"style":213},[212],[39,88897,88899],{"className":88898},[217,218,219,220],[39,88900,73578],{"className":88901},[83,84,220],[39,88903,673],{"className":88904},[672],[39,88906,88908],{"className":88907},[200],[39,88909,88911],{"className":88910,"style":35396},[204],[39,88912],{}," 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ame":90470},[200],[39,90472,90474],{"className":90473,"style":680},[204],[39,90475],{},[39,90477,90479,90482],{"className":90478},[83],[39,90480,89893],{"className":90481,"style":2114},[83,84],[39,90483,90485],{"className":90484},[192],[39,90486,90488,90508],{"className":90487},[196,649],[39,90489,90491,90505],{"className":90490},[200],[39,90492,90494],{"className":90493,"style":963},[204],[39,90495,90496,90499],{"style":41316},[39,90497],{"className":90498,"style":213},[212],[39,90500,90502],{"className":90501},[217,218,219,220],[39,90503,17359],{"className":90504},[83,220],[39,90506,673],{"className":90507},[672],[39,90509,90511],{"className":90510},[200],[39,90512,90514],{"className":90513,"style":680},[204],[39,90515],{},[39,90517,150],{"className":90518},[83],[39,90520,161],{"className":90521},[263],[39,90523],{"className":90524,"style":231},[230],[39,90526,793],{"className":90527},[235],[39,90529],{"className":90530,"style":231},[230],[39,90532,90534,90537,90540,90580,90583,90586],{"className":90533},[74],[39,90535],{"className":90536,"style":245},[78],[39,90538,136],{"className":90539},[182],[39,90541,90543,90546],{"className":90542},[83],[39,90544,89893],{"className":90545,"style":2114},[83,84],[39,90547,90549],{"className":90548},[192],[39,90550,90552,90572],{"className":90551},[196,649],[39,90553,90555,90569],{"className":90554},[200],[39,90556,90558],{"className":90557,"style":963},[204],[39,90559,90560,90563],{"style":41316},[39,90561],{"className":90562,"style":213},[212],[39,90564,90566],{"className":90565},[217,218,219,220],[39,90567,546],{"className":90568},[83,220],[39,90570,673],{"className":90571},[672],[39,90573,90575],{"className":90574},[200],[39,90576,90578],{"className":90577,"style":680},[204],[39,90579],{},[39,90581],{"className":90582,"style":461},[230],[39,90584,5378],{"className":90585},[465],[39,90587],{"className":90588,"style":461},[230],[39,90590,90592,90595,90635],{"className":90591},[74],[39,90593],{"className":90594,"style":84615},[78],[39,90596,90598,90601],{"className":90597},[83],[39,90599,89893],{"className":90600,"style":2114},[83,84],[39,90602,90604],{"className":90603},[192],[39,90605,90607,90627],{"className":90606},[196,649],[39,90608,90610,90624],{"className":90609},[200],[39,90611,90613],{"className":90612,"style":963},[204],[39,90614,90615,90618],{"style":41316},[39,90616],{"className":90617,"style":213},[212],[39,90619,90621],{"className":90620},[217,218,219,220],[39,90622,17359],{"className":90623},[83,220],[39,90625,673],{"className":90626},[672],[39,90628,90630],{"className":90629},[200],[39,90631,90633],{"className":90632,"style":680},[204],[39,90634],{},[39,90636,90638,90641],{"className":90637},[263],[39,90639,161],{"className":90640},[263],[39,90642,90644],{"className":90643},[192],[39,90645,90647],{"className":90646},[196],[39,90648,90650],{"className":90649},[200],[39,90651,90653],{"className":90652,"style":19710},[204],[39,90654,90655,90658],{"style":208},[39,90656],{"className":90657,"style":213},[212],[39,90659,90661],{"className":90660},[217,218,219,220],[39,90662,17359],{"className":90663},[83,220],"，对得上。取 ",[39,90666,90668,90692],{"className":90667,"translate":43},[42],[39,90669,90671],{"className":90670},[47],[49,90672,90673],{"xmlns":51},[53,90674,90675,90689],{},[56,90676,90677,90679,90681,90683,90685,90687],{},[59,90678,77559],{"mathvariant":617},[133,90680,136],{"stretchy":135},[539,90682,541],{},[133,90684,155],{"separator":70},[539,90686,546],{},[133,90688,161],{"stretchy":135},[63,90690,90691],{"encoding":65},"\\mathcal N(0,1)",[39,90693,90695],{"className":90694,"ariaHidden":70},[69],[39,90696,90698,90701,90704,90707,90710,90713,90716,90719],{"className":90697},[74],[39,90699],{"className":90700,"style":245},[78],[39,90702,77559],{"className":90703,"style":77627},[83,641],[39,90705,136],{"className":90706},[182],[39,90708,541],{"className":90709},[83],[39,90711,155],{"className":90712},[252],[39,90714],{"className":90715,"style":256},[230],[39,90717,546],{"className":90718},[83],[39,90720,161],{"className":90721},[263],[39,90723,90725,90749],{"className":90724,"translate":43},[42],[39,90726,90728],{"className":90727},[47],[49,90729,90730],{"xmlns":51},[53,90731,90732,90746],{},[56,90733,90734,90736,90738,90740,90742,90744],{},[59,90735,77559],{"mathvariant":617},[133,90737,136],{"stretchy":135},[539,90739,541],{},[133,90741,155],{"separator":70},[539,90743,59223],{},[133,90745,161],{"stretchy":135},[63,90747,90748],{"encoding":65},"\\mathcal N(0,4)",[39,90750,90752],{"className":90751,"ariaHidden":70},[69],[39,90753,90755,90758,90761,90764,90767,90770,90773,90776],{"className":90754},[74],[39,90756],{"className":90757,"style":245},[78],[39,90759,77559],{"className":90760,"style":77627},[83,641],[39,90762,136],{"className":90763},[182],[39,90765,541],{"className":90766},[83],[39,90768,155],{"className":90769},[252],[39,90771],{"className":90772,"style":256},[230],[39,90774,59223],{"className":90775},[83],[39,90777,161],{"className":90778},[263],"，得 ",[39,90781,90783,90805],{"className":90782,"translate":43},[42],[39,90784,90786],{"className":90785},[47],[49,90787,90788],{"xmlns":51},[53,90789,90790,90802],{},[56,90791,90792,90798,90800],{},[613,90793,90794,90796],{},[59,90795,23703],{},[539,90797,17359],{},[133,90799,793],{},[539,90801,546],{},[63,90803,90804],{"encoding":65},"W_2=1",[39,90806,90808,90863],{"className":90807,"ariaHidden":70},[69],[39,90809,90811,90814,90854,90857,90860],{"className":90810},[74],[39,90812],{"className":90813,"style":634},[78],[39,90815,90817,90820],{"className":90816},[83],[39,90818,23703],{"className":90819,"style":178},[83,84],[39,90821,90823],{"className":90822},[192],[39,90824,90826,90846],{"className":90825},[196,649],[39,90827,90829,90843],{"className":90828},[200],[39,90830,90832],{"className":90831,"style":963},[204],[39,90833,90834,90837],{"style":73765},[39,90835],{"className":90836,"style":213},[212],[39,90838,90840],{"className":90839},[217,218,219,220],[39,90841,17359],{"className":90842},[83,220],[39,90844,673],{"className":90845},[672],[39,90847,90849],{"className":90848},[200],[39,90850,90852],{"className":90851,"style":680},[204],[39,90853],{},[39,90855],{"className":90856,"style":231},[230],[39,90858,793],{"className":90859},[235],[39,90861],{"className":90862,"style":231},[230],[39,90864,90866,90869],{"className":90865},[74],[39,90867],{"className":90868,"style":2473},[78],[39,90870,546],{"className":90871},[83],"，正好是标准差之差；而最优映射是 ",[39,90874,90876,90896],{"className":90875,"translate":43},[42],[39,90877,90879],{"className":90878},[47],[49,90880,90881],{"xmlns":51},[53,90882,90883,90893],{},[56,90884,90885,90887,90889,90891],{},[59,90886,1755],{},[133,90888,69813],{},[539,90890,17359],{},[59,90892,1755],{},[63,90894,90895],{"encoding":65},"x\\mapsto2x",[39,90897,90899,90917],{"className":90898,"ariaHidden":70},[69],[39,90900,90902,90905,90908,90911,90914],{"className":90901},[74],[39,90903],{"className":90904,"style":69838},[78],[39,90906,1755],{"className":90907},[83,84],[39,90909],{"className":90910,"style":231},[230],[39,90912,69813],{"className":90913},[235],[39,90915],{"className":90916,"style":231},[230],[39,90918,90920,90923,90926],{"className":90919},[74],[39,90921],{"className":90922,"style":2473},[78],[39,90924,17359],{"className":90925},[83],[39,90927,1755],{"className":90928},[83,84],"，位移的均方根也正好是 ",[39,90931,90933,90946],{"className":90932,"translate":43},[42],[39,90934,90936],{"className":90935},[47],[49,90937,90938],{"xmlns":51},[53,90939,90940,90944],{},[56,90941,90942],{},[539,90943,546],{},[63,90945,546],{"encoding":65},[39,90947,90949],{"className":90948,"ariaHidden":70},[69],[39,90950,90952,90955],{"className":90951},[74],[39,90953],{"className":90954,"style":2473},[78],[39,90956,546],{"className":90957},[83],"。两个数在这里是同一个数。",[11,90960,90961,90962,91169],{},"最优映射还是仿射的：",[39,90963,90965,91011],{"className":90964,"translate":43},[42],[39,90966,90968],{"className":90967},[47],[49,90969,90970],{"xmlns":51},[53,90971,90972,91008],{},[56,90973,90974,90976,90978,90980,90982,90984,90990,90992,90994,90996,90998,91000,91006],{},[59,90975,11772],{},[133,90977,136],{"stretchy":135},[59,90979,1755],{},[133,90981,161],{"stretchy":135},[133,90983,793],{},[613,90985,90986,90988],{},[59,90987,1748],{},[539,90989,17359],{},[133,90991,788],{},[59,90993,101],{},[133,90995,136],{"stretchy":135},[59,90997,1755],{},[133,90999,5378],{},[613,91001,91002,91004],{},[59,91003,1748],{},[539,91005,546],{},[133,91007,161],{"stretchy":135},[63,91009,91010],{"encoding":65},"T(x)=m_2+A(x-m_1)",[39,91012,91014,91041,91096,91120],{"className":91013,"ariaHidden":70},[69],[39,91015,91017,91020,91023,91026,91029,91032,91035,91038],{"className":91016},[74],[39,91018],{"className":91019,"style":245},[78],[39,91021,11772],{"className":91022,"style":178},[83,84],[39,91024,136],{"className":91025},[182],[39,91027,1755],{"className":91028},[83,84],[39,91030,161],{"className":91031},[263],[39,91033],{"className":91034,"style":231},[230],[39,91036,793],{"className":91037},[235],[39,91039],{"className":91040,"style":231},[230],[39,91042,91044,91047,91087,91090,91093],{"className":91043},[74],[39,91045],{"className":91046,"style":7528},[78],[39,91048,91050,91053],{"className":91049},[83],[39,91051,1748],{"className":91052},[83,84],[39,91054,91056],{"className":91055},[192],[39,91057,91059,91079],{"className":91058},[196,649],[39,91060,91062,91076],{"className":91061},[200],[39,91063,91065],{"className":91064,"style":963},[204],[39,91066,91067,91070],{"style":966},[39,91068],{"className":91069,"style":213},[212],[39,91071,91073],{"className":91072},[217,218,219,220],[39,91074,17359],{"className":91075},[83,220],[39,91077,673],{"className":91078},[672],[39,91080,91082],{"className":91081},[200],[39,91083,91085],{"className":91084,"style":680},[204],[39,91086],{},[39,91088],{"className":91089,"style":461},[230],[39,91091,788],{"className":91092},[465],[39,91094],{"className":91095,"style":461},[230],[39,91097,91099,91102,91105,91108,91111,91114,91117],{"className":91098},[74],[39,91100],{"className":91101,"style":245},[78],[39,91103,101],{"className":91104},[83,84],[39,91106,136],{"className":91107},[182],[39,91109,1755],{"className":91110},[83,84],[39,91112],{"className":91113,"style":461},[230],[39,91115,5378],{"className":91116},[465],[39,91118],{"className":91119,"style":461},[230],[39,91121,91123,91126,91166],{"className":91122},[74],[39,91124],{"className":91125,"style":245},[78],[39,91127,91129,91132],{"className":91128},[83],[39,91130,1748],{"className":91131},[83,84],[39,91133,91135],{"className":91134},[192],[39,91136,91138,91158],{"className":91137},[196,649],[39,91139,91141,91155],{"className":91140},[200],[39,91142,91144],{"className":91143,"style":963},[204],[39,91145,91146,91149],{"style":966},[39,91147],{"className":91148,"style":213},[212],[39,91150,91152],{"className":91151},[217,218,219,220],[39,91153,546],{"className":91154},[83,220],[39,91156,673],{"className":91157},[672],[39,91159,91161],{"className":91160},[200],[39,91162,91164],{"className":91163,"style":680},[204],[39,91165],{},[39,91167,161],{"className":91168},[263],"，其中",[39,91171,91173],{"className":91172,"translate":43},[746],[39,91174,91176,91274],{"className":91175,"translate":43},[42],[39,91177,91179],{"className":91178},[47],[49,91180,91181],{"xmlns":51,"display":755},[53,91182,91183,91271],{},[56,91184,91185,91187,91189,91205,91207,91221,91227,91241,91253,91269],{},[59,91186,101],{},[133,91188,793],{},[9932,91190,91191,91193,91195],{},[59,91192,88958],{"mathvariant":145},[539,91194,546],{},[56,91196,91197,91199,91201,91203],{},[133,91198,5378],{},[539,91200,546],{},[59,91202,45691],{"mathvariant":145},[539,91204,17359],{},[133,91206,136],{"fence":70,"stretchy":70,"minsize":73875,"maxsize":73875},[9932,91208,91209,91211,91213],{},[59,91210,88958],{"mathvariant":145},[539,91212,546],{},[56,91214,91215,91217,91219],{},[539,91216,546],{},[59,91218,45691],{"mathvariant":145},[539,91220,17359],{},[613,91222,91223,91225],{},[59,91224,88958],{"mathvariant":145},[539,91226,17359],{},[9932,91228,91229,91231,91233],{},[59,91230,88958],{"mathvariant":145},[539,91232,546],{},[56,91234,91235,91237,91239],{},[539,91236,546],{},[59,91238,45691],{"mathvariant":145},[539,91240,17359],{},[138,91242,91243,91245],{},[133,91244,161],{"fence":70,"stretchy":70,"minsize":73875,"maxsize":73875},[56,91246,91247,91249,91251],{},[539,91248,546],{},[59,91250,45691],{"mathvariant":145},[539,91252,17359],{},[9932,91254,91255,91257,91259],{},[59,91256,88958],{"mathvariant":145},[539,91258,546],{},[56,91260,91261,91263,91265,91267],{},[133,91262,5378],{},[539,91264,546],{},[59,91266,45691],{"mathvariant":145},[539,91268,17359],{},[59,91270,2903],{"mathvariant":145},[63,91272,91273],{"encoding":65},"A=\\Sigma_1^{-1\u002F2}\\bigl(\\Sigma_1^{1\u002F2}\\Sigma_2\\Sigma_1^{1\u002F2}\\bigr)^{1\u002F2}\\Sigma_1^{-1\u002F2}.",[39,91275,91277,91295],{"className":91276,"ariaHidden":70},[69],[39,91278,91280,91283,91286,91289,91292],{"className":91279},[74],[39,91281],{"className":91282,"style":79},[78],[39,91284,101],{"className":91285},[83,84],[39,91287],{"className":91288,"style":231},[230],[39,91290,793],{"className":91291},[235],[39,91293],{"className":91294,"style":231},[230],[39,91296,91298,91302,91359,91365,91419,91459,91513,91548,91605],{"className":91297},[74],[39,91299],{"className":91300,"style":91301},[78],"height:1.4779em;vertical-align:-0.35em;",[39,91303,91305,91308],{"className":91304},[83],[39,91306,88958],{"className":91307},[83],[39,91309,91311],{"className":91310},[192],[39,91312,91314,91351],{"className":91313},[196,649],[39,91315,91317,91348],{"className":91316},[200],[39,91318,91320,91331],{"className":91319,"style":89669},[204],[39,91321,91322,91325],{"style":89672},[39,91323],{"className":91324,"style":213},[212],[39,91326,91328],{"className":91327},[217,218,219,220],[39,91329,546],{"className":91330},[83,220],[39,91332,91333,91336],{"style":89684},[39,91334],{"className":91335,"style":213},[212],[39,91337,91339],{"className":91338},[217,218,219,220],[39,91340,91342,91345],{"className":91341},[83,220],[39,91343,5378],{"className":91344},[83,220],[39,91346,45696],{"className":91347},[83,220],[39,91349,673],{"className":91350},[672],[39,91352,91354],{"className":91353},[200],[39,91355,91357],{"className":91356,"style":89706},[204],[39,91358],{},[39,91360,91362],{"className":91361},[182],[39,91363,136],{"className":91364},[18520,4009],[39,91366,91368,91371],{"className":91367},[83],[39,91369,88958],{"className":91370},[83],[39,91372,91374],{"className":91373},[192],[39,91375,91377,91411],{"className":91376},[196,649],[39,91378,91380,91408],{"className":91379},[200],[39,91381,91383,91394],{"className":91382,"style":89669},[204],[39,91384,91385,91388],{"style":89672},[39,91386],{"className":91387,"style":213},[212],[39,91389,91391],{"className":91390},[217,218,219,220],[39,91392,546],{"className":91393},[83,220],[39,91395,91396,91399],{"style":89684},[39,91397],{"className":91398,"style":213},[212],[39,91400,91402],{"className":91401},[217,218,219,220],[39,91403,91405],{"className":91404},[83,220],[39,91406,45696],{"className":91407},[83,220],[39,91409,673],{"className":91410},[672],[39,91412,91414],{"className":91413},[200],[39,91415,91417],{"className":91416,"style":89706},[204],[39,91418],{},[39,91420,91422,91425],{"className":91421},[83],[39,91423,88958],{"className":91424},[83],[39,91426,91428],{"className":91427},[192],[39,91429,91431,91451],{"className":91430},[196,649],[39,91432,91434,91448],{"className":91433},[200],[39,91435,91437],{"className":91436,"style":963},[204],[39,91438,91439,91442],{"style":966},[39,91440],{"className":91441,"style":213},[212],[39,91443,91445],{"className":91444},[217,218,219,220],[39,91446,17359],{"className":91447},[83,220],[39,91449,673],{"className":91450},[672],[39,91452,91454],{"className":91453},[200],[39,91455,91457],{"className":91456,"style":680},[204],[39,91458],{},[39,91460,91462,91465],{"className":91461},[83],[39,91463,88958],{"className":91464},[83],[39,91466,91468],{"className":91467},[192],[39,91469,91471,91505],{"className":91470},[196,649],[39,91472,91474,91502],{"className":91473},[200],[39,91475,91477,91488],{"className":91476,"style":89669},[204],[39,91478,91479,91482],{"style":89672},[39,91480],{"className":91481,"style":213},[212],[39,91483,91485],{"className":91484},[217,218,219,220],[39,91486,546],{"className":91487},[83,220],[39,91489,91490,91493],{"style":89684},[39,91491],{"className":91492,"style":213},[212],[39,91494,91496],{"className":91495},[217,218,219,220],[39,91497,91499],{"className":91498},[83,220],[39,91500,45696],{"className":91501},[83,220],[39,91503,673],{"className":91504},[672],[39,91506,91508],{"className":91507},[200],[39,91509,91511],{"className":91510,"style":89706},[204],[39,91512],{},[39,91514,91516,91522],{"className":91515},[263],[39,91517,91519],{"className":91518},[263],[39,91520,161],{"className":91521},[18520,4009],[39,91523,91525],{"className":91524},[192],[39,91526,91528],{"className":91527},[196],[39,91529,91531],{"className":91530},[200],[39,91532,91534],{"className":91533,"style":89824},[204],[39,91535,91536,91539],{"style":86136},[39,91537],{"className":91538,"style":213},[212],[39,91540,91542],{"className":91541},[217,218,219,220],[39,91543,91545],{"className":91544},[83,220],[39,91546,45696],{"className":91547},[83,220],[39,91549,91551,91554],{"className":91550},[83],[39,91552,88958],{"className":91553},[83],[39,91555,91557],{"className":91556},[192],[39,91558,91560,91597],{"className":91559},[196,649],[39,91561,91563,91594],{"className":91562},[200],[39,91564,91566,91577],{"className":91565,"style":89669},[204],[39,91567,91568,91571],{"style":89672},[39,91569],{"className":91570,"style":213},[212],[39,91572,91574],{"className":91573},[217,218,219,220],[39,91575,546],{"className":91576},[83,220],[39,91578,91579,91582],{"style":89684},[39,91580],{"className":91581,"style":213},[212],[39,91583,91585],{"className":91584},[217,218,219,220],[39,91586,91588,91591],{"className":91587},[83,220],[39,91589,5378],{"className":91590},[83,220],[39,91592,45696],{"className":91593},[83,220],[39,91595,673],{"className":91596},[672],[39,91598,91600],{"className":91599},[200],[39,91601,91603],{"className":91602,"style":89706},[204],[39,91604],{},[39,91606,2903],{"className":91607},[83],[11,91609,91610],{},"写成操作就是“先白化再上色”，风格迁移里的 AdaIN 做的正是这个（对角版本）。所以“高斯之间的最优传输”这件事，工程界早就在做了，只是没有这么叫。",[11,91612,91613,91614,91684,91685,91913,91914,91983],{},"高斯集合配上 ",[39,91615,91617,91635],{"className":91616,"translate":43},[42],[39,91618,91620],{"className":91619},[47],[49,91621,91622],{"xmlns":51},[53,91623,91624,91632],{},[56,91625,91626],{},[613,91627,91628,91630],{},[59,91629,23703],{},[539,91631,17359],{},[63,91633,91634],{"encoding":65},"W_2",[39,91636,91638],{"className":91637,"ariaHidden":70},[69],[39,91639,91641,91644],{"className":91640},[74],[39,91642],{"className":91643,"style":634},[78],[39,91645,91647,91650],{"className":91646},[83],[39,91648,23703],{"className":91649,"style":178},[83,84],[39,91651,91653],{"className":91652},[192],[39,91654,91656,91676],{"className":91655},[196,649],[39,91657,91659,91673],{"className":91658},[200],[39,91660,91662],{"className":91661,"style":963},[204],[39,91663,91664,91667],{"style":73765},[39,91665],{"className":91666,"style":213},[212],[39,91668,91670],{"className":91669},[217,218,219,220],[39,91671,17359],{"className":91672},[83,220],[39,91674,673],{"className":91675},[672],[39,91677,91679],{"className":91678},[200],[39,91680,91682],{"className":91681,"style":680},[204],[39,91683],{}," 构成一个黎曼流形（Bures–Wasserstein 空间），两点间的测地线就是位移插值 ",[39,91686,91688,91740],{"className":91687,"translate":43},[42],[39,91689,91691],{"className":91690},[47],[49,91692,91693],{"xmlns":51},[53,91694,91695,91737],{},[56,91696,91697,91703,91705,91707,91709,91711,91713,91715,91717,91723,91725,91727,91729,91735],{},[613,91698,91699,91701],{},[59,91700,73578],{},[59,91702,621],{},[133,91704,793],{},[133,91706,136],{"fence":70,"stretchy":70,"minsize":73875,"maxsize":73875},[133,91708,136],{"stretchy":135},[539,91710,546],{},[133,91712,5378],{},[59,91714,621],{},[133,91716,161],{"stretchy":135},[56,91718,91719,91721],{},[59,91720,13377],{"mathvariant":145},[59,91722,29425],{"mathvariant":145},[133,91724,788],{},[59,91726,621],{},[59,91728,11772],{},[613,91730,91731,91733],{},[133,91732,161],{"fence":70,"stretchy":70,"minsize":73875,"maxsize":73875},[59,91734,73726],{"mathvariant":145},[59,91736,73578],{},[63,91738,91739],{"encoding":65},"\\mu_t=\\bigl((1-t)\\mathrm{id}+tT\\bigr)_\\#\\mu",[39,91741,91743,91798,91825,91853],{"className":91742,"ariaHidden":70},[69],[39,91744,91746,91749,91789,91792,91795],{"className":91745},[74],[39,91747],{"className":91748,"style":2654},[78],[39,91750,91752,91755],{"className":91751},[83],[39,91753,73578],{"className":91754},[83,84],[39,91756,91758],{"className":91757},[192],[39,91759,91761,91781],{"className":91760},[196,649],[39,91762,91764,91778],{"className":91763},[200],[39,91765,91767],{"className":91766,"style":656},[204],[39,91768,91769,91772],{"style":966},[39,91770],{"className":91771,"style":213},[212],[39,91773,91775],{"className":91774},[217,218,219,220],[39,91776,621],{"className":91777},[83,84,220],[39,91779,673],{"className":91780},[672],[39,91782,91784],{"className":91783},[200],[39,91785,91787],{"className":91786,"style":680},[204],[39,91788],{},[39,91790],{"className":91791,"style":231},[230],[39,91793,793],{"className":91794},[235],[39,91796],{"className":91797,"style":231},[230],[39,91799,91801,91804,91810,91813,91816,91819,91822],{"className":91800},[74],[39,91802],{"className":91803,"style":908},[78],[39,91805,91807],{"className":91806},[182],[39,91808,136],{"className":91809},[18520,4009],[39,91811,136],{"className":91812},[182],[39,91814,546],{"className":91815},[83],[39,91817],{"className":91818,"style":461},[230],[39,91820,5378],{"className":91821},[465],[39,91823],{"className":91824,"style":461},[230],[39,91826,91828,91831,91834,91837,91844,91847,91850],{"className":91827},[74],[39,91829],{"className":91830,"style":245},[78],[39,91832,621],{"className":91833},[83,84],[39,91835,161],{"className":91836},[263],[39,91838,91840],{"className":91839},[83],[39,91841,91843],{"className":91842},[83,35077],"id",[39,91845],{"className":91846,"style":461},[230],[39,91848,788],{"className":91849},[465],[39,91851],{"className":91852,"style":461},[230],[39,91854,91856,91860,91864,91910],{"className":91855},[74],[39,91857],{"className":91858,"style":91859},[78],"height:1.3858em;vertical-align:-0.5358em;",[39,91861,91863],{"className":91862,"style":178},[83,84],"tT",[39,91865,91867,91873],{"className":91866},[263],[39,91868,91870],{"className":91869},[263],[39,91871,161],{"className":91872},[18520,4009],[39,91874,91876],{"className":91875},[192],[39,91877,91879,91901],{"className":91878},[196,649],[39,91880,91882,91898],{"className":91881},[200],[39,91883,91886],{"className":91884,"style":91885},[204],"height:0.0864em;",[39,91887,91889,91892],{"style":91888},"top:-2.3003em;margin-right:0.05em;",[39,91890],{"className":91891,"style":213},[212],[39,91893,91895],{"className":91894},[217,218,219,220],[39,91896,73726],{"className":91897},[83,220],[39,91899,673],{"className":91900},[672],[39,91902,91904],{"className":91903},[200],[39,91905,91908],{"className":91906,"style":91907},[204],"height:0.5358em;",[39,91909],{},[39,91911,73578],{"className":91912},[83,84],"。这意味着我可以在分布空间里插值和求平均。我在写 Fréchet 均值那篇时说过，在没有向量结构的空间里“平均”要重新定义；",[39,91915,91917,91934],{"className":91916,"translate":43},[42],[39,91918,91920],{"className":91919},[47],[49,91921,91922],{"xmlns":51},[53,91923,91924,91932],{},[56,91925,91926],{},[613,91927,91928,91930],{},[59,91929,23703],{},[539,91931,17359],{},[63,91933,91634],{"encoding":65},[39,91935,91937],{"className":91936,"ariaHidden":70},[69],[39,91938,91940,91943],{"className":91939},[74],[39,91941],{"className":91942,"style":634},[78],[39,91944,91946,91949],{"className":91945},[83],[39,91947,23703],{"className":91948,"style":178},[83,84],[39,91950,91952],{"className":91951},[192],[39,91953,91955,91975],{"className":91954},[196,649],[39,91956,91958,91972],{"className":91957},[200],[39,91959,91961],{"className":91960,"style":963},[204],[39,91962,91963,91966],{"style":73765},[39,91964],{"className":91965,"style":213},[212],[39,91967,91969],{"className":91968},[217,218,219,220],[39,91970,17359],{"className":91971},[83,220],[39,91973,673],{"className":91974},[672],[39,91976,91978],{"className":91977},[200],[39,91979,91981],{"className":91980,"style":680},[204],[39,91982],{}," 空间是最典型的例子——两个高斯的中点要沿测地线走一半，不是把参数取平均。",[21,91985,91986],{"id":91986},"计算的代价",[11,91988,91989,92017],{},[39,91990,91992,92005],{"className":91991,"translate":43},[42],[39,91993,91995],{"className":91994},[47],[49,91996,91997],{"xmlns":51},[53,91998,91999,92003],{},[56,92000,92001],{},[59,92002,23703],{},[63,92004,23703],{"encoding":65},[39,92006,92008],{"className":92007,"ariaHidden":70},[69],[39,92009,92011,92014],{"className":92010},[74],[39,92012],{"className":92013,"style":79},[78],[39,92015,23703],{"className":92016,"style":178},[83,84]," 定义了“是什么”，但没告诉我“怎么算”。",[11,92019,92020,92104,92105,92167,92168,92231,92232,92260],{},[39,92021,92023,92045],{"className":92022,"translate":43},[42],[39,92024,92026],{"className":92025},[47],[49,92027,92028],{"xmlns":51},[53,92029,92030,92042],{},[56,92031,92032,92034,92036],{},[59,92033,12962],{},[133,92035,793],{},[138,92037,92038,92040],{},[539,92039,48919],{},[539,92041,59223],{},[63,92043,92044],{"encoding":65},"n=10^4",[39,92046,92048,92066],{"className":92047,"ariaHidden":70},[69],[39,92049,92051,92054,92057,92060,92063],{"className":92050},[74],[39,92052],{"className":92053,"style":290},[78],[39,92055,12962],{"className":92056},[83,84],[39,92058],{"className":92059,"style":231},[230],[39,92061,793],{"className":92062},[235],[39,92064],{"className":92065,"style":231},[230],[39,92067,92069,92072,92075],{"className":92068},[74],[39,92070],{"className":92071,"style":19710},[78],[39,92073,546],{"className":92074},[83],[39,92076,92078,92081],{"className":92077},[83],[39,92079,541],{"className":92080},[83],[39,92082,92084],{"className":92083},[192],[39,92085,92087],{"className":92086},[196],[39,92088,92090],{"className":92089},[200],[39,92091,92093],{"className":92092,"style":19710},[204],[39,92094,92095,92098],{"style":208},[39,92096],{"className":92097,"style":213},[212],[39,92099,92101],{"className":92100},[217,218,219,220],[39,92102,59223],{"className":92103},[83,220]," 个样本对 ",[39,92106,92108,92126],{"className":92107,"translate":43},[42],[39,92109,92111],{"className":92110},[47],[49,92112,92113],{"xmlns":51},[53,92114,92115,92123],{},[56,92116,92117],{},[138,92118,92119,92121],{},[539,92120,48919],{},[539,92122,59223],{},[63,92124,92125],{"encoding":65},"10^4",[39,92127,92129],{"className":92128,"ariaHidden":70},[69],[39,92130,92132,92135,92138],{"className":92131},[74],[39,92133],{"className":92134,"style":19710},[78],[39,92136,546],{"className":92137},[83],[39,92139,92141,92144],{"className":92140},[83],[39,92142,541],{"className":92143},[83],[39,92145,92147],{"className":92146},[192],[39,92148,92150],{"className":92149},[196],[39,92151,92153],{"className":92152},[200],[39,92154,92156],{"className":92155,"style":19710},[204],[39,92157,92158,92161],{"style":208},[39,92159],{"className":92160,"style":213},[212],[39,92162,92164],{"className":92163},[217,218,219,220],[39,92165,59223],{"className":92166},[83,220]," 个样本，代价矩阵就是 ",[39,92169,92171,92190],{"className":92170,"translate":43},[42],[39,92172,92174],{"className":92173},[47],[49,92175,92176],{"xmlns":51},[53,92177,92178,92187],{},[56,92179,92180],{},[138,92181,92182,92184],{},[539,92183,48919],{},[539,92185,92186],{},"8",[63,92188,92189],{"encoding":65},"10^8",[39,92191,92193],{"className":92192,"ariaHidden":70},[69],[39,92194,92196,92199,92202],{"className":92195},[74],[39,92197],{"className":92198,"style":19710},[78],[39,92200,546],{"className":92201},[83],[39,92203,92205,92208],{"className":92204},[83],[39,92206,541],{"className":92207},[83],[39,92209,92211],{"className":92210},[192],[39,92212,92214],{"className":92213},[196],[39,92215,92217],{"className":92216},[200],[39,92218,92220],{"className":92219,"style":19710},[204],[39,92221,92222,92225],{"style":208},[39,92223],{"className":92224,"style":213},[212],[39,92226,92228],{"className":92227},[217,218,219,220],[39,92229,92186],{"className":92230},[83,220]," 个数，float32 存下来 400 MB，还没开始迭代。我最早在自己机器上试 OT，把 ",[39,92233,92235,92248],{"className":92234,"translate":43},[42],[39,92236,92238],{"className":92237},[47],[49,92239,92240],{"xmlns":51},[53,92241,92242,92246],{},[56,92243,92244],{},[59,92245,12962],{},[63,92247,12962],{"encoding":65},[39,92249,92251],{"className":92250,"ariaHidden":70},[69],[39,92252,92254,92257],{"className":92253},[74],[39,92255],{"className":92256,"style":290},[78],[39,92258,12962],{"className":92259},[83,84]," 开到 5000 就被内存劝退了。",[11,92262,92263,92264,92339,92340,92368,92369,92397],{},"离散 OT 是线性规划，实践上是 ",[39,92265,92267,92292],{"className":92266,"translate":43},[42],[39,92268,92270],{"className":92269},[47],[49,92271,92272],{"xmlns":51},[53,92273,92274,92289],{},[56,92275,92276,92279,92281,92287],{},[59,92277,92278],{},"O",[133,92280,136],{"stretchy":135},[138,92282,92283,92285],{},[59,92284,12962],{},[539,92286,59187],{},[133,92288,161],{"stretchy":135},[63,92290,92291],{"encoding":65},"O(n^3)",[39,92293,92295],{"className":92294,"ariaHidden":70},[69],[39,92296,92298,92301,92304,92307,92336],{"className":92297},[74],[39,92299],{"className":92300,"style":84615},[78],[39,92302,92278],{"className":92303,"style":438},[83,84],[39,92305,136],{"className":92306},[182],[39,92308,92310,92313],{"className":92309},[83],[39,92311,12962],{"className":92312},[83,84],[39,92314,92316],{"className":92315},[192],[39,92317,92319],{"className":92318},[196],[39,92320,92322],{"className":92321},[200],[39,92323,92325],{"className":92324,"style":19710},[204],[39,92326,92327,92330],{"style":208},[39,92328],{"className":92329,"style":213},[212],[39,92331,92333],{"className":92332},[217,218,219,220],[39,92334,59187],{"className":92335},[83,220],[39,92337,161],{"className":92338},[263]," 量级的算法。更麻烦的是解落在多面体的顶点上，",[39,92341,92343,92356],{"className":92342,"translate":43},[42],[39,92344,92346],{"className":92345},[47],[49,92347,92348],{"xmlns":51},[53,92349,92350,92354],{},[56,92351,92352],{},[59,92353,2098],{},[63,92355,2101],{"encoding":65},[39,92357,92359],{"className":92358,"ariaHidden":70},[69],[39,92360,92362,92365],{"className":92361},[74],[39,92363],{"className":92364,"style":290},[78],[39,92366,2098],{"className":92367,"style":2114},[83,84]," 很稀疏，而 ",[39,92370,92372,92385],{"className":92371,"translate":43},[42],[39,92373,92375],{"className":92374},[47],[49,92376,92377],{"xmlns":51},[53,92378,92379,92383],{},[56,92380,92381],{},[59,92382,23703],{},[63,92384,23703],{"encoding":65},[39,92386,92388],{"className":92387,"ariaHidden":70},[69],[39,92389,92391,92394],{"className":92390},[74],[39,92392],{"className":92393,"style":79},[78],[39,92395,23703],{"className":92396,"style":178},[83,84]," 作为代价矩阵的函数是分段线性的，梯度分段常数、在切换点跳变。这种“硬”信号在优化里很难用。",[11,92399,92400],{},"Cuturi 在 2013 年的做法是加熵正则：",[39,92402,92404],{"className":92403,"translate":43},[746],[39,92405,92407,92501],{"className":92406,"translate":43},[42],[39,92408,92410],{"className":92409},[47],[49,92411,92412],{"xmlns":51,"display":755},[53,92413,92414,92498],{},[56,92415,92416,92443,92445,92448,92450,92452,92454,92457,92459,92461,92471,92481,92484,92486,92496],{},[6556,92417,92418,92425],{},[56,92419,92420,92423],{},[59,92421,92422],{},"min",[133,92424,1690],{},[56,92426,92427,92429,92431,92433,92435,92437,92439,92441],{},[59,92428,2098],{},[133,92430,534],{},[59,92432,75153],{"mathvariant":145},[133,92434,136],{"stretchy":135},[59,92436,73578],{},[133,92438,155],{"separator":70},[59,92440,73609],{},[133,92442,161],{"stretchy":135},[35841,92444,74223],{},[133,92446,92447],{"stretchy":135},"⟨",[59,92449,48444],{},[133,92451,155],{"separator":70},[59,92453,2098],{},[133,92455,92456],{"stretchy":135},"⟩",[133,92458,788],{},[59,92460,58009],{},[6556,92462,92463,92465],{},[133,92464,3191],{},[56,92466,92467,92469],{},[59,92468,13377],{},[59,92470,13446],{},[613,92472,92473,92475],{},[59,92474,2098],{},[56,92476,92477,92479],{},[59,92478,13377],{},[59,92480,13446],{},[59,92482,92483],{},"log",[133,92485,1690],{},[613,92487,92488,92490],{},[59,92489,2098],{},[56,92491,92492,92494],{},[59,92493,13377],{},[59,92495,13446],{},[59,92497,2903],{"mathvariant":145},[63,92499,92500],{"encoding":65},"\\min_{\\pi\\in\\Pi(\\mu,\\nu)}\\ \\langle C,\\pi\\rangle+\\varepsilon\\sum_{ij}\\pi_{ij}\\log\\pi_{ij}.",[39,92502,92504,92610],{"className":92503,"ariaHidden":70},[69],[39,92505,92507,92511,92580,92583,92586,92589,92592,92595,92598,92601,92604,92607],{"className":92506},[74],[39,92508],{"className":92509,"style":92510},[78],"height:1.716em;vertical-align:-0.966em;",[39,92512,92514],{"className":92513},[1744,3294],[39,92515,92517,92572],{"className":92516},[196,649],[39,92518,92520,92569],{"className":92519},[200],[39,92521,92524,92559],{"className":92522,"style":92523},[204],"height:0.6679em;",[39,92525,92526,92529],{"style":75939},[39,92527],{"className":92528,"style":935},[212],[39,92530,92532],{"className":92531},[217,218,219,220],[39,92533,92535,92538,92541,92544,92547,92550,92553,92556],{"className":92534},[83,220],[39,92536,2098],{"className":92537,"style":2114},[83,84,220],[39,92539,534],{"className":92540},[235,220],[39,92542,75153],{"className":92543},[83,220],[39,92545,136],{"className":92546},[182,220],[39,92548,73578],{"className":92549},[83,84,220],[39,92551,155],{"className":92552},[252,220],[39,92554,73609],{"className":92555,"style":73625},[83,84,220],[39,92557,161],{"className":92558},[263,220],[39,92560,92561,92564],{"style":21599},[39,92562],{"className":92563,"style":935},[212],[39,92565,92566],{},[39,92567,92422],{"className":92568},[1744],[39,92570,673],{"className":92571},[672],[39,92573,92575],{"className":92574},[200],[39,92576,92578],{"className":92577,"style":75994},[204],[39,92579],{},[39,92581,74223],{"className":92582},[230],[39,92584,92447],{"className":92585},[182],[39,92587,48444],{"className":92588,"style":48487},[83,84],[39,92590,155],{"className":92591},[252],[39,92593],{"className":92594,"style":256},[230],[39,92596,2098],{"className":92597,"style":2114},[83,84],[39,92599,92456],{"className":92600},[263],[39,92602],{"className":92603,"style":461},[230],[39,92605,788],{"className":92606},[465],[39,92608],{"className":92609,"style":461},[230],[39,92611,92613,92616,92619,92622,92669,92672,92715,92718,92724,92727,92770],{"className":92612},[74],[39,92614],{"className":92615,"style":83038},[78],[39,92617,58009],{"className":92618},[83,84],[39,92620],{"className":92621,"style":256},[230],[39,92623,92625],{"className":92624},[1744,3294],[39,92626,92628,92661],{"className":92627},[196,649],[39,92629,92631,92658],{"className":92630},[200],[39,92632,92634,92648],{"className":92633,"style":6670},[204],[39,92635,92636,92639],{"style":13926},[39,92637],{"className":92638,"style":3311},[212],[39,92640,92642],{"className":92641},[217,218,219,220],[39,92643,92645],{"className":92644},[83,220],[39,92646,13732],{"className":92647,"style":13731},[83,84,220],[39,92649,92650,92653],{"style":3330},[39,92651],{"className":92652,"style":3311},[212],[39,92654,92655],{},[39,92656,3191],{"className":92657},[1744,3339,3340],[39,92659,673],{"className":92660},[672],[39,92662,92664],{"className":92663},[200],[39,92665,92667],{"className":92666,"style":13958},[204],[39,92668],{},[39,92670],{"className":92671,"style":256},[230],[39,92673,92675,92678],{"className":92674},[83],[39,92676,2098],{"className":92677,"style":2114},[83,84],[39,92679,92681],{"className":92680},[192],[39,92682,92684,92707],{"className":92683},[196,649],[39,92685,92687,92704],{"className":92686},[200],[39,92688,92690],{"className":92689,"style":13604},[204],[39,92691,92692,92695],{"style":41316},[39,92693],{"className":92694,"style":213},[212],[39,92696,92698],{"className":92697},[217,218,219,220],[39,92699,92701],{"className":92700},[83,220],[39,92702,13732],{"className":92703,"style":13731},[83,84,220],[39,92705,673],{"className":92706},[672],[39,92708,92710],{"className":92709},[200],[39,92711,92713],{"className":92712,"style":35396},[204],[39,92714],{},[39,92716],{"className":92717,"style":256},[230],[39,92719,92721,92722],{"className":92720},[1744],"lo",[39,92723,39209],{"style":35078},[39,92725],{"className":92726,"style":256},[230],[39,92728,92730,92733],{"className":92729},[83],[39,92731,2098],{"className":92732,"style":2114},[83,84],[39,92734,92736],{"className":92735},[192],[39,92737,92739,92762],{"className":92738},[196,649],[39,92740,92742,92759],{"className":92741},[200],[39,92743,92745],{"className":92744,"style":13604},[204],[39,92746,92747,92750],{"style":41316},[39,92748],{"className":92749,"style":213},[212],[39,92751,92753],{"className":92752},[217,218,219,220],[39,92754,92756],{"className":92755},[83,220],[39,92757,13732],{"className":92758,"style":13731},[83,84,220],[39,92760,673],{"className":92761},[672],[39,92763,92765],{"className":92764},[200],[39,92766,92768],{"className":92767,"style":35396},[204],[39,92769],{},[39,92771,2903],{"className":92772},[83],[11,92774,92775],{},"熵项把解从顶点“推”到内部，变成严格正的光滑解。用拉格朗日乘子可以解出",[39,92777,92779],{"className":92778,"translate":43},[746],[39,92780,92782,92871],{"className":92781,"translate":43},[42],[39,92783,92785],{"className":92784},[47],[49,92786,92787],{"xmlns":51,"display":755},[53,92788,92789,92868],{},[56,92790,92791,92801,92803,92809,92819,92825,92827,92829,92839,92841,92844,92846,92848,92850,92860,92862,92864,92866],{},[613,92792,92793,92795],{},[59,92794,2098],{},[56,92796,92797,92799],{},[59,92798,13377],{},[59,92800,13446],{},[133,92802,793],{},[613,92804,92805,92807],{},[59,92806,35321],{},[59,92808,13377],{},[613,92810,92811,92813],{},[59,92812,77187],{},[56,92814,92815,92817],{},[59,92816,13377],{},[59,92818,13446],{},[613,92820,92821,92823],{},[59,92822,77971],{},[59,92824,13446],{},[133,92826,155],{"separator":70},[230,92828],{"width":10251},[613,92830,92831,92833],{},[59,92832,77187],{},[56,92834,92835,92837],{},[59,92836,13377],{},[59,92838,13446],{},[133,92840,793],{},[59,92842,92843],{},"exp",[133,92845,1690],{},[133,92847,136],{"stretchy":135},[133,92849,5378],{},[613,92851,92852,92854],{},[59,92853,48444],{},[56,92855,92856,92858],{},[59,92857,13377],{},[59,92859,13446],{},[59,92861,45691],{"mathvariant":145},[59,92863,58009],{},[133,92865,161],{"stretchy":135},[133,92867,155],{"separator":70},[63,92869,92870],{"encoding":65},"\\pi_{ij}=u_iK_{ij}v_j,\\qquad K_{ij}=\\exp(-C_{ij}\u002F\\varepsilon),",[39,92872,92874,92932,93122],{"className":92873,"ariaHidden":70},[69],[39,92875,92877,92880,92923,92926,92929],{"className":92876},[74],[39,92878],{"className":92879,"style":41295},[78],[39,92881,92883,92886],{"className":92882},[83],[39,92884,2098],{"className":92885,"style":2114},[83,84],[39,92887,92889],{"className":92888},[192],[39,92890,92892,92915],{"className":92891},[196,649],[39,92893,92895,92912],{"className":92894},[200],[39,92896,92898],{"className":92897,"style":13604},[204],[39,92899,92900,92903],{"style":41316},[39,92901],{"className":92902,"style":213},[212],[39,92904,92906],{"className":92905},[217,218,219,220],[39,92907,92909],{"className":92908},[83,220],[39,92910,13732],{"className":92911,"style":13731},[83,84,220],[39,92913,673],{"className":92914},[672],[39,92916,92918],{"className":92917},[200],[39,92919,92921],{"className":92920,"style":35396},[204],[39,92922],{},[39,92924],{"className":92925,"style":231},[230],[39,92927,793],{"className":92928},[235],[39,92930],{"className":92931,"style":231},[230],[39,92933,92935,92938,92978,93021,93061,93064,93067,93070,93113,93116,93119],{"className":92934},[74],[39,92936],{"className":92937,"style":35350},[78],[39,92939,92941,92944],{"className":92940},[83],[39,92942,35321],{"className":92943},[83,84],[39,92945,92947],{"className":92946},[192],[39,92948,92950,92970],{"className":92949},[196,649],[39,92951,92953,92967],{"className":92952},[200],[39,92954,92956],{"className":92955,"style":13604},[204],[39,92957,92958,92961],{"style":966},[39,92959],{"className":92960,"style":213},[212],[39,92962,92964],{"className":92963},[217,218,219,220],[39,92965,13377],{"className":92966},[83,84,220],[39,92968,673],{"className":92969},[672],[39,92971,92973],{"className":92972},[200],[39,92974,92976],{"className":92975,"style":680},[204],[39,92977],{},[39,92979,92981,92984],{"className":92980},[83],[39,92982,77187],{"className":92983,"style":48487},[83,84],[39,92985,92987],{"className":92986},[192],[39,92988,92990,93013],{"className":92989},[196,649],[39,92991,92993,93010],{"className":92992},[200],[39,92994,92996],{"className":92995,"style":13604},[204],[39,92997,92998,93001],{"style":81127},[39,92999],{"className":93000,"style":213},[212],[39,93002,93004],{"className":93003},[217,218,219,220],[39,93005,93007],{"className":93006},[83,220],[39,93008,13732],{"className":93009,"style":13731},[83,84,220],[39,93011,673],{"className":93012},[672],[39,93014,93016],{"className":93015},[200],[39,93017,93019],{"className":93018,"style":35396},[204],[39,93020],{},[39,93022,93024,93027],{"className":93023},[83],[39,93025,77971],{"className":93026,"style":2114},[83,84],[39,93028,93030],{"className":93029},[192],[39,93031,93033,93053],{"className":93032},[196,649],[39,93034,93036,93050],{"className":93035},[200],[39,93037,93039],{"className":93038,"style":13604},[204],[39,93040,93041,93044],{"style":41316},[39,93042],{"className":93043,"style":213},[212],[39,93045,93047],{"className":93046},[217,218,219,220],[39,93048,13446],{"className":93049,"style":13731},[83,84,220],[39,93051,673],{"className":93052},[672],[39,93054,93056],{"className":93055},[200],[39,93057,93059],{"className":93058,"style":35396},[204],[39,93060],{},[39,93062,155],{"className":93063},[252],[39,93065],{"className":93066,"style":10787},[230],[39,93068],{"className":93069,"style":256},[230],[39,93071,93073,93076],{"className":93072},[83],[39,93074,77187],{"className":93075,"style":48487},[83,84],[39,93077,93079],{"className":93078},[192],[39,93080,93082,93105],{"className":93081},[196,649],[39,93083,93085,93102],{"className":93084},[200],[39,93086,93088],{"className":93087,"style":13604},[204],[39,93089,93090,93093],{"style":81127},[39,93091],{"className":93092,"style":213},[212],[39,93094,93096],{"className":93095},[217,218,219,220],[39,93097,93099],{"className":93098},[83,220],[39,93100,13732],{"className":93101,"style":13731},[83,84,220],[39,93103,673],{"className":93104},[672],[39,93106,93108],{"className":93107},[200],[39,93109,93111],{"className":93110,"style":35396},[204],[39,93112],{},[39,93114],{"className":93115,"style":231},[230],[39,93117,793],{"className":93118},[235],[39,93120],{"className":93121,"style":231},[230],[39,93123,93125,93128,93131,93134,93137,93180,93183,93186,93189],{"className":93124},[74],[39,93126],{"className":93127,"style":41560},[78],[39,93129,92843],{"className":93130},[1744],[39,93132,136],{"className":93133},[182],[39,93135,5378],{"className":93136},[83],[39,93138,93140,93143],{"className":93139},[83],[39,93141,48444],{"className":93142,"style":48487},[83,84],[39,93144,93146],{"className":93145},[192],[39,93147,93149,93172],{"className":93148},[196,649],[39,93150,93152,93169],{"className":93151},[200],[39,93153,93155],{"className":93154,"style":13604},[204],[39,93156,93157,93160],{"style":81127},[39,93158],{"className":93159,"style":213},[212],[39,93161,93163],{"className":93162},[217,218,219,220],[39,93164,93166],{"className":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由边缘约束定出，交替更新 ",[39,93237,93239,93268],{"className":93238,"translate":43},[42],[39,93240,93242],{"className":93241},[47],[49,93243,93244],{"xmlns":51},[53,93245,93246,93265],{},[56,93247,93248,93250,93253,93255,93257,93259,93261,93263],{},[59,93249,35321],{},[133,93251,93252],{},"←",[59,93254,158],{},[59,93256,45691],{"mathvariant":145},[133,93258,136],{"stretchy":135},[59,93260,77187],{},[59,93262,77971],{},[133,93264,161],{"stretchy":135},[63,93266,93267],{"encoding":65},"u\\leftarrow a\u002F(Kv)",[39,93269,93271,93289],{"className":93270,"ariaHidden":70},[69],[39,93272,93274,93277,93280,93283,93286],{"className":93273},[74],[39,93275],{"className":93276,"style":290},[78],[39,93278,35321],{"className":93279},[83,84],[39,93281],{"className":93282,"style":231},[230],[39,93284,93252],{"className":93285},[235],[39,93287],{"className":93288,"style":231},[230],[39,93290,93292,93295,93298,93301,93304,93307,93310],{"className":93291},[74],[39,93293],{"className":93294,"style":245},[78],[39,93296,158],{"className":93297},[83,84],[39,93299,45691],{"className":93300},[83],[39,93302,136],{"className":93303},[182],[39,93305,77187],{"className":93306,"style":48487},[83,84],[39,93308,77971],{"className":93309,"style":2114},[83,84],[39,93311,161],{"className":93312},[263],[39,93314,93316,93349],{"className":93315,"translate":43},[42],[39,93317,93319],{"className":93318},[47],[49,93320,93321],{"xmlns":51},[53,93322,93323,93346],{},[56,93324,93325,93327,93329,93331,93333,93335,93342,93344],{},[59,93326,77971],{},[133,93328,93252],{},[59,93330,78435],{},[59,93332,45691],{"mathvariant":145},[133,93334,136],{"stretchy":135},[138,93336,93337,93339],{},[59,93338,77187],{},[59,93340,93341],{"mathvariant":145},"⊤",[59,93343,35321],{},[133,93345,161],{"stretchy":135},[63,93347,93348],{"encoding":65},"v\\leftarrow b\u002F(K^\\top u)",[39,93350,93352,93370],{"className":93351,"ariaHidden":70},[69],[39,93353,93355,93358,93361,93364,93367],{"className":93354},[74],[39,93356],{"className":93357,"style":290},[78],[39,93359,77971],{"className":93360,"style":2114},[83,84],[39,93362],{"className":93363,"style":231},[230],[39,93365,93252],{"className":93366},[235],[39,93368],{"className":93369,"style":231},[230],[39,93371,93373,93376,93379,93382,93385,93414,93417],{"className":93372},[74],[39,93374],{"className":93375,"style":15555},[78],[39,93377,78435],{"className":93378},[83,84],[39,93380,45691],{"className":93381},[83],[39,93383,136],{"className":93384},[182],[39,93386,93388,93391],{"className":93387},[83],[39,93389,77187],{"className":93390,"style":48487},[83,84],[39,93392,93394],{"className":93393},[192],[39,93395,93397],{"className":93396},[196],[39,93398,93400],{"className":93399},[200],[39,93401,93403],{"className":93402,"style":15577},[204],[39,93404,93405,93408],{"style":208},[39,93406],{"className":93407,"style":213},[212],[39,93409,93411],{"className":93410},[217,218,219,220],[39,93412,93341],{"className":93413},[83,220],[39,93415,35321],{"className":93416},[83,84],[39,93418,161],{"className":93419},[263],"，每次都是矩阵—向量乘法。这就是 Sinkhorn 迭代。它的意义不只是快，而是可微、可并到 GPU 上，于是能塞进神经网络当一层用。",[11,93422,93423,93451,93452,93480,93481,93509,93510,93538,93539,93590,93591,93619],{},[39,93424,93426,93439],{"className":93425,"translate":43},[42],[39,93427,93429],{"className":93428},[47],[49,93430,93431],{"xmlns":51},[53,93432,93433,93437],{},[56,93434,93435],{},[59,93436,58009],{},[63,93438,77422],{"encoding":65},[39,93440,93442],{"className":93441,"ariaHidden":70},[69],[39,93443,93445,93448],{"className":93444},[74],[39,93446],{"className":93447,"style":290},[78],[39,93449,58009],{"className":93450},[83,84]," 可以读作一个旋钮。取三个点搬到三个点：",[39,93453,93455,93468],{"className":93454,"translate":43},[42],[39,93456,93458],{"className":93457},[47],[49,93459,93460],{"xmlns":51},[53,93461,93462,93466],{},[56,93463,93464],{},[59,93465,58009],{},[63,93467,77422],{"encoding":65},[39,93469,93471],{"className":93470,"ariaHidden":70},[69],[39,93472,93474,93477],{"className":93473},[74],[39,93475],{"className":93476,"style":290},[78],[39,93478,58009],{"className":93479},[83,84]," 很大时，",[39,93482,93484,93497],{"className":93483,"translate":43},[42],[39,93485,93487],{"className":93486},[47],[49,93488,93489],{"xmlns":51},[53,93490,93491,93495],{},[56,93492,93493],{},[59,93494,77187],{},[63,93496,77187],{"encoding":65},[39,93498,93500],{"className":93499,"ariaHidden":70},[69],[39,93501,93503,93506],{"className":93502},[74],[39,93504],{"className":93505,"style":79},[78],[39,93507,77187],{"className":93508,"style":48487},[83,84]," 接近全 ",[39,93511,93513,93526],{"className":93512,"translate":43},[42],[39,93514,93516],{"className":93515},[47],[49,93517,93518],{"xmlns":51},[53,93519,93520,93524],{},[56,93521,93522],{},[539,93523,546],{},[63,93525,546],{"encoding":65},[39,93527,93529],{"className":93528,"ariaHidden":70},[69],[39,93530,93532,93535],{"className":93531},[74],[39,93533],{"className":93534,"style":2473},[78],[39,93536,546],{"className":93537},[83]," 矩阵，每个点都往每处送一点，解是稠密的、模糊的；",[39,93540,93542,93560],{"className":93541,"translate":43},[42],[39,93543,93545],{"className":93544},[47],[49,93546,93547],{"xmlns":51},[53,93548,93549,93557],{},[56,93550,93551,93553,93555],{},[59,93552,58009],{},[133,93554,418],{},[539,93556,541],{},[63,93558,93559],{"encoding":65},"\\varepsilon\\to0",[39,93561,93563,93581],{"className":93562,"ariaHidden":70},[69],[39,93564,93566,93569,93572,93575,93578],{"className":93565},[74],[39,93567],{"className":93568,"style":290},[78],[39,93570,58009],{"className":93571},[83,84],[39,93573],{"className":93574,"style":231},[230],[39,93576,418],{"className":93577},[235],[39,93579],{"className":93580,"style":231},[230],[39,93582,93584,93587],{"className":93583},[74],[39,93585],{"className":93586,"style":2473},[78],[39,93588,541],{"className":93589},[83]," 时 ",[39,93592,93594,93607],{"className":93593,"translate":43},[42],[39,93595,93597],{"className":93596},[47],[49,93598,93599],{"xmlns":51},[53,93600,93601,93605],{},[56,93602,93603],{},[59,93604,77187],{},[63,93606,77187],{"encoding":65},[39,93608,93610],{"className":93609,"ariaHidden":70},[69],[39,93611,93613,93616],{"className":93612},[74],[39,93614],{"className":93615,"style":79},[78],[39,93617,77187],{"className":93618,"style":48487},[83,84]," 接近单位矩阵，解退化成“就近硬匹配”，也就是原始 OT。中间是连续插值。",[11,93621,74083,93622,93673,93674,93752,93753,94097],{},[39,93623,93625,93643],{"className":93624,"translate":43},[42],[39,93626,93628],{"className":93627},[47],[49,93629,93630],{"xmlns":51},[53,93631,93632,93640],{},[56,93633,93634,93636,93638],{},[59,93635,58009],{},[133,93637,31514],{},[539,93639,541],{},[63,93641,93642],{"encoding":65},"\\varepsilon>0",[39,93644,93646,93664],{"className":93645,"ariaHidden":70},[69],[39,93647,93649,93652,93655,93658,93661],{"className":93648},[74],[39,93650],{"className":93651,"style":24094},[78],[39,93653,58009],{"className":93654},[83,84],[39,93656],{"className":93657,"style":231},[230],[39,93659,31514],{"className":93660},[235],[39,93662],{"className":93663,"style":231},[230],[39,93665,93667,93670],{"className":93666},[74],[39,93668],{"className":93669,"style":2473},[78],[39,93671,541],{"className":93672},[83]," 时它不再是度量。最直接的一点：把 ",[39,93675,93677,93709],{"className":93676,"translate":43},[42],[39,93678,93680],{"className":93679},[47],[49,93681,93682],{"xmlns":51},[53,93683,93684,93706],{},[56,93685,93686,93688,93690,93700,93702,93704],{},[59,93687,2098],{},[133,93689,793],{},[56,93691,93692,93694,93696,93698],{},[59,93693,29425],{"mathvariant":145},[59,93695,13377],{"mathvariant":145},[59,93697,158],{"mathvariant":145},[59,93699,39209],{"mathvariant":145},[133,93701,136],{"stretchy":135},[59,93703,158],{},[133,93705,161],{"stretchy":135},[63,93707,93708],{"encoding":65},"\\pi=\\mathrm{diag}(a)",[39,93710,93712,93730],{"className":93711,"ariaHidden":70},[69],[39,93713,93715,93718,93721,93724,93727],{"className":93714},[74],[39,93716],{"className":93717,"style":290},[78],[39,93719,2098],{"className":93720,"style":2114},[83,84],[39,93722],{"className":93723,"style":231},[230],[39,93725,793],{"className":93726},[235],[39,93728],{"className":93729,"style":231},[230],[39,93731,93733,93736,93743,93746,93749],{"className":93732},[74],[39,93734],{"className":93735,"style":245},[78],[39,93737,93739],{"className":93738},[83],[39,93740,93742],{"className":93741,"style":35078},[83,35077],"diag",[39,93744,136],{"className":93745},[182],[39,93747,158],{"className":93748},[83,84],[39,93750,161],{"className":93751},[263]," 代入，",[39,93754,93756,93815],{"className":93755,"translate":43},[42],[39,93757,93759],{"className":93758},[47],[49,93760,93761],{"xmlns":51},[53,93762,93763,93812],{},[56,93764,93765,93771,93773,93775,93777,93779,93781,93783,93785,93791,93797,93799,93801,93807,93810],{},[613,93766,93767,93769],{},[59,93768,23703],{},[59,93770,58009],{},[133,93772,136],{"stretchy":135},[59,93774,73578],{},[133,93776,155],{"separator":70},[59,93778,73578],{},[133,93780,161],{"stretchy":135},[133,93782,793],{},[59,93784,58009],{},[613,93786,93787,93789],{},[133,93788,3191],{},[59,93790,13377],{},[613,93792,93793,93795],{},[59,93794,158],{},[59,93796,13377],{},[59,93798,92483],{},[133,93800,1690],{},[613,93802,93803,93805],{},[59,93804,158],{},[59,93806,13377],{},[133,93808,93809],{"mathvariant":145},"≠",[539,93811,541],{},[63,93813,93814],{"encoding":65},"W_\\varepsilon(\\mu,\\mu)=\\varepsilon\\sum_i a_i\\log a_i\\ne0",[39,93816,93818,93891,94088],{"className":93817,"ariaHidden":70},[69],[39,93819,93821,93824,93864,93867,93870,93873,93876,93879,93882,93885,93888],{"className":93820},[74],[39,93822],{"className":93823,"style":245},[78],[39,93825,93827,93830],{"className":93826},[83],[39,93828,23703],{"className":93829,"style":178},[83,84],[39,93831,93833],{"className":93832},[192],[39,93834,93836,93856],{"className":93835},[196,649],[39,93837,93839,93853],{"className":93838},[200],[39,93840,93842],{"className":93841,"style":1728},[204],[39,93843,93844,93847],{"style":73765},[39,93845],{"className":93846,"style":213},[212],[39,93848,93850],{"className":93849},[217,218,219,220],[39,93851,58009],{"className":93852},[83,84,220],[39,93854,673],{"className":93855},[672],[39,93857,93859],{"className":93858},[200],[39,93860,93862],{"className":93861,"style":680},[204],[39,93863],{},[39,93865,136],{"className":93866},[182],[39,93868,73578],{"className":93869},[83,84],[39,93871,155],{"className":93872},[252],[39,93874],{"className":93875,"style":256},[230],[39,93877,73578],{"className":93878},[83,84],[39,93880,161],{"className":93881},[263],[39,93883],{"className":93884,"style":231},[230],[39,93886,793],{"className":93887},[235],[39,93889],{"className":93890,"style":231},[230],[39,93892,93894,93898,93901,93904,93945,93948,93988,93991,93996,93999,94039,94042,94085],{"className":93893},[74],[39,93895],{"className":93896,"style":93897},[78],"height:1.0497em;vertical-align:-0.2997em;",[39,93899,58009],{"className":93900},[83,84],[39,93902],{"className":93903,"style":256},[230],[39,93905,93907,93910],{"className":93906},[1744],[39,93908,3191],{"className":93909,"style":3953},[1744,3339,3952],[39,93911,93913],{"className":93912},[192],[39,93914,93916,93937],{"className":93915},[196,649],[39,93917,93919,93934],{"className":93918},[200],[39,93920,93923],{"className":93921,"style":93922},[204],"height:0.162em;",[39,93924,93925,93928],{"style":3969},[39,93926],{"className":93927,"style":213},[212],[39,93929,93931],{"className":93930},[217,218,219,220],[39,93932,13377],{"className":93933},[83,84,220],[39,93935,673],{"className":93936},[672],[39,93938,93940],{"className":93939},[200],[39,93941,93943],{"className":93942,"style":4025},[204],[39,93944],{},[39,93946],{"className":93947,"style":256},[230],[39,93949,93951,93954],{"className":93950},[83],[39,93952,158],{"className":93953},[83,84],[39,93955,93957],{"className":93956},[192],[39,93958,93960,93980],{"className":93959},[196,649],[39,93961,93963,93977],{"className":93962},[200],[39,93964,93966],{"className":93965,"style":13604},[204],[39,93967,93968,93971],{"style":966},[39,93969],{"className":93970,"style":213},[212],[39,93972,93974],{"className":93973},[217,218,219,220],[39,93975,13377],{"className":93976},[83,84,220],[39,93978,673],{"className":93979},[672],[39,93981,93983],{"className":93982},[200],[39,93984,93986],{"className":93985,"style":680},[204],[39,93987],{},[39,93989],{"className":93990,"style":256},[230],[39,93992,92721,93994],{"className":93993},[1744],[39,93995,39209],{"style":35078},[39,93997],{"className":93998,"style":256},[230],[39,94000,94002,94005],{"className":94001},[83],[39,94003,158],{"className":94004},[83,84],[39,94006,94008],{"className":94007},[192],[39,94009,94011,94031],{"className":94010},[196,649],[39,94012,94014,94028],{"className":94013},[200],[39,94015,94017],{"className":94016,"style":13604},[204],[39,94018,94019,94022],{"style":966},[39,94020],{"className":94021,"style":213},[212],[39,94023,94025],{"className":94024},[217,218,219,220],[39,94026,13377],{"className":94027},[83,84,220],[39,94029,673],{"className":94030},[672],[39,94032,94034],{"className":94033},[200],[39,94035,94037],{"className":94036,"style":680},[204],[39,94038],{},[39,94040],{"className":94041,"style":231},[230],[39,94043,94045,94078,94082],{"className":94044},[235],[39,94046,94048],{"className":94047},[235],[39,94049,94052],{"className":94050},[83,94051],"vbox",[39,94053,94056],{"className":94054},[94055],"thinbox",[39,94057,94060,94063,94074],{"className":94058},[94059],"rlap",[39,94061],{"className":94062,"style":51280},[78],[39,94064,94067],{"className":94065},[94066],"inner",[39,94068,94070],{"className":94069},[83],[39,94071,94073],{"className":94072},[235],"",[39,94075],{"className":94076},[94077],"fix",[39,94079],{"className":94080},[230,94081],"nobreak",[39,94083,793],{"className":94084},[235],[39,94086],{"className":94087,"style":231},[230],[39,94089,94091,94094],{"className":94090},[74],[39,94092],{"className":94093,"style":2473},[78],[39,94095,541],{"className":94096},[83],"。所以工程上常用的 Sinkhorn 散度要先去偏：",[39,94099,94101],{"className":94100,"translate":43},[746],[39,94102,94104,94216],{"className":94103,"translate":43},[42],[39,94105,94107],{"className":94106},[47],[49,94108,94109],{"xmlns":51,"display":755},[53,94110,94111,94213],{},[56,94112,94113,94119,94121,94123,94125,94127,94129,94131,94141,94143,94145,94147,94149,94151,94153,94161,94171,94173,94175,94177,94179,94181,94183,94191,94201,94203,94205,94207,94209,94211],{},[613,94114,94115,94117],{},[59,94116,61],{},[59,94118,58009],{},[133,94120,136],{"stretchy":135},[59,94122,73578],{},[133,94124,155],{"separator":70},[59,94126,73609],{},[133,94128,161],{"stretchy":135},[133,94130,793],{},[613,94132,94133,94139],{},[56,94134,94135,94137],{},[59,94136,92278],{"mathvariant":145},[59,94138,11772],{"mathvariant":145},[59,94140,58009],{},[133,94142,136],{"stretchy":135},[59,94144,73578],{},[133,94146,155],{"separator":70},[59,94148,73609],{},[133,94150,161],{"stretchy":135},[133,94152,5378],{},[770,94154,94155],{"scriptlevel":541,"displaystyle":135},[5362,94156,94157,94159],{},[539,94158,546],{},[539,94160,17359],{},[613,94162,94163,94169],{},[56,94164,94165,94167],{},[59,94166,92278],{"mathvariant":145},[59,94168,11772],{"mathvariant":145},[59,94170,58009],{},[133,94172,136],{"stretchy":135},[59,94174,73578],{},[133,94176,155],{"separator":70},[59,94178,73578],{},[133,94180,161],{"stretchy":135},[133,94182,5378],{},[770,94184,94185],{"scriptlevel":541,"displaystyle":135},[5362,94186,94187,94189],{},[539,94188,546],{},[539,94190,17359],{},[613,94192,94193,94199],{},[56,94194,94195,94197],{},[59,94196,92278],{"mathvariant":145},[59,94198,11772],{"mathvariant":145},[59,94200,58009],{},[133,94202,136],{"stretchy":135},[59,94204,73609],{},[133,94206,155],{"separator":70},[59,94208,73609],{},[133,94210,161],{"stretchy":135},[59,94212,2903],{"mathvariant":145},[63,94214,94215],{"encoding":65},"S_\\varepsilon(\\mu,\\nu)=\\mathrm{OT}_\\varepsilon(\\mu,\\nu)-\\tfrac12\\mathrm{OT}_\\varepsilon(\\mu,\\mu)-\\tfrac12\\mathrm{OT}_\\varepsilon(\\nu,\\nu).",[39,94217,94219,94293,94370,94514],{"className":94218,"ariaHidden":70},[69],[39,94220,94222,94225,94266,94269,94272,94275,94278,94281,94284,94287,94290],{"className":94221},[74],[39,94223],{"className":94224,"style":245},[78],[39,94226,94228,94231],{"className":94227},[83],[39,94229,61],{"className":94230,"style":85},[83,84],[39,94232,94234],{"className":94233},[192],[39,94235,94237,94258],{"className":94236},[196,649],[39,94238,94240,94255],{"className":94239},[200],[39,94241,94243],{"className":94242,"style":1728},[204],[39,94244,94246,94249],{"style":94245},"top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;",[39,94247],{"className":94248,"style":213},[212],[39,94250,94252],{"className":94251},[217,218,219,220],[39,94253,58009],{"className":94254},[83,84,220],[39,94256,673],{"className":94257},[672],[39,94259,94261],{"className":94260},[200],[39,94262,94264],{"className":94263,"style":680},[204],[39,94265],{},[39,94267,136],{"className":94268},[182],[39,94270,73578],{"className":94271},[83,84],[39,94273,155],{"className":94274},[252],[39,94276],{"className":94277,"style":256},[230],[39,94279,73609],{"className":94280,"style":73625},[83,84],[39,94282,161],{"className":94283},[263],[39,94285],{"className":94286,"style":231},[230],[39,94288,793],{"className":94289},[235],[39,94291],{"className":94292,"style":231},[230],[39,94294,94296,94299,94343,94346,94349,94352,94355,94358,94361,94364,94367],{"className":94295},[74],[39,94297],{"className":94298,"style":245},[78],[39,94300,94302,94309],{"className":94301},[83],[39,94303,94305],{"className":94304},[83],[39,94306,94308],{"className":94307},[83,35077],"OT",[39,94310,94312],{"className":94311},[192],[39,94313,94315,94335],{"className":94314},[196,649],[39,94316,94318,94332],{"className":94317},[200],[39,94319,94321],{"className":94320,"style":1728},[204],[39,94322,94323,94326],{"style":44021},[39,94324],{"className":94325,"style":213},[212],[39,94327,94329],{"className":94328},[217,218,219,220],[39,94330,58009],{"className":94331},[83,84,220],[39,94333,673],{"className":94334},[672],[39,94336,94338],{"className":94337},[200],[39,94339,94341],{"className":94340,"style":680},[204],[39,94342],{},[39,94344,136],{"className":94345},[182],[39,94347,73578],{"className":94348},[83,84],[39,94350,155],{"className":94351},[252],[39,94353],{"className":94354,"style":256},[230],[39,94356,73609],{"className":94357,"style":73625},[83,84],[39,94359,161],{"className":94360},[263],[39,94362],{"className":94363,"style":461},[230],[39,94365,5378],{"className":94366},[465],[39,94368],{"className":94369,"style":461},[230],[39,94371,94373,94376,94444,94487,94490,94493,94496,94499,94502,94505,94508,94511],{"className":94372},[74],[39,94374],{"className":94375,"style":74794},[78],[39,94377,94379,94382,94441],{"className":94378},[83],[39,94380],{"className":94381},[182,5463],[39,94383,94385],{"className":94384},[5362],[39,94386,94388,94433],{"className":94387},[196,649],[39,94389,94391,94430],{"className":94390},[200],[39,94392,94394,94408,94416],{"className":94393,"style":74813},[204],[39,94395,94396,94399],{"style":74816},[39,94397],{"className":94398,"style":935},[212],[39,94400,94402],{"className":94401},[217,218,219,220],[39,94403,94405],{"className":94404},[83,220],[39,94406,17359],{"className":94407},[83,220],[39,94409,94410,94413],{"style":5503},[39,94411],{"className":94412,"style":935},[212],[39,94414],{"className":94415,"style":5511},[5510],[39,94417,94418,94421],{"style":74839},[39,94419],{"className":94420,"style":935},[212],[39,94422,94424],{"className":94423},[217,218,219,220],[39,94425,94427],{"className":94426},[83,220],[39,94428,546],{"className":94429},[83,220],[39,94431,673],{"className":94432},[672],[39,94434,94436],{"className":94435},[200],[39,94437,94439],{"className":94438,"style":74861},[204],[39,94440],{},[39,94442],{"className":94443},[263,5463],[39,94445,94447,94453],{"className":94446},[83],[39,94448,94450],{"className":94449},[83],[39,94451,94308],{"className":94452},[83,35077],[39,94454,94456],{"className":94455},[192],[39,94457,94459,94479],{"className":94458},[196,649],[39,94460,94462,94476],{"className":94461},[200],[39,94463,94465],{"className":94464,"style":1728},[204],[39,94466,94467,94470],{"style":44021},[39,94468],{"className":94469,"style":213},[212],[39,94471,94473],{"className":94472},[217,218,219,220],[39,94474,58009],{"className":94475},[83,84,220],[39,94477,673],{"className":94478},[672],[39,94480,94482],{"className":94481},[200],[39,94483,94485],{"className":94484,"style":680},[204],[39,94486],{},[39,94488,136],{"className":94489},[182],[39,94491,73578],{"className":94492},[83,84],[39,94494,155],{"className":94495},[252],[39,94497],{"className":94498,"style":256},[230],[39,94500,73578],{"className":94501},[83,84],[39,94503,161],{"className":94504},[263],[39,94506],{"className":94507,"style":461},[230],[39,94509,5378],{"className":94510},[465],[39,94512],{"className":94513,"style":461},[230],[39,94515,94517,94520,94588,94631,94634,94637,94640,94643,94646,94649],{"className":94516},[74],[39,94518],{"className":94519,"style":74794},[78],[39,94521,94523,94526,94585],{"className":94522},[83],[39,94524],{"className":94525},[182,5463],[39,94527,94529],{"className":94528},[5362],[39,94530,94532,94577],{"className":94531},[196,649],[39,94533,94535,94574],{"className":94534},[200],[39,94536,94538,94552,94560],{"className":94537,"style":74813},[204],[39,94539,94540,94543],{"style":74816},[39,94541],{"className":94542,"style":935},[212],[39,94544,94546],{"className":94545},[217,218,219,220],[39,94547,94549],{"className":94548},[83,220],[39,94550,17359],{"className":94551},[83,220],[39,94553,94554,94557],{"style":5503},[39,94555],{"className":94556,"style":935},[212],[39,94558],{"className":94559,"style":5511},[5510],[39,94561,94562,94565],{"style":74839},[39,94563],{"className":94564,"style":935},[212],[39,94566,94568],{"className":94567},[217,218,219,220],[39,94569,94571],{"className":94570},[83,220],[39,94572,546],{"className":94573},[83,220],[39,94575,673],{"className":94576},[672],[39,94578,94580],{"className":94579},[200],[39,94581,94583],{"className":94582,"style":74861},[204],[39,94584],{},[39,94586],{"className":94587},[263,5463],[39,94589,94591,94597],{"className":94590},[83],[39,94592,94594],{"className":94593},[83],[39,94595,94308],{"className":94596},[83,35077],[39,94598,94600],{"className":94599},[192],[39,94601,94603,94623],{"className":94602},[196,649],[39,94604,94606,94620],{"className":94605},[200],[39,94607,94609],{"className":94608,"style":1728},[204],[39,94610,94611,94614],{"style":44021},[39,94612],{"className":94613,"style":213},[212],[39,94615,94617],{"className":94616},[217,218,219,220],[39,94618,58009],{"className":94619},[83,84,220],[39,94621,673],{"className":94622},[672],[39,94624,94626],{"className":94625},[200],[39,94627,94629],{"className":94628,"style":680},[204],[39,94630],{},[39,94632,136],{"className":94633},[182],[39,94635,73609],{"className":94636,"style":73625},[83,84],[39,94638,155],{"className":94639},[252],[39,94641],{"className":94642,"style":256},[230],[39,94644,73609],{"className":94645,"style":73625},[83,84],[39,94647,161],{"className":94648},[263],[39,94650,2903],{"className":94651},[83],[11,94653,94654,94655,94706],{},"去偏之后才非负、才在 ",[39,94656,94658,94676],{"className":94657,"translate":43},[42],[39,94659,94661],{"className":94660},[47],[49,94662,94663],{"xmlns":51},[53,94664,94665,94673],{},[56,94666,94667,94669,94671],{},[59,94668,73578],{},[133,94670,793],{},[59,94672,73609],{},[63,94674,94675],{"encoding":65},"\\mu=\\nu",[39,94677,94679,94697],{"className":94678,"ariaHidden":70},[69],[39,94680,94682,94685,94688,94691,94694],{"className":94681},[74],[39,94683],{"className":94684,"style":2654},[78],[39,94686,73578],{"className":94687},[83,84],[39,94689],{"className":94690,"style":231},[230],[39,94692,793],{"className":94693},[235],[39,94695],{"className":94696,"style":231},[230],[39,94698,94700,94703],{"className":94699},[74],[39,94701],{"className":94702,"style":290},[78],[39,94704,73609],{"className":94705,"style":73625},[83,84]," 处为零。",[11,94708,94709,94710,94780,94781,94809,94810,94838],{},"还有一个更根本的代价，来自样本。经验分布 ",[39,94711,94713,94731],{"className":94712,"translate":43},[42],[39,94714,94716],{"className":94715},[47],[49,94717,94718],{"xmlns":51},[53,94719,94720,94728],{},[56,94721,94722],{},[613,94723,94724,94726],{},[59,94725,73578],{},[59,94727,12962],{},[63,94729,94730],{"encoding":65},"\\mu_n",[39,94732,94734],{"className":94733,"ariaHidden":70},[69],[39,94735,94737,94740],{"className":94736},[74],[39,94738],{"className":94739,"style":2654},[78],[39,94741,94743,94746],{"className":94742},[83],[39,94744,73578],{"className":94745},[83,84],[39,94747,94749],{"className":94748},[192],[39,94750,94752,94772],{"className":94751},[196,649],[39,94753,94755,94769],{"className":94754},[200],[39,94756,94758],{"className":94757,"style":1728},[204],[39,94759,94760,94763],{"style":966},[39,94761],{"className":94762,"style":213},[212],[39,94764,94766],{"className":94765},[217,218,219,220],[39,94767,12962],{"className":94768},[83,84,220],[39,94770,673],{"className":94771},[672],[39,94773,94775],{"className":94774},[200],[39,94776,94778],{"className":94777,"style":680},[204],[39,94779],{}," 到真分布 ",[39,94782,94784,94797],{"className":94783,"translate":43},[42],[39,94785,94787],{"className":94786},[47],[49,94788,94789],{"xmlns":51},[53,94790,94791,94795],{},[56,94792,94793],{},[59,94794,73578],{},[63,94796,73581],{"encoding":65},[39,94798,94800],{"className":94799,"ariaHidden":70},[69],[39,94801,94803,94806],{"className":94802},[74],[39,94804],{"className":94805,"style":2654},[78],[39,94807,73578],{"className":94808},[83,84]," 的收敛速度，在 ",[39,94811,94813,94826],{"className":94812,"translate":43},[42],[39,94814,94816],{"className":94815},[47],[49,94817,94818],{"xmlns":51},[53,94819,94820,94824],{},[56,94821,94822],{},[59,94823,29425],{},[63,94825,29425],{"encoding":65},[39,94827,94829],{"className":94828,"ariaHidden":70},[69],[39,94830,94832,94835],{"className":94831},[74],[39,94833],{"className":94834,"style":28824},[78],[39,94836,29425],{"className":94837},[83,84]," 维空间里大致是",[39,94840,94842],{"className":94841,"translate":43},[746],[39,94843,94845,94918],{"className":94844,"translate":43},[42],[39,94846,94848],{"className":94847},[47],[49,94849,94850],{"xmlns":51,"display":755},[53,94851,94852,94915],{},[56,94853,94854,94856,94858,94864,94866,94872,94874,94876,94878,94880,94883,94885,94899,94901,94903,94905,94907,94909,94911,94913],{},[59,94855,4253],{"mathvariant":421},[35841,94857,35843],{},[613,94859,94860,94862],{},[59,94861,23703],{},[59,94863,11],{},[133,94865,136],{"stretchy":135},[613,94867,94868,94870],{},[59,94869,73578],{},[59,94871,12962],{},[133,94873,155],{"separator":70},[59,94875,73578],{},[133,94877,161],{"stretchy":135},[35841,94879,74223],{},[133,94881,94882],{},"≲",[35841,94884,74223],{},[138,94886,94887,94889],{},[59,94888,12962],{},[56,94890,94891,94893,94895,94897],{},[133,94892,5378],{},[539,94894,546],{},[59,94896,45691],{"mathvariant":145},[59,94898,29425],{},[230,94900],{"width":10251},[133,94902,136],{"stretchy":135},[59,94904,29425],{},[133,94906,31514],{},[539,94908,17359],{},[59,94910,11],{},[133,94912,161],{"stretchy":135},[59,94914,2903],{"mathvariant":145},[63,94916,94917],{"encoding":65},"\\mathbb E\\,W_p(\\mu_n,\\mu)\\ \\lesssim\\ n^{-1\u002Fd}\\qquad(d>2p).",[39,94919,94921,95044,95108],{"className":94920,"ariaHidden":70},[69],[39,94922,94924,94927,94930,94933,94973,94976,95016,95019,95022,95025,95028,95031,95034,95038,95041],{"className":94923},[74],[39,94925],{"className":94926,"style":41560},[78],[39,94928,4253],{"className":94929},[83,515],[39,94931],{"className":94932,"style":256},[230],[39,94934,94936,94939],{"className":94935},[83],[39,94937,23703],{"className":94938,"style":178},[83,84],[39,94940,94942],{"className":94941},[192],[39,94943,94945,94965],{"className":94944},[196,649],[39,94946,94948,94962],{"className":94947},[200],[39,94949,94951],{"className":94950,"style":1728},[204],[39,94952,94953,94956],{"style":73765},[39,94954],{"className":94955,"style":213},[212],[39,94957,94959],{"className":94958},[217,218,219,220],[39,94960,11],{"className":94961},[83,84,220],[39,94963,673],{"className":94964},[672],[39,94966,94968],{"className":94967},[200],[39,94969,94971],{"className":94970,"style":35396},[204],[39,94972],{},[39,94974,136],{"className":94975},[182],[39,94977,94979,94982],{"className":94978},[83],[39,94980,73578],{"className":94981},[83,84],[39,94983,94985],{"className":94984},[192],[39,94986,94988,95008],{"className":94987},[196,649],[39,94989,94991,95005],{"className":94990},[200],[39,94992,94994],{"className":94993,"style":1728},[204],[39,94995,94996,94999],{"style":966},[39,94997],{"className":94998,"style":213},[212],[39,95000,95002],{"className":95001},[217,218,219,220],[39,95003,12962],{"className":95004},[83,84,220],[39,95006,673],{"className":95007},[672],[39,95009,95011],{"className":95010},[200],[39,95012,95014],{"className":95013,"style":680},[204],[39,95015],{},[39,95017,155],{"className":95018},[252],[39,95020],{"className":95021,"style":256},[230],[39,95023,73578],{"className":95024},[83,84],[39,95026,161],{"className":95027},[263],[39,95029,74223],{"className":95030},[230],[39,95032],{"className":95033,"style":231},[230],[39,95035,94882],{"className":95036},[235,95037],"amsrm",[39,95039,74223],{"className":95040},[230],[39,95042],{"className":95043,"style":231},[230],[39,95045,95047,95051,95090,95093,95096,95099,95102,95105],{"className":95046},[74],[39,95048],{"className":95049,"style":95050},[78],"height:1.188em;vertical-align:-0.25em;",[39,95052,95054,95057],{"className":95053},[83],[39,95055,12962],{"className":95056},[83,84],[39,95058,95060],{"className":95059},[192],[39,95061,95063],{"className":95062},[196],[39,95064,95066],{"className":95065},[200],[39,95067,95070],{"className":95068,"style":95069},[204],"height:0.938em;",[39,95071,95072,95075],{"style":1027},[39,95073],{"className":95074,"style":213},[212],[39,95076,95078],{"className":95077},[217,218,219,220],[39,95079,95081,95084,95087],{"className":95080},[83,220],[39,95082,5378],{"className":95083},[83,220],[39,95085,76672],{"className":95086},[83,220],[39,95088,29425],{"className":95089},[83,84,220],[39,95091],{"className":95092,"style":10787},[230],[39,95094,136],{"className":95095},[182],[39,95097,29425],{"className":95098},[83,84],[39,95100],{"className":95101,"style":231},[230],[39,95103,31514],{"className":95104},[235],[39,95106],{"className":95107,"style":231},[230],[39,95109,95111,95114,95117,95120,95123],{"className":95110},[74],[39,95112],{"className":95113,"style":245},[78],[39,95115,17359],{"className":95116},[83],[39,95118,11],{"className":95119},[83,84],[39,95121,161],{"className":95122},[263],[39,95124,2903],{"className":95125},[83],[11,95127,95128,95129,95180,95181,95251,95252,95314],{},"指数太糟糕了。",[39,95130,95132,95150],{"className":95131,"translate":43},[42],[39,95133,95135],{"className":95134},[47],[49,95136,95137],{"xmlns":51},[53,95138,95139,95147],{},[56,95140,95141,95143,95145],{},[59,95142,29425],{},[133,95144,793],{},[539,95146,28724],{},[63,95148,95149],{"encoding":65},"d=100",[39,95151,95153,95171],{"className":95152,"ariaHidden":70},[69],[39,95154,95156,95159,95162,95165,95168],{"className":95155},[74],[39,95157],{"className":95158,"style":28824},[78],[39,95160,29425],{"className":95161},[83,84],[39,95163],{"className":95164,"style":231},[230],[39,95166,793],{"className":95167},[235],[39,95169],{"className":95170,"style":231},[230],[39,95172,95174,95177],{"className":95173},[74],[39,95175],{"className":95176,"style":2473},[78],[39,95178,28724],{"className":95179},[83]," 时误差按 ",[39,95182,95184,95207],{"className":95183,"translate":43},[42],[39,95185,95187],{"className":95186},[47],[49,95188,95189],{"xmlns":51},[53,95190,95191,95204],{},[56,95192,95193],{},[138,95194,95195,95197],{},[59,95196,12962],{},[56,95198,95199,95201],{},[133,95200,5378],{},[539,95202,95203],{},"0.01",[63,95205,95206],{"encoding":65},"n^{-0.01}",[39,95208,95210],{"className":95209,"ariaHidden":70},[69],[39,95211,95213,95216],{"className":95212},[74],[39,95214],{"className":95215,"style":19710},[78],[39,95217,95219,95222],{"className":95218},[83],[39,95220,12962],{"className":95221},[83,84],[39,95223,95225],{"className":95224},[192],[39,95226,95228],{"className":95227},[196],[39,95229,95231],{"className":95230},[200],[39,95232,95234],{"className":95233,"style":19710},[204],[39,95235,95236,95239],{"style":208},[39,95237],{"className":95238,"style":213},[212],[39,95240,95242],{"className":95241},[217,218,219,220],[39,95243,95245,95248],{"className":95244},[83,220],[39,95246,5378],{"className":95247},[83,220],[39,95249,95203],{"className":95250},[83,220]," 衰减，想把误差减半，样本量要乘 ",[39,95253,95255,95273],{"className":95254,"translate":43},[42],[39,95256,95258],{"className":95257},[47],[49,95259,95260],{"xmlns":51},[53,95261,95262,95270],{},[56,95263,95264],{},[138,95265,95266,95268],{},[539,95267,17359],{},[539,95269,28724],{},[63,95271,95272],{"encoding":65},"2^{100}",[39,95274,95276],{"className":95275,"ariaHidden":70},[69],[39,95277,95279,95282],{"className":95278},[74],[39,95280],{"className":95281,"style":19710},[78],[39,95283,95285,95288],{"className":95284},[83],[39,95286,17359],{"className":95287},[83],[39,95289,95291],{"className":95290},[192],[39,95292,95294],{"className":95293},[196],[39,95295,95297],{"className":95296},[200],[39,95298,95300],{"className":95299,"style":19710},[204],[39,95301,95302,95305],{"style":208},[39,95303],{"className":95304,"style":213},[212],[39,95306,95308],{"className":95307},[217,218,219,220],[39,95309,95311],{"className":95310},[83,220],[39,95312,28724],{"className":95313},[83,220],"。所以“直接在高维空间算 Wasserstein”在绝大多数真实问题上不可行。",[11,95316,95317,95318,95346,95347,95421],{},"三条出路我都用过：把数据先降到低维再算；用 sliced Wasserstein，把分布投影到随机一维方向上算 ",[39,95319,95321,95334],{"className":95320,"translate":43},[42],[39,95322,95324],{"className":95323},[47],[49,95325,95326],{"xmlns":51},[53,95327,95328,95332],{},[56,95329,95330],{},[59,95331,23703],{},[63,95333,23703],{"encoding":65},[39,95335,95337],{"className":95336,"ariaHidden":70},[69],[39,95338,95340,95343],{"className":95339},[74],[39,95341],{"className":95342,"style":79},[78],[39,95344,23703],{"className":95345,"style":178},[83,84],"，再对方向平均（一维的收敛是 ",[39,95348,95350,95376],{"className":95349,"translate":43},[42],[39,95351,95353],{"className":95352},[47],[49,95354,95355],{"xmlns":51},[53,95356,95357,95373],{},[56,95358,95359],{},[138,95360,95361,95363],{},[59,95362,12962],{},[56,95364,95365,95367,95369,95371],{},[133,95366,5378],{},[539,95368,546],{},[59,95370,45691],{"mathvariant":145},[539,95372,17359],{},[63,95374,95375],{"encoding":65},"n^{-1\u002F2}",[39,95377,95379],{"className":95378,"ariaHidden":70},[69],[39,95380,95382,95386],{"className":95381},[74],[39,95383],{"className":95384,"style":95385},[78],"height:0.888em;",[39,95387,95389,95392],{"className":95388},[83],[39,95390,12962],{"className":95391},[83,84],[39,95393,95395],{"className":95394},[192],[39,95396,95398],{"className":95397},[196],[39,95399,95401],{"className":95400},[200],[39,95402,95404],{"className":95403,"style":95385},[204],[39,95405,95406,95409],{"style":208},[39,95407],{"className":95408,"style":213},[212],[39,95410,95412],{"className":95411},[217,218,219,220],[39,95413,95415,95418],{"className":95414},[83,220],[39,95416,5378],{"className":95417},[83,220],[39,95419,45696],{"className":95420},[83,220],"，一下就正常了）；或者用低秩分解把代价矩阵近似成低秩。共同点是不要在原始高维空间里做全秩的 OT。",[21,95423,95424],{"id":95424},"单纯形上的尺子",[11,95426,95427,95428,75103,95482,95535,95536,75103,95589,1607],{},"我拿两个模型对同一张图跑一遍，输出分别是 ",[39,95429,95431,95455],{"className":95430,"translate":43},[42],[39,95432,95434],{"className":95433},[47],[49,95435,95436],{"xmlns":51},[53,95437,95438,95452],{},[56,95439,95440,95442,95445,95447,95450],{},[133,95441,136],{"stretchy":135},[539,95443,95444],{},"0.90",[133,95446,155],{"separator":70},[539,95448,95449],{},"0.10",[133,95451,161],{"stretchy":135},[63,95453,95454],{"encoding":65},"(0.90,0.10)",[39,95456,95458],{"className":95457,"ariaHidden":70},[69],[39,95459,95461,95464,95467,95470,95473,95476,95479],{"className":95460},[74],[39,95462],{"className":95463,"style":245},[78],[39,95465,136],{"className":95466},[182],[39,95468,95444],{"className":95469},[83],[39,95471,155],{"className":95472},[252],[39,95474],{"className":95475,"style":256},[230],[39,95477,95449],{"className":95478},[83],[39,95480,161],{"className":95481},[263],[39,95483,95485,95508],{"className":95484,"translate":43},[42],[39,95486,95488],{"className":95487},[47],[49,95489,95490],{"xmlns":51},[53,95491,95492,95505],{},[56,95493,95494,95496,95499,95501,95503],{},[133,95495,136],{"stretchy":135},[539,95497,95498],{},"0.99",[133,95500,155],{"separator":70},[539,95502,95203],{},[133,95504,161],{"stretchy":135},[63,95506,95507],{"encoding":65},"(0.99,0.01)",[39,95509,95511],{"className":95510,"ariaHidden":70},[69],[39,95512,95514,95517,95520,95523,95526,95529,95532],{"className":95513},[74],[39,95515],{"className":95516,"style":245},[78],[39,95518,136],{"className":95519},[182],[39,95521,95498],{"className":95522},[83],[39,95524,155],{"className":95525},[252],[39,95527],{"className":95528,"style":256},[230],[39,95530,95203],{"className":95531},[83],[39,95533,161],{"className":95534},[263],"。另一张图上，两个模型的输出是 ",[39,95537,95539,95562],{"className":95538,"translate":43},[42],[39,95540,95542],{"className":95541},[47],[49,95543,95544],{"xmlns":51},[53,95545,95546,95559],{},[56,95547,95548,95550,95553,95555,95557],{},[133,95549,136],{"stretchy":135},[539,95551,95552],{},"0.50",[133,95554,155],{"separator":70},[539,95556,95552],{},[133,95558,161],{"stretchy":135},[63,95560,95561],{"encoding":65},"(0.50,0.50)",[39,95563,95565],{"className":95564,"ariaHidden":70},[69],[39,95566,95568,95571,95574,95577,95580,95583,95586],{"className":95567},[74],[39,95569],{"className":95570,"style":245},[78],[39,95572,136],{"className":95573},[182],[39,95575,95552],{"className":95576},[83],[39,95578,155],{"className":95579},[252],[39,95581],{"className":95582,"style":256},[230],[39,95584,95552],{"className":95585},[83],[39,95587,161],{"className":95588},[263],[39,95590,95592,95616],{"className":95591,"translate":43},[42],[39,95593,95595],{"className":95594},[47],[49,95596,95597],{"xmlns":51},[53,95598,95599,95613],{},[56,95600,95601,95603,95606,95608,95611],{},[133,95602,136],{"stretchy":135},[539,95604,95605],{},"0.59",[133,95607,155],{"separator":70},[539,95609,95610],{},"0.41",[133,95612,161],{"stretchy":135},[63,95614,95615],{"encoding":65},"(0.59,0.41)",[39,95617,95619],{"className":95618,"ariaHidden":70},[69],[39,95620,95622,95625,95628,95631,95634,95637,95640],{"className":95621},[74],[39,95623],{"className":95624,"style":245},[78],[39,95626,136],{"className":95627},[182],[39,95629,95605],{"className":95630},[83],[39,95632,155],{"className":95633},[252],[39,95635],{"className":95636,"style":256},[230],[39,95638,95610],{"className":95639},[83],[39,95641,161],{"className":95642},[263],[11,95644,95645,95646,95675],{},"两组变化的欧氏距离都是 ",[39,95647,95649,95663],{"className":95648,"translate":43},[42],[39,95650,95652],{"className":95651},[47],[49,95653,95654],{"xmlns":51},[53,95655,95656,95661],{},[56,95657,95658],{},[539,95659,95660],{},"0.09",[63,95662,95660],{"encoding":65},[39,95664,95666],{"className":95665,"ariaHidden":70},[69],[39,95667,95669,95672],{"className":95668},[74],[39,95670],{"className":95671,"style":2473},[78],[39,95673,95660],{"className":95674},[83],"。但我的直觉说它们不是一回事：前者是从“犹豫”变成“笃定”，后者只是从不偏不倚稍微偏了一点。",[11,95677,95678,95679,38249,95732,742,95761,38249,95813,95842,95843,34856,95871,95899],{},"这种直觉能不能写成数？能。把 Bernoulli 分布按 Fisher–Rao 距离量一下：",[39,95680,95682,95705],{"className":95681,"translate":43},[42],[39,95683,95685],{"className":95684},[47],[49,95686,95687],{"xmlns":51},[53,95688,95689,95702],{},[56,95690,95691,95693,95696,95698,95700],{},[133,95692,136],{"stretchy":135},[539,95694,95695],{},"0.5",[133,95697,155],{"separator":70},[539,95699,95605],{},[133,95701,161],{"stretchy":135},[63,95703,95704],{"encoding":65},"(0.5,0.59)",[39,95706,95708],{"className":95707,"ariaHidden":70},[69],[39,95709,95711,95714,95717,95720,95723,95726,95729],{"className":95710},[74],[39,95712],{"className":95713,"style":245},[78],[39,95715,136],{"className":95716},[182],[39,95718,95695],{"className":95719},[83],[39,95721,155],{"className":95722},[252],[39,95724],{"className":95725,"style":256},[230],[39,95727,95605],{"className":95728},[83],[39,95730,161],{"className":95731},[263],[39,95733,95735,95749],{"className":95734,"translate":43},[42],[39,95736,95738],{"className":95737},[47],[49,95739,95740],{"xmlns":51},[53,95741,95742,95747],{},[56,95743,95744],{},[539,95745,95746],{},"0.18",[63,95748,95746],{"encoding":65},[39,95750,95752],{"className":95751,"ariaHidden":70},[69],[39,95753,95755,95758],{"className":95754},[74],[39,95756],{"className":95757,"style":2473},[78],[39,95759,95746],{"className":95760},[83],[39,95762,95764,95786],{"className":95763,"translate":43},[42],[39,95765,95767],{"className":95766},[47],[49,95768,95769],{"xmlns":51},[53,95770,95771,95783],{},[56,95772,95773,95775,95777,95779,95781],{},[133,95774,136],{"stretchy":135},[539,95776,17739],{},[133,95778,155],{"separator":70},[539,95780,95498],{},[133,95782,161],{"stretchy":135},[63,95784,95785],{"encoding":65},"(0.9,0.99)",[39,95787,95789],{"className":95788,"ariaHidden":70},[69],[39,95790,95792,95795,95798,95801,95804,95807,95810],{"className":95791},[74],[39,95793],{"className":95794,"style":245},[78],[39,95796,136],{"className":95797},[182],[39,95799,17739],{"className":95800},[83],[39,95802,155],{"className":95803},[252],[39,95805],{"className":95806,"style":256},[230],[39,95808,95498],{"className":95809},[83],[39,95811,161],{"className":95812},[263],[39,95814,95816,95830],{"className":95815,"translate":43},[42],[39,95817,95819],{"className":95818},[47],[49,95820,95821],{"xmlns":51},[53,95822,95823,95828],{},[56,95824,95825],{},[539,95826,95827],{},"0.44",[63,95829,95827],{"encoding":65},[39,95831,95833],{"className":95832,"ariaHidden":70},[69],[39,95834,95836,95839],{"className":95835},[74],[39,95837],{"className":95838,"style":2473},[78],[39,95840,95827],{"className":95841},[83],"，后者是前者的两倍半。同样的坐标变化，在概率接近 ",[39,95844,95846,95859],{"className":95845,"translate":43},[42],[39,95847,95849],{"className":95848},[47],[49,95850,95851],{"xmlns":51},[53,95852,95853,95857],{},[56,95854,95855],{},[539,95856,541],{},[63,95858,541],{"encoding":65},[39,95860,95862],{"className":95861,"ariaHidden":70},[69],[39,95863,95865,95868],{"className":95864},[74],[39,95866],{"className":95867,"style":2473},[78],[39,95869,541],{"className":95870},[83],[39,95872,95874,95887],{"className":95873,"translate":43},[42],[39,95875,95877],{"className":95876},[47],[49,95878,95879],{"xmlns":51},[53,95880,95881,95885],{},[56,95882,95883],{},[539,95884,546],{},[63,95886,546],{"encoding":65},[39,95888,95890],{"className":95889,"ariaHidden":70},[69],[39,95891,95893,95896],{"className":95892},[74],[39,95894],{"className":95895,"style":2473},[78],[39,95897,546],{"className":95898},[83]," 的地方要走更远的路——那里是单纯形的“边缘”，流形在那边被拉长了。",[11,95901,95902],{},"那么这把尺子是怎么来的？",[11,95904,95905],{},"模型的输出住在单纯形",[39,95907,95909],{"className":95908,"translate":43},[746],[39,95910,95912,95986],{"className":95911,"translate":43},[42],[39,95913,95915],{"className":95914},[47],[49,95916,95917],{"xmlns":51,"display":755},[53,95918,95919,95983],{},[56,95920,95921,95933,95935,95937,95939,95941,95947,95949,95951,95957,95959,95961,95963,95965,95971,95977,95979,95981],{},[138,95922,95923,95925],{},[59,95924,2134],{"mathvariant":145},[56,95926,95927,95929,95931],{},[59,95928,77187],{},[133,95930,5378],{},[539,95932,546],{},[133,95934,793],{},[133,95936,12951],{"fence":70,"stretchy":70,"minsize":65017,"maxsize":65017},[59,95938,11],{},[133,95940,534],{},[138,95942,95943,95945],{},[59,95944,422],{"mathvariant":421},[59,95946,77187],{},[133,95948,404],{},[35841,95950,74223],{},[613,95952,95953,95955],{},[59,95954,11],{},[59,95956,13377],{},[133,95958,25173],{},[539,95960,541],{},[133,95962,155],{"separator":70},[35841,95964,74223],{},[6556,95966,95967,95969],{},[133,95968,3191],{},[59,95970,13377],{},[613,95972,95973,95975],{},[59,95974,11],{},[59,95976,13377],{},[133,95978,793],{},[539,95980,546],{},[133,95982,12965],{"fence":70,"stretchy":70,"minsize":65017,"maxsize":65017},[63,95984,95985],{"encoding":65},"\\Delta^{K-1}=\\Bigl\\{p\\in\\mathbb R^K:\\ p_i\\ge0,\\ \\sum_i p_i=1\\Bigr\\}",[39,95987,95989,96043,96067,96114,96170,96284],{"className":95988,"ariaHidden":70},[69],[39,95990,95992,95996,96034,96037,96040],{"className":95991},[74],[39,95993],{"className":95994,"style":95995},[78],"height:0.8913em;",[39,95997,95999,96002],{"className":95998},[83],[39,96000,2134],{"className":96001},[83],[39,96003,96005],{"className":96004},[192],[39,96006,96008],{"className":96007},[196],[39,96009,96011],{"className":96010},[200],[39,96012,96014],{"className":96013,"style":95995},[204],[39,96015,96016,96019],{"style":1027},[39,96017],{"className":96018,"style":213},[212],[39,96020,96022],{"className":96021},[217,218,219,220],[39,96023,96025,96028,96031],{"className":96024},[83,220],[39,96026,77187],{"className":96027,"style":48487},[83,84,220],[39,96029,5378],{"className":96030},[465,220],[39,96032,546],{"className":96033},[83,220],[39,96035],{"className":96036,"style":231},[230],[39,96038,793],{"className":96039},[235],[39,96041],{"className":96042,"style":231},[230],[39,96044,96046,96049,96055,96058,96061,96064],{"className":96045},[74],[39,96047],{"className":96048,"style":75302},[78],[39,96050,96052],{"className":96051},[182],[39,96053,12951],{"className":96054},[18520,33204],[39,96056,11],{"className":96057},[83,84],[39,96059],{"className":96060,"style":231},[230],[39,96062,534],{"className":96063},[235],[39,96065],{"className":96066,"style":231},[230],[39,96068,96070,96073,96102,96105,96108,96111],{"className":96069},[74],[39,96071],{"className":96072,"style":95995},[78],[39,96074,96076,96079],{"className":96075},[83],[39,96077,422],{"className":96078},[83,515],[39,96080,96082],{"className":96081},[192],[39,96083,96085],{"className":96084},[196],[39,96086,96088],{"className":96087},[200],[39,96089,96091],{"className":96090,"style":95995},[204],[39,96092,96093,96096],{"style":1027},[39,96094],{"className":96095,"style":213},[212],[39,96097,96099],{"className":96098},[217,218,219,220],[39,96100,77187],{"className":96101,"style":48487},[83,84,220],[39,96103],{"className":96104,"style":231},[230],[39,96106,404],{"className":96107},[235],[39,96109,74223],{"className":96110},[230],[39,96112],{"className":96113,"style":231},[230],[39,96115,96117,96121,96161,96164,96167],{"className":96116},[74],[39,96118],{"className":96119,"style":96120},[78],"height:0.8304em;vertical-align:-0.1944em;",[39,96122,96124,96127],{"className":96123},[83],[39,96125,11],{"className":96126},[83,84],[39,96128,96130],{"className":96129},[192],[39,96131,96133,96153],{"className":96132},[196,649],[39,96134,96136,96150],{"className":96135},[200],[39,96137,96139],{"className":96138,"style":13604},[204],[39,96140,96141,96144],{"style":966},[39,96142],{"className":96143,"style":213},[212],[39,96145,96147],{"className":96146},[217,218,219,220],[39,96148,13377],{"className":96149},[83,84,220],[39,96151,673],{"className":96152},[672],[39,96154,96156],{"className":96155},[200],[39,96157,96159],{"className":96158,"style":680},[204],[39,96160],{},[39,96162],{"className":96163,"style":231},[230],[39,96165,25173],{"className":96166},[235],[39,96168],{"className":96169,"style":231},[230],[39,96171,96173,96176,96179,96182,96185,96188,96232,96235,96275,96278,96281],{"className":96172},[74],[39,96174],{"className":96175,"style":82834},[78],[39,96177,541],{"className":96178},[83],[39,96180,155],{"className":96181},[252],[39,96183,74223],{"className":96184},[230],[39,96186],{"className":96187,"style":256},[230],[39,96189,96191],{"className":96190},[1744,3294],[39,96192,96194,96224],{"className":96193},[196,649],[39,96195,96197,96221],{"className":96196},[200],[39,96198,96200,96211],{"className":96199,"style":6670},[204],[39,96201,96202,96205],{"style":13926},[39,96203],{"className":96204,"style":3311},[212],[39,96206,96208],{"className":96207},[217,218,219,220],[39,96209,13377],{"className":96210},[83,84,220],[39,96212,96213,96216],{"style":3330},[39,96214],{"className":96215,"style":3311},[212],[39,96217,96218],{},[39,96219,3191],{"className":96220},[1744,3339,3340],[39,96222,673],{"className":96223},[672],[39,96225,96227],{"className":96226},[200],[39,96228,96230],{"className":96229,"style":78935},[204],[39,96231],{},[39,96233],{"className":96234,"style":256},[230],[39,96236,96238,96241],{"className":96237},[83],[39,96239,11],{"className":96240},[83,84],[39,96242,96244],{"className":96243},[192],[39,96245,96247,96267],{"className":96246},[196,649],[39,96248,96250,96264],{"className":96249},[200],[39,96251,96253],{"className":96252,"style":13604},[204],[39,96254,96255,96258],{"style":966},[39,96256],{"className":96257,"style":213},[212],[39,96259,96261],{"className":96260},[217,218,219,220],[39,96262,13377],{"className":96263},[83,84,220],[39,96265,673],{"className":96266},[672],[39,96268,96270],{"className":96269},[200],[39,96271,96273],{"className":96272,"style":680},[204],[39,96274],{},[39,96276],{"className":96277,"style":231},[230],[39,96279,793],{"className":96280},[235],[39,96282],{"className":96283,"style":231},[230],[39,96285,96287,96290,96293],{"className":96286},[74],[39,96288],{"className":96289,"style":75302},[78],[39,96291,546],{"className":96292},[83],[39,96294,96296],{"className":96295},[263],[39,96297,12965],{"className":96298},[18520,33204],[11,96300,96301],{},"里。问题在于这个空间有边界：越靠近顶点，同样的坐标变化对应的“实质变化”越大。欧氏距离看不见这件事。",[11,96303,96304,96305,96386],{},"Rao 在 1945 年的想法是：把参数化的分布族 ",[39,96306,96308,96331],{"className":96307,"translate":43},[42],[39,96309,96311],{"className":96310},[47],[49,96312,96313],{"xmlns":51},[53,96314,96315,96328],{},[56,96316,96317,96319,96326],{},[133,96318,12951],{"stretchy":135},[613,96320,96321,96323],{},[59,96322,11],{},[59,96324,96325],{},"θ",[133,96327,12965],{"stretchy":135},[63,96329,96330],{"encoding":65},"\\{p_\\theta\\}",[39,96332,96334],{"className":96333,"ariaHidden":70},[69],[39,96335,96337,96340,96343,96383],{"className":96336},[74],[39,96338],{"className":96339,"style":245},[78],[39,96341,12951],{"className":96342},[182],[39,96344,96346,96349],{"className":96345},[83],[39,96347,11],{"className":96348},[83,84],[39,96350,96352],{"className":96351},[192],[39,96353,96355,96375],{"className":96354},[196,649],[39,96356,96358,96372],{"className":96357},[200],[39,96359,96361],{"className":96360,"style":3419},[204],[39,96362,96363,96366],{"style":966},[39,96364],{"className":96365,"style":213},[212],[39,96367,96369],{"className":96368},[217,218,219,220],[39,96370,96325],{"className":96371,"style":438},[83,84,220],[39,96373,673],{"className":96374},[672],[39,96376,96378],{"className":96377},[200],[39,96379,96381],{"className":96380,"style":680},[204],[39,96382],{},[39,96384,12965],{"className":96385},[263]," 本身看成一个流形，在上面装一把由数据决定的尺子。这把尺子就是 Fisher 信息矩阵",[39,96388,96390],{"className":96389,"translate":43},[746],[39,96391,96393,96514],{"className":96392,"translate":43},[42],[39,96394,96396],{"className":96395},[47],[49,96397,96398],{"xmlns":51,"display":755},[53,96399,96400,96511],{},[56,96401,96402,96412,96414,96416,96418,96420,96430,96432,96439,96441,96443,96449,96451,96457,96459,96461,96467,96469,96471,96473,96483,96485,96491,96497,96499,96501,96507,96509],{},[613,96403,96404,96406],{},[59,96405,39209],{},[56,96407,96408,96410],{},[59,96409,13377],{},[59,96411,13446],{},[133,96413,136],{"stretchy":135},[59,96415,96325],{},[133,96417,161],{"stretchy":135},[133,96419,793],{},[613,96421,96422,96424],{},[59,96423,4253],{"mathvariant":421},[613,96425,96426,96428],{},[59,96427,11],{},[59,96429,96325],{},[133,96431,537],{"fence":70,"stretchy":70,"minsize":73875,"maxsize":73875},[613,96433,96434,96437],{},[59,96435,96436],{"mathvariant":145},"∂",[59,96438,13377],{},[59,96440,92483],{},[133,96442,1690],{},[613,96444,96445,96447],{},[59,96446,11],{},[59,96448,96325],{},[133,96450,8885],{},[613,96452,96453,96455],{},[59,96454,96436],{"mathvariant":145},[59,96456,13446],{},[59,96458,92483],{},[133,96460,1690],{},[613,96462,96463,96465],{},[59,96464,11],{},[59,96466,96325],{},[133,96468,4280],{"fence":70,"stretchy":70,"minsize":73875,"maxsize":73875},[133,96470,793],{},[133,96472,5378],{},[613,96474,96475,96477],{},[59,96476,4253],{"mathvariant":421},[613,96478,96479,96481],{},[59,96480,11],{},[59,96482,96325],{},[133,96484,537],{"fence":70,"stretchy":70,"minsize":73875,"maxsize":73875},[613,96486,96487,96489],{},[59,96488,96436],{"mathvariant":145},[59,96490,13377],{},[613,96492,96493,96495],{},[59,96494,96436],{"mathvariant":145},[59,96496,13446],{},[59,96498,92483],{},[133,96500,1690],{},[613,96502,96503,96505],{},[59,96504,11],{},[59,96506,96325],{},[133,96508,4280],{"fence":70,"stretchy":70,"minsize":73875,"maxsize":73875},[133,96510,155],{"separator":70},[63,96512,96513],{"encoding":65},"g_{ij}(\\theta)=\\mathbb E_{p_\\theta}\\bigl[\\partial_i\\log p_\\theta\\cdot\\partial_j\\log p_\\theta\\bigr]\n=-\\mathbb E_{p_\\theta}\\bigl[\\partial_i\\partial_j\\log p_\\theta\\bigr],",[39,96515,96517,96584,96779,96891],{"className":96516,"ariaHidden":70},[69],[39,96518,96520,96523,96566,96569,96572,96575,96578,96581],{"className":96519},[74],[39,96521],{"className":96522,"style":41560},[78],[39,96524,96526,96529],{"className":96525},[83],[39,96527,39209],{"className":96528,"style":2114},[83,84],[39,96530,96532],{"className":96531},[192],[39,96533,96535,96558],{"className":96534},[196,649],[39,96536,96538,96555],{"className":96537},[200],[39,96539,96541],{"className":96540,"style":13604},[204],[39,96542,96543,96546],{"style":41316},[39,96544],{"className":96545,"style":213},[212],[39,96547,96549],{"className":96548},[217,218,219,220],[39,96550,96552],{"className":96551},[83,220],[39,96553,13732],{"className":96554,"style":13731},[83,84,220],[39,96556,673],{"className":96557},[672],[39,96559,96561],{"className":96560},[200],[39,96562,96564],{"className":96563,"style":35396},[204],[39,96565],{},[39,96567,136],{"className":96568},[182],[39,96570,96325],{"className":96571,"style":438},[83,84],[39,96573,161],{"className":96574},[263],[39,96576],{"className":96577,"style":231},[230],[39,96579,793],{"className":96580},[235],[39,96582],{"className":96583,"style":231},[230],[39,96585,96587,96590,96672,96678,96719,96722,96727,96730,96770,96773,96776],{"className":96586},[74],[39,96588],{"className":96589,"style":908},[78],[39,96591,96593,96596],{"className":96592},[83],[39,96594,4253],{"className":96595},[83,515],[39,96597,96599],{"className":96598},[192],[39,96600,96602,96664],{"className":96601},[196,649],[39,96603,96605,96661],{"className":96604},[200],[39,96606,96608],{"className":96607,"style":1728},[204],[39,96609,96610,96613],{"style":966},[39,96611],{"className":96612,"style":213},[212],[39,96614,96616],{"className":96615},[217,218,219,220],[39,96617,96619],{"className":96618},[83,220],[39,96620,96622,96625],{"className":96621},[83,220],[39,96623,11],{"className":96624},[83,84,220],[39,96626,96628],{"className":96627},[192],[39,96629,96631,96652],{"className":96630},[196,649],[39,96632,96634,96649],{"className":96633},[200],[39,96635,96637],{"className":96636,"style":84344},[204],[39,96638,96640,96643],{"style":96639},"top:-2.3488em;margin-left:0em;margin-right:0.0714em;",[39,96641],{"className":96642,"style":4004},[212],[39,96644,96646],{"className":96645},[217,4008,4009,220],[39,96647,96325],{"className":96648,"style":438},[83,84,220],[39,96650,673],{"className":96651},[672],[39,96653,96655],{"className":96654},[200],[39,96656,96659],{"className":96657,"style":96658},[204],"height:0.1512em;",[39,96660],{},[39,96662,673],{"className":96663},[672],[39,96665,96667],{"className":96666},[200],[39,96668,96670],{"className":96669,"style":35396},[204],[39,96671],{},[39,96673,96675],{"className":96674},[182],[39,96676,537],{"className":96677},[18520,4009],[39,96679,96681,96684],{"className":96680},[83],[39,96682,96436],{"className":96683,"style":565},[83],[39,96685,96687],{"className":96686},[192],[39,96688,96690,96711],{"className":96689},[196,649],[39,96691,96693,96708],{"className":96692},[200],[39,96694,96696],{"className":96695,"style":13604},[204],[39,96697,96699,96702],{"style":96698},"top:-2.55em;margin-left:-0.0556em;margin-right:0.05em;",[39,96700],{"className":96701,"style":213},[212],[39,96703,96705],{"className":96704},[217,218,219,220],[39,96706,13377],{"className":96707},[83,84,220],[39,96709,673],{"className":96710},[672],[39,96712,96714],{"className":96713},[200],[39,96715,96717],{"className":96716,"style":680},[204],[39,96718],{},[39,96720],{"className":96721,"style":256},[230],[39,96723,92721,96725],{"className":96724},[1744],[39,96726,39209],{"style":35078},[39,96728],{"className":96729,"style":256},[230],[39,96731,96733,96736],{"className":96732},[83],[39,96734,11],{"className":96735},[83,84],[39,96737,96739],{"className":96738},[192],[39,96740,96742,96762],{"className":96741},[196,649],[39,96743,96745,96759],{"className":96744},[200],[39,96746,96748],{"className":96747,"style":3419},[204],[39,96749,96750,96753],{"style":966},[39,96751],{"className":96752,"style":213},[212],[39,96754,96756],{"className":96755},[217,218,219,220],[39,96757,96325],{"className":96758,"style":438},[83,84,220],[39,96760,673],{"className":96761},[672],[39,96763,96765],{"className":96764},[200],[39,96766,96768],{"className":96767,"style":680},[204],[39,96769],{},[39,96771],{"className":96772,"style":461},[230],[39,96774,8885],{"className":96775},[465],[39,96777],{"className":96778,"style":461},[230],[39,96780,96782,96785,96825,96828,96833,96836,96876,96882,96885,96888],{"className":96781},[74],[39,96783],{"className":96784,"style":908},[78],[39,96786,96788,96791],{"className":96787},[83],[39,96789,96436],{"className":96790,"style":565},[83],[39,96792,96794],{"className":96793},[192],[39,96795,96797,96817],{"className":96796},[196,649],[39,96798,96800,96814],{"className":96799},[200],[39,96801,96803],{"className":96802,"style":13604},[204],[39,96804,96805,96808],{"style":96698},[39,9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",[39,97130,97132,97166],{"className":97131,"translate":43},[42],[39,97133,97135],{"className":97134},[47],[49,97136,97137],{"xmlns":51},[53,97138,97139,97163],{},[56,97140,97141,97143,97145,97151,97153,97155,97157,97159,97161],{},[59,97142,4253],{"mathvariant":421},[133,97144,537],{"stretchy":135},[613,97146,97147,97149],{},[59,97148,96436],{"mathvariant":145},[59,97150,13377],{},[59,97152,92483],{},[133,97154,1690],{},[59,97156,11],{},[133,97158,4280],{"stretchy":135},[133,97160,793],{},[539,97162,541],{},[63,97164,97165],{"encoding":65},"\\mathbb E[\\partial_i\\log p]=0",[39,97167,97169,97247],{"className":97168,"ariaHidden":70},[69],[39,97170,97172,97175,97178,97181,97221,97224,97229,97232,97235,97238,97241,97244],{"className":97171},[74],[39,97173],{"className":97174,"style":245},[78],[39,97176,4253],{"className":97177},[83,515],[39,97179,537],{"className":97180},[182],[39,97182,97184,97187],{"className":97183},[83],[39,97185,96436],{"className":97186,"style":565},[83],[39,97188,97190],{"className":97189},[192],[39,97191,97193,97213],{"className":97192},[196,649],[39,97194,97196,97210],{"className":97195},[200],[39,97197,97199],{"className":97198,"style":13604},[204],[39,97200,97201,97204],{"style":96698},[39,97202],{"className":97203,"style":213},[212],[39,97205,97207],{"className":97206},[217,218,219,220],[39,97208,13377],{"className":97209},[83,84,220],[39,97211,673],{"className":97212},[672],[39,97214,97216],{"className":97215},[200],[39,97217,97219],{"className":97218,"style":680},[204],[39,97220],{},[39,97222],{"className":97223,"style":256},[230],[39,97225,92721,97227],{"className":97226},[1744],[39,97228,39209],{"style":35078},[39,97230],{"className":97231,"style":256},[230],[39,97233,11],{"className":97234},[83,84],[39,97236,4280],{"className":97237},[263],[39,97239],{"className":97240,"style":231},[230],[39,97242,793],{"className":97243},[235],[39,97245],{"className":97246,"style":231},[230],[39,97248,97250,97253],{"className":97249},[74],[39,97251],{"className":97252,"style":2473},[78],[39,97254,541],{"className":97255},[83],"，因为对 ",[39,97258,97260,97274],{"className":97259,"translate":43},[42],[39,97261,97263],{"className":97262},[47],[49,97264,97265],{"xmlns":51},[53,97266,97267,97271],{},[56,97268,97269],{},[59,97270,96325],{},[63,97272,97273],{"encoding":65},"\\theta",[39,97275,97277],{"className":97276,"ariaHidden":70},[69],[39,97278,97280,97283],{"className":97279},[74],[39,97281],{"className":97282,"style":28824},[78],[39,97284,96325],{"className":97285,"style":438},[83,84]," 求导再积分，",[39,97288,97290,97340],{"className":97289,"translate":43},[42],[39,97291,97293],{"className":97292},[47],[49,97294,97295],{"xmlns":51},[53,97296,97297,97337],{},[56,97298,97299,97301,97307,97309,97311,97313,97315,97317,97323,97325,97327,97329,97331,97333,97335],{},[133,97300,69087],{},[613,97302,97303,97305],{},[59,97304,96436],{"mathvariant":145},[59,97306,13377],{},[59,97308,11],{},[35841,97310,35843],{},[59,97312,29425],{},[59,97314,1755],{},[133,97316,793],{},[613,97318,97319,97321],{},[59,97320,96436],{"mathvariant":145},[59,97322,13377],{},[133,97324,69087],{},[59,97326,11],{},[35841,97328,35843],{},[59,97330,29425],{},[59,97332,1755],{},[133,97334,793],{},[539,97336,541],{},[63,97338,97339],{"encoding":65},"\\int\\partial_i p\\,dx=\\partial_i\\int 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KL 虽然整体上不对称，局部却退化成一个二次型，二次型的矩阵就是 Fisher。反过来说，任何“局部像 KL”的距离都会给出同一个度量。",[11,98588,98589],{},"第二，Chentsov 定理说：单纯形上的黎曼度量，若在充分统计量（更准确地说，Markov 态射）下保持不变，则差一个常数因子必为 Fisher–Rao。换句话说，这不是我随手挑的度量，而是在“只依赖分布、不依赖参数化”这个要求下唯一的选择。",[11,98591,98592],{},"第三，对指数族它恰好是势函数的 Hessian，下一节说。",[11,98594,98595,98596,742],{},"把 Bernoulli 族算一遍。",[39,98597,98599,98651],{"className":98598,"translate":43},[42],[39,98600,98602],{"className":98601},[47],[49,98603,98604],{"xmlns":51},[53,98605,98606,98648],{},[56,98607,98608,98610,98612,98614,98616,98618,98620,98622,98628,98630,98632,98634,98636],{},[59,98609,11],{},[133,98611,136],{"stretchy":135},[59,98613,1755],{},[133,98615,150],{},[59,98617,96325],{},[133,98619,161],{"stretchy":135},[133,98621,793],{},[138,98623,98624,98626],{},[59,98625,96325],{},[59,98627,1755],{},[133,98629,136],{"stretchy":135},[539,98631,546],{},[133,98633,5378],{},[59,98635,96325],{},[138,98637,98638,98640],{},[133,98639,161],{"stretchy":135},[56,98641,98642,98644,98646],{},[539,98643,546],{},[133,98645,5378],{},[59,98647,1755],{},[63,98649,98650],{"encoding":65},"p(x\\mid\\theta)=\\theta^x(1-\\theta)^{1-x}",[39,98652,98654,98678,98699,98749],{"className":98653,"ariaHidden":70},[69],[39,98655,98657,98660,98663,98666,98669,98672,98675],{"className":98656},[74],[39,98658],{"className":98659,"style":245},[78],[39,98661,11],{"className":98662},[83,84],[39,98664,136],{"className":98665},[182],[39,98667,1755],{"className":98668},[83,84],[39,98670],{"className":98671,"style":231},[230],[39,98673,150],{"className":98674},[235],[39,98676],{"className":98677,"style":231},[230],[39,98679,98681,98684,98687,98690,98693,98696],{"className":98680},[74],[39,98682],{"className":98683,"style":245},[78],[39,98685,96325],{"className":98686,"style":438},[83,84],[39,98688,161],{"className":98689},[263],[39,98691],{"className":98692,"style":231},[230],[39,98694,793],{"className":98695},[235],[39,98697],{"className":98698,"style":231},[230],[39,98700,98702,98705,98734,98737,98740,98743,98746],{"className":98701},[74],[39,98703],{"className":98704,"style":245},[78],[39,98706,98708,98711],{"className":98707},[83],[39,98709,96325],{"className":98710,"style":438},[83,84],[39,98712,98714],{"className":98713},[192],[39,98715,98717],{"className":98716},[196],[39,98718,98720],{"className":98719},[200],[39,98721,98723],{"className":98722,"style":6410},[204],[39,98724,98725,98728],{"style":208},[39,98726],{"className":98727,"style":213},[212],[39,98729,98731],{"className":98730},[217,218,219,220],[39,98732,1755],{"className":98733},[83,84,220],[39,98735,136],{"className":98736},[182],[39,98738,546],{"className":98739},[83],[39,98741],{"className":98742,"style":461},[230],[39,98744,5378],{"className":98745},[465],[39,98747],{"className":98748,"style":461},[230],[39,98750,98752,98755,98758],{"className":98751},[74],[39,98753],{"className":98754,"style":84615},[78],[39,98756,96325],{"className":98757,"style":438},[83,84],[39,98759,98761,98764],{"className":98760},[263],[39,98762,161],{"className":98763},[263],[39,98765,98767],{"className":98766},[192],[39,98768,98770],{"className":98769},[196],[39,98771,98773],{"className":98772},[200],[39,98774,98776],{"className":98775,"style":19710},[204],[39,98777,98778,98781],{"style":208},[39,98779],{"className":98780,"style":213},[212],[39,98782,98784],{"className":98783},[217,218,219,220],[39,98785,98787,98790,98793],{"className":98786},[83,220],[39,98788,546],{"className":98789},[83,220],[39,98791,5378],{"className":98792},[465,220],[39,98794,1755],{"className":98795},[83,84,220],[39,98797,98799],{"className":98798,"translate":43},[746],[39,98800,98802,98844],{"className":98801,"translate":43},[42],[39,98803,98805],{"className":98804},[47],[49,98806,98807],{"xmlns":51,"display":755},[53,98808,98809,98841],{},[56,98810,98811,98813,98815,98817,98819,98821,98839],{},[59,98812,39209],{},[133,98814,136],{"stretchy":135},[59,98816,96325],{},[133,98818,161],{"stretchy":135},[133,98820,793],{},[5362,98822,98823,98825],{},[539,98824,546],{},[56,98826,98827,98829,98831,98833,98835,98837],{},[59,98828,96325],{},[133,98830,136],{"stretchy":135},[539,98832,546],{},[133,98834,5378],{},[59,98836,96325],{},[133,98838,161],{"stretchy":135},[59,98840,2903],{"mathvariant":145},[63,98842,98843],{"encoding":65},"g(\\theta)=\\frac{1}{\\theta(1-\\theta)} 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的距离是 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",[39,99810,99812,99830],{"className":99811,"translate":43},[42],[39,99813,99815],{"className":99814},[47],[49,99816,99817],{"xmlns":51},[53,99818,99819,99827],{},[56,99820,99821,99823,99825],{},[59,99822,96325],{},[133,99824,418],{},[539,99826,541],{},[63,99828,99829],{"encoding":65},"\\theta\\to0",[39,99831,99833,99851],{"className":99832,"ariaHidden":70},[69],[39,99834,99836,99839,99842,99845,99848],{"className":99835},[74],[39,99837],{"className":99838,"style":28824},[78],[39,99840,96325],{"className":99841,"style":438},[83,84],[39,99843],{"className":99844,"style":231},[230],[39,99846,418],{"className":99847},[235],[39,99849],{"className":99850,"style":231},[230],[39,99852,99854,99857],{"className":99853},[74],[39,99855],{"className":99856,"style":2473},[78],[39,99858,541],{"className":99859},[83],[39,99861,99863,99881],{"className":99862,"translate":43},[42],[39,99864,99866],{"className":99865},[47],[49,99867,99868],{"xmlns":51},[53,99869,99870,99878],{},[56,99871,99872,99874,99876],{},[59,99873,96325],{},[133,99875,418],{},[539,99877,546],{},[63,99879,99880],{"encoding":65},"\\theta\\to1",[39,99882,99884,99902],{"className":99883,"ariaHidden":70},[69],[39,99885,99887,99890,99893,99896,99899],{"className":99886},[74],[39,99888],{"className":99889,"style":28824},[78],[39,99891,96325],{"className":99892,"style":438},[83,84],[39,99894],{"className":99895,"style":231},[230],[39,99897,418],{"className":99898},[235],[39,99900],{"className":99901,"style":231},[230],[39,99903,99905,99908],{"className":99904},[74],[39,99906],{"className":99907,"style":2473},[78],[39,99909,546],{"className":99910},[83],[39,99912,99914,99927],{"className":99913,"translate":43},[42],[39,99915,99917],{"className":99916},[47],[49,99918,99919],{"xmlns":51},[53,99920,99921,99925],{},[56,99922,99923],{},[59,99924,2098],{},[63,99926,2101],{"encoding":65},[39,99928,99930],{"className":99929,"ariaHidden":70},[69],[39,99931,99933,99936],{"className":99932},[74],[39,99934],{"className":99935,"style":290},[78],[39,99937,2098],{"className":99938,"style":2114},[83,84],"，有限。这一点我很喜欢：从“几乎确定是 0”走到“几乎确定是 1”，不管两端有多极端，路径长度都是 ",[39,99941,99943,99956],{"className":99942,"translate":43},[42],[39,99944,99946],{"className":99945},[47],[49,99947,99948],{"xmlns":51},[53,99949,99950,99954],{},[56,99951,99952],{},[59,99953,2098],{},[63,99955,2101],{"encoding":65},[39,99957,99959],{"className":99958,"ariaHidden":70},[69],[39,99960,99962,99965],{"className":99961},[74],[39,99963],{"className":99964,"style":290},[78],[39,99966,2098],{"className":99967,"style":2114},[83,84],"；而 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并且随极端程度一起发散。几何距离和信息量在这里分道扬镳。",[11,100052,100053,100054,100153],{},"那个 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不是巧合。对一般的离散分布，",[39,100155,100157],{"className":100156,"translate":43},[746],[39,100158,100160,100230],{"className":100159,"translate":43},[42],[39,100161,100163],{"className":100162},[47],[49,100164,100165],{"xmlns":51,"display":755},[53,100166,100167,100227],{},[56,100168,100169,100179,100181,100183,100185,100188,100190,100192,100194,100197,100199,100201,100207,100223,100225],{},[613,100170,100171,100173],{},[59,100172,29425],{},[56,100174,100175,100177],{},[59,100176,618],{"mathvariant":145},[59,100178,422],{"mathvariant":145},[133,100180,136],{"stretchy":135},[59,100182,11],{},[133,100184,155],{"separator":70},[59,100186,100187],{},"q",[133,100189,161],{"stretchy":135},[133,100191,793],{},[539,100193,17359],{},[59,100195,100196],{},"arccos",[133,100198,1690],{},[133,100200,136],{"fence":70,"stretchy":70,"minsize":65017,"maxsize":65017},[6556,100202,100203,100205],{},[133,100204,3191],{},[59,100206,13377],{},[98996,100208,100209],{},[56,100210,100211,100217],{},[613,100212,100213,100215],{},[59,100214,11],{},[59,100216,13377],{},[613,100218,100219,100221],{},[59,100220,100187],{},[59,100222,13377],{},[133,100224,161],{"fence":70,"stretchy":70,"minsize":65017,"maxsize":65017},[133,100226,155],{"separator":70},[63,100228,100229],{"encoding":65},"d_{\\mathrm{FR}}(p,q)=2\\arccos\\Bigl(\\sum_i\\sqrt{p_iq_i}\\Bigr),",[39,100231,100233,100313],{"className":100232,"ariaHidden":70},[69],[39,100234,100236,100239,100286,100289,100292,100295,100298,100301,100304,100307,100310],{"className":100235},[74],[39,100237],{"className":100238,"style":245},[78],[39,100240,100242,100245],{"className":100241},[83],[39,100243,29425],{"className":100244},[83,84],[39,100246,100248],{"className":100247},[192],[39,100249,100251,100278],{"className":100250},[196,649],[39,100252,100254,100275],{"className":100253},[200],[39,100255,100257],{"className":100256,"style":62921},[204],[39,100258,100259,100262],{"style":966},[39,100260],{"className":100261,"style":213},[212],[39,100263,100265],{"className":100264},[217,218,219,220],[39,100266,100268],{"className":100267},[83,220],[39,100269,100271],{"className":100270},[83,220],[39,100272,100274],{"className":100273},[83,35077,220],"FR",[39,100276,673],{"className":100277},[672],[39,100279,100281],{"className":100280},[200],[39,100282,100284],{"className":100283,"style":680},[204],[39,100285],{},[39,100287,136],{"className":100288},[182],[39,100290,11],{"className":100291},[83,84],[39,100293,155],{"className":100294},[252],[39,100296],{"className":100297,"style":256},[230],[39,100299,100187],{"className":100300,"style":2114},[83,84],[39,100302,161],{"className":100303},[263],[39,100305],{"className":100306,"style":231},[230],[39,100308,793],{"className":100309},[235],[39,100311],{"className":100312,"style":231},[230],[39,100314,100316,100320,100323,100326,100329,100335,100379,100382,100509,100515],{"className":100315},[74],[39,100317],{"className":100318,"style":100319},[78],"height:2.4277em;vertical-align:-1.2777em;",[39,100321,17359],{"className":100322},[83],[39,100324],{"className":100325,"style":256},[230],[39,100327,100196],{"className":100328},[1744],[39,100330,100332],{"className":100331},[182],[39,100333,136],{"className":100334},[18520,33204],[39,100336,100338],{"className":100337},[1744,3294],[39,100339,100341,100371],{"className":100340},[196,649],[39,100342,100344,100368],{"className":100343},[200],[39,100345,100347,100358],{"className":100346,"style":6670},[204],[39,100348,100349,100352],{"style":13926},[39,100350],{"className":100351,"style":3311},[212],[39,100353,100355],{"className":100354},[217,218,219,220],[39,100356,13377],{"className":100357},[83,84,220],[39,100359,100360,100363],{"style":3330},[39,100361],{"className":100362,"style":3311},[212],[39,100364,100365],{},[39,100366,3191],{"className":100367},[1744,3339,3340],[39,100369,673],{"className":100370},[672],[39,100372,100374],{"className":100373},[200],[39,100375,100377],{"className":100376,"style":78935},[204],[39,100378],{},[39,100380],{"className":100381,"style":256},[230],[39,100383,100385],{"className":100384},[83,99058],[39,100386,100388,100500],{"className":100387},[196,649],[39,100389,100391,100497],{"className":100390},[200],[39,100392,100394,100484],{"className":100393,"style":205},[204],[39,100395,100397,100400],{"className":100396,"style":21599},[99072],[39,100398],{"className":100399,"style":935},[212],[39,100401,100404,100444],{"className":100402,"style":100403},[83],"padding-left:0.833em;",[39,100405,100407,100410],{"className":100406},[83],[39,100408,11],{"className":100409},[83,84],[39,100411,100413],{"className":100412},[192],[39,100414,100416,100436],{"className":100415},[196,649],[39,100417,100419,100433],{"className":100418},[200],[39,100420,100422],{"className":100421,"style":13604},[204],[39,100423,100424,100427],{"style":966},[39,100425],{"className":100426,"style":213},[212],[39,100428,100430],{"className":100429},[217,218,219,220],[39,100431,13377],{"className":100432},[83,84,220],[39,1004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",[39,100522,100524,100550],{"className":100523,"translate":43},[42],[39,100525,100527],{"className":100526},[47],[49,100528,100529],{"xmlns":51},[53,100530,100531,100547],{},[56,100532,100533,100535,100537,100539,100541,100543],{},[59,100534,11],{},[133,100536,69813],{},[59,100538,35321],{},[133,100540,793],{},[539,100542,17359],{},[98996,100544,100545],{},[59,100546,11],{},[63,100548,100549],{"encoding":65},"p\\mapsto u=2\\sqrt p",[39,100551,100553,100572,100590],{"className":100552,"ariaHidden":70},[69],[39,100554,100556,100560,100563,100566,100569],{"className":100555},[74],[39,100557],{"className":100558,"style":100559},[78],"height:0.7054em;vertical-align:-0.1944em;",[39,100561,11],{"className":100562},[83,84],[39,100564],{"className":100565,"style":231},[230],[39,100567,69813],{"className":100568},[235],[39,100570],{"className":100571,"style":231},[230],[39,100573,100575,100578,100581,100584,100587],{"className":100574},[74],[39,100576],{"className":100577,"style":290},[78],[39,100579,35321],{"className":100580},[83,84],[39,100582],{"className":100583,"style":231},[230],[39,100585,793],{"className":100586},[235],[39,100588],{"className":100589,"style":231},[230],[39,100591,100593,100597,100600],{"className":100592},[74],[39,100594],{"className":100595,"style":100596},[78],"height:1.04em;vertical-align:-0.3369em;",[39,100598,17359],{"className":100599},[83],[39,100601,100603],{"className":100602},[83,99058],[39,100604,100606,100638],{"className":100605},[196,649],[39,100607,100609,100635],{"className":100608},[200],[39,100610,100613,100622],{"className":100611,"style":100612},[204],"height:0.7031em;",[39,100614,100616,100619],{"className":100615,"style":21599},[99072],[39,100617],{"className":100618,"style":935},[212],[39,100620,11],{"className":100621,"style":100403},[83,84],[39,100623,100625,100628],{"style":100624},"top:-2.6631em;",[39,100626],{"className":100627,"style":935},[212],[39,100629,100631],{"className":100630,"style":100133},[99114],[85913,100632,100633],{"xmlns":85915,"width":99118,"height":100136,"viewBox":100137,"preserveAspectRatio":99121},[85919,100634],{"d":100140},[39,100636,673],{"className":100637},[672],[39,100639,100641],{"className":100640},[200],[39,100642,100645],{"className":100643,"style":100644},[204],"height:0.3369em;",[39,100646],{}," 把单纯形嵌入到 ",[39,100649,100651,100669],{"className":100650,"translate":43},[42],[39,100652,100654],{"className":100653},[47],[49,100655,100656],{"xmlns":51},[53,100657,100658,100666],{},[56,100659,100660],{},[138,100661,100662,100664],{},[59,100663,422],{"mathvariant":421},[59,100665,77187],{},[63,100667,100668],{"encoding":65},"\\mathbb R^K",[39,100670,100672],{"className":100671,"ariaHidden":70},[69],[39,100673,100675,100679],{"className":100674},[74],[39,100676],{"className":100677,"style":100678},[78],"height:0.8413em;",[39,100680,100682,100685],{"className":100681},[83],[39,100683,422],{"className":100684},[83,515],[39,100686,100688],{"className":100687},[192],[39,100689,100691],{"className":100690},[196],[39,100692,100694],{"className":100693},[200],[39,100695,100697],{"className":100696,"style":100678},[204],[39,100698,100699,100702],{"style":208},[39,100700],{"className":100701,"style":213},[212],[39,100703,100705],{"className":100704},[217,218,219,220],[39,100706,77187],{"className":100707,"style":48487},[83,84,220]," 中半径为 ",[39,100710,100712,100725],{"className":100711,"translate":43},[42],[39,100713,100715],{"className":100714},[47],[49,100716,100717],{"xmlns":51},[53,100718,100719,100723],{},[56,100720,100721],{},[539,100722,17359],{},[63,100724,17359],{"encoding":65},[39,100726,100728],{"className":100727,"ariaHidden":70},[69],[39,100729,100731,100734],{"className":100730},[74],[39,100732],{"className":100733,"style":2473},[78],[39,100735,17359],{"className":100736},[83]," 的球面上（因为 ",[39,100739,100741,100771],{"className":100740,"translate":43},[42],[39,100742,100744],{"className":100743},[47],[49,100745,100746],{"xmlns":51},[53,100747,100748,100768],{},[56,100749,100750,100756,100758,100764,100766],{},[613,100751,100752,100754],{},[133,100753,3191],{},[59,100755,13377],{},[539,100757,59223],{},[613,100759,100760,100762],{},[59,100761,11],{},[59,100763,13377],{},[133,100765,793],{},[539,100767,59223],{},[63,100769,100770],{"encoding":65},"\\sum_i4p_i=4",[39,100772,100774,100875],{"className":100773,"ariaHidden":70},[69],[39,100775,100777,100780,100820,100823,100826,100866,100869,100872],{"className":100776},[74],[39,100778],{"className":100779,"style":93897},[78],[39,100781,100783,100786],{"className":100782},[1744],[39,100784,3191],{"className":100785,"style":3953},[1744,3339,3952],[39,100787,100789],{"className":100788},[192],[39,100790,100792,100812],{"className":100791},[196,649],[39,100793,100795,100809],{"className":100794},[200],[39,100796,100798],{"className":100797,"style":93922},[204],[39,100799,100800,100803],{"style":3969},[39,100801],{"className":100802,"style":213},[212],[39,100804,100806],{"className":100805},[217,218,219,220],[39,100807,13377],{"className":100808},[83,84,220],[39,100810,673],{"className":100811},[672],[39,100813,100815],{"className":100814},[200],[39,100816,100818],{"className":100817,"style":4025},[204],[39,100819],{},[39,100821],{"className":100822,"style":256},[230],[39,100824,59223],{"className":100825},[83],[39,100827,100829,100832],{"className":100828},[83],[39,100830,11],{"className":100831},[83,84],[39,100833,100835],{"className":100834},[192],[39,100836,100838,100858],{"className":100837},[196,649],[39,100839,100841,100855],{"className":100840},[200],[39,100842,100844],{"className":100843,"style":13604},[204],[39,100845,100846,100849],{"style":966},[39,100847],{"className":100848,"style":213},[212],[39,100850,100852],{"className":100851},[217,218,219,220],[39,100853,13377],{"className":100854},[83,84,220],[39,100856,673],{"className":100857},[672],[39,100859,100861],{"className":100860},[200],[39,100862,100864],{"className":100863,"style":680},[204],[39,100865],{},[39,100867],{"className":100868,"style":231},[230],[39,100870,793],{"className":100871},[235],[39,100873],{"className":100874,"style":231},[230],[39,100876,100878,100881],{"className":100877},[74],[39,100879],{"className":100880,"style":2473},[78],[39,100882,59223],{"className":100883},[83],"），Fisher–Rao 距离正好是球面上的大圆距离。这个嵌入叫 Hellinger 嵌入，",[39,100886,100888,100922],{"className":100887,"translate":43},[42],[39,100889,100891],{"className":100890},[47],[49,100892,100893],{"xmlns":51},[53,100894,100895,100919],{},[56,100896,100897,100903],{},[613,100898,100899,100901],{},[133,100900,3191],{},[59,100902,13377],{},[98996,100904,100905],{},[56,100906,100907,100913],{},[613,100908,100909,100911],{},[59,100910,11],{},[59,100912,13377],{},[613,100914,100915,100917],{},[59,100916,100187],{},[59,100918,13377],{},[63,100920,100921],{"encoding":65},"\\sum_i\\sqrt{p_iq_i}",[39,100923,100925],{"className":100924,"ariaHidden":70},[69],[39,100926,100928,100932,100972,100975],{"className":100927},[74],[39,100929],{"className":100930,"style":100931},[78],"height:1.0869em;vertical-align:-0.3369em;",[39,100933,100935,100938],{"className":100934},[1744],[39,100936,3191],{"className":100937,"style":3953},[1744,3339,3952],[39,100939,100941],{"className":100940},[192],[39,100942,100944,100964],{"className":100943},[196,649],[39,100945,100947,100961],{"className":100946},[200],[39,100948,100950],{"className":100949,"style":93922},[204],[39,100951,100952,100955],{"style":3969},[39,100953],{"className":100954,"style":213},[212],[39,100956,100958],{"className":100957},[217,218,219,220],[39,100959,13377],{"className":100960},[83,84,220],[39,100962,673],{"className":100963},[672],[39,100965,100967],{"className":100966},[200],[39,100968,100970],{"className":100969,"style":4025},[204],[39,100971],{},[39,100973],{"className":100974,"style":256},[230],[39,100976,100978],{"className":100977},[83,99058],[39,100979,100981,101091],{"className":100980},[196,649],[39,100982,100984,101088],{"className":100983},[200],[39,100985,100987,101076],{"className":100986,"style":100612},[204],[39,100988,100990,100993],{"className":100989,"style":21599},[99072],[39,100991],{"className":100992,"style":935},[212],[39,100994,100996,101036],{"className":100995,"style":100403},[83],[39,100997,100999,101002],{"className":100998},[83],[39,101000,11],{"className":101001},[83,84],[39,101003,101005],{"className":101004},[192],[39,101006,101008,101028],{"className":101007},[196,649],[39,101009,101011,101025],{"className":101010},[200],[39,101012,101014],{"className":101013,"style":13604},[204],[39,101015,101016,101019],{"style":966},[39,101017],{"className":101018,"style":213},[212],[39,101020,101022],{"className":101021},[217,218,219,220],[39,101023,13377],{"className":101024},[83,84,220],[39,101026,673],{"className":101027},[672],[39,101029,101031],{"className":101030},[200],[39,101032,101034],{"className":101033,"style":680},[204],[39,101035],{},[39,101037,101039,101042],{"className":101038},[83],[39,101040,100187],{"className":101041,"style":2114},[83,84],[39,101043,101045],{"className":101044},[192],[39,101046,101048,101068],{"className":101047},[196,649],[39,101049,101051,101065],{"className":101050},[200],[39,101052,101054],{"className":101053,"style":13604},[204],[39,101055,101056,101059],{"style":41316},[39,101057],{"className":101058,"style":213},[212],[39,101060,101062],{"className":101061},[217,218,219,220],[39,101063,13377],{"className":101064},[83,84,220],[39,101066,673],{"className":101067},[672],[39,101069,101071],{"className":101070},[200],[39,101072,101074],{"className":101073,"style":680},[204],[39,101075],{},[39,101077,101078,101081],{"style":100624},[39,101079],{"className":101080,"style":935},[212],[39,101082,101084],{"className":101083,"style":100133},[99114],[85913,101085,101086],{"xmlns":85915,"width":99118,"height":100136,"viewBox":100137,"preserveAspectRatio":99121},[85919,101087],{"d":100140},[39,101089,673],{"className":101090},[672],[39,101092,101094],{"className":101093},[200],[39,101095,101097],{"className":101096,"style":100644},[204],[39,101098],{}," 叫 Bhattacharyya 系数。",[11,101101,101102],{},"有了球面，测地线就清楚了。两点之间的大圆弧写回单纯形是",[39,101104,101106],{"className":101105,"translate":43},[746],[39,101107,101109,101228],{"className":101108,"translate":43},[42],[39,101110,101112],{"className":101111},[47],[49,101113,101114],{"xmlns":51,"display":755},[53,101115,101116,101225],{},[56,101117,101118,101124,101126,101167,101169,101171,101177,101179,101185,101187,101189,101191,101193,101195,101197,101205,101207,101209,101217,101223],{},[613,101119,101120,101122],{},[59,101121,11],{},[59,101123,621],{},[133,101125,793],{},[5362,101127,101128,101160],{},[56,101129,101130,101132,101134,101136,101138,101140,101142,101146,101148,101150,101154],{},[133,101131,136],{"fence":70,"stretchy":70,"minsize":73875,"maxsize":73875},[133,101133,136],{"stretchy":135},[539,101135,546],{},[133,101137,5378],{},[59,101139,621],{},[133,101141,161],{"stretchy":135},[98996,101143,101144],{},[59,101145,11],{},[133,101147,788],{},[59,101149,621],{},[98996,101151,101152],{},[59,101153,100187],{},[138,101155,101156,101158],{},[133,101157,161],{"fence":70,"stretchy":70,"minsize":73875,"maxsize":73875},[539,101159,17359],{},[613,101161,101162,101165],{},[59,101163,101164],{},"Z",[59,101166,621],{},[133,101168,155],{"separator":70},[230,101170],{"width":10251},[613,101172,101173,101175],{},[59,101174,101164],{},[59,101176,621],{},[133,101178,793],{},[6556,101180,101181,101183],{},[133,101182,3191],{},[59,101184,13377],{},[133,101186,136],{"fence":70,"stretchy":70,"minsize":73875,"maxsize":73875},[133,101188,136],{"stretchy":135},[539,101190,546],{},[133,101192,5378],{},[59,101194,621],{},[133,101196,161],{"stretchy":135},[98996,101198,101199],{},[613,101200,101201,101203],{},[59,101202,11],{},[59,101204,13377],{},[133,101206,788],{},[59,101208,621],{},[98996,101210,101211],{},[613,101212,101213,101215],{},[59,101214,100187],{},[59,101216,13377],{},[138,101218,101219,101221],{},[133,101220,161],{"fence":70,"stretchy":70,"minsize":73875,"maxsize":73875},[539,101222,17359],{},[59,101224,2903],{"mathvariant":145},[63,101226,101227],{"encoding":65},"p_t=\\frac{\\bigl((1-t)\\sqrt p+t\\sqrt q\\bigr)^2}{Z_t},\\qquad\nZ_t=\\sum_i\\bigl((1-t)\\sqrt{p_i}+t\\sqrt{q_i}\\bigr)^2 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100 次，60 次正面。这句话里真正有用的信息只有两个数：100 和 60。给定正面概率 ",[39,101930,101932,101945],{"className":101931,"translate":43},[42],[39,101933,101935],{"className":101934},[47],[49,101936,101937],{"xmlns":51},[53,101938,101939,101943],{},[56,101940,101941],{},[59,101942,96325],{},[63,101944,97273],{"encoding":65},[39,101946,101948],{"className":101947,"ariaHidden":70},[69],[39,101949,101951,101954],{"className":101950},[74],[39,101952],{"className":101953,"style":28824},[78],[39,101955,96325],{"className":101956,"style":438},[83,84],"，正面次数服从二项分布，而样本对 ",[39,101959,101961,101974],{"className":101960,"translate":43},[42],[39,101962,101964],{"className":101963},[47],[49,101965,101966],{"xmlns":51},[53,101967,101968,101972],{},[56,101969,101970],{},[59,101971,96325],{},[63,101973,97273],{"encoding":65},[39,101975,101977],{"className":101976,"ariaHidden":70},[69],[39,101978,101980,101983],{"className":101979},[74],[39,101981],{"className":101982,"style":28824},[78],[39,101984,96325],{"className":101985,"style":438},[83,84]," 的全部影响都通过“正面次数”这个线性统计量进去——这就是充分统计量。",[11,101988,101989],{},"指数族把这个结构写成一个模板：",[39,101991,101993],{"className":101992,"translate":43},[746],[39,101994,101996,102064],{"className":101995,"translate":43},[42],[39,101997,101999],{"className":101998},[47],[49,102000,102001],{"xmlns":51,"display":755},[53,102002,102003,102061],{},[56,102004,102005,102007,102009,102011,102013,102015,102017,102019,102021,102023,102025,102027,102029,102031,102033,102039,102041,102043,102045,102047,102049,102051,102053,102055,102057,102059],{},[59,102006,11],{},[133,102008,136],{"stretchy":135},[59,102010,1755],{},[133,102012,150],{},[59,102014,96325],{},[133,102016,161],{"stretchy":135},[133,102018,793],{},[59,102020,69998],{},[133,102022,136],{"stretchy":135},[59,102024,1755],{},[133,102026,161],{"stretchy":135},[59,102028,92843],{},[133,102030,1690],{},[133,102032,136],{"fence":70,"stretchy":70,"minsize":73875,"maxsize":73875},[138,102034,102035,102037],{},[59,102036,96325],{},[59,102038,93341],{"mathvariant":145},[59,102040,11772],{},[133,102042,136],{"stretchy":135},[59,102044,1755],{},[133,102046,161],{"stretchy":135},[133,102048,5378],{},[59,102050,80653],{},[133,102052,136],{"stretchy":135},[59,102054,96325],{},[133,102056,161],{"stretchy":135},[133,102058,161],{"fence":70,"stretchy":70,"minsize":73875,"maxsize":73875},[59,102060,2903],{"mathvariant":145},[63,102062,102063],{"encoding":65},"p(x\\mid\\theta)=h(x)\\exp\\bigl(\\theta^\\top T(x)-\\psi(\\theta)\\bigr).",[39,102065,102067,102091,102112,102193],{"className":102066,"ariaHidden":70},[69],[39,102068,102070,102073,102076,102079,102082,102085,102088],{"className":102069},[74],[39,102071],{"className":102072,"style":245},[78],[39,102074,11],{"className":102075},[83,84],[39,102077,136],{"className":102078},[182],[39,102080,1755],{"className":102081},[83,84],[39,102083],{"className":102084,"style":231},[230],[39,102086,150],{"className":102087},[235],[39,102089],{"className":102090,"style":231},[230],[39,102092,102094,102097,102100,102103,102106,102109],{"className":102093},[74],[39,102095],{"className":102096,"style":245},[78],[39,102098,96325],{"className":102099,"style":438},[83,84],[39,102101,161],{"className":102102},[263],[39,102104],{"className":102105,"style":231},[230],[39,102107,793],{"className":102108},[235],[39,102110],{"className":102111,"style":231},[230],[39,102113,102115,102119,102122,102125,102128,102131,102134,102137,102143,102172,102175,102178,102181,102184,102187,102190],{"className":102114},[74],[39,102116],{"className":102117,"style":102118},[78],"height:1.2491em;vertical-align:-0.35em;",[39,102120,69998],{"className":102121},[83,84],[39,102123,136],{"className":102124},[182],[39,102126,1755],{"className":102127},[83,84],[39,102129,161],{"className":102130},[263],[39,102132],{"className":102133,"style":256},[230],[39,102135,92843],{"className":102136},[1744],[39,102138,102140],{"className":102139},[182],[39,102141,136],{"className":102142},[18520,4009],[39,102144,102146,102149],{"className":102145},[83],[39,102147,96325],{"className":102148,"style":438},[83,84],[39,102150,102152],{"className":102151},[192],[39,102153,102155],{"className":102154},[196],[39,102156,102158],{"className":102157},[200],[39,102159,102161],{"className":102160,"style":3389},[204],[39,102162,102163,102166],{"style":1027},[39,102164],{"className":102165,"style":213},[212],[39,102167,102169],{"className":102168},[217,218,219,220],[39,102170,93341],{"className":102171},[83,220],[39,102173,11772],{"className":102174,"style":178},[83,84],[39,102176,136],{"className":102177},[182],[39,102179,1755],{"className":102180},[83,84],[39,102182,161],{"className":102183},[263],[39,102185],{"className":102186,"style":461},[230],[39,102188,5378],{"className":102189},[465],[39,102191],{"className":102192,"style":461},[230],[39,102194,102196,102199,102202,102205,102208,102211,102217],{"className":102195},[74],[39,102197],{"className":102198,"style":908},[78],[39,102200,80653],{"className":102201,"style":2114},[83,84],[39,102203,136],{"className":102204},[182],[39,102206,96325],{"className":102207,"style":438},[83,84],[39,102209,161],{"className":102210},[263],[39,102212,102214],{"className":102213},[263],[39,102215,161],{"className":102216},[18520,4009],[39,102218,2903],{"className":102219},[83],[11,102221,102222,102250,102251,102279,102280,102309,102310,87840,102376,87840,102524,1607],{},[39,102223,102225,102238],{"className":102224,"translate":43},[42],[39,102226,102228],{"className":102227},[47],[49,102229,102230],{"xmlns":51},[53,102231,102232,102236],{},[56,102233,102234],{},[59,102235,11772],{},[63,102237,11772],{"encoding":65},[39,102239,102241],{"className":102240,"ariaHidden":70},[69],[39,102242,102244,102247],{"className":102243},[74],[39,102245],{"className":102246,"style":79},[78],[39,102248,11772],{"className":102249,"style":178},[83,84]," 是充分统计量，",[39,102252,102254,102267],{"className":102253,"translate":43},[42],[39,102255,102257],{"className":102256},[47],[49,102258,102259],{"xmlns":51},[53,102260,102261,102265],{},[56,102262,102263],{},[59,102264,96325],{},[63,102266,97273],{"encoding":65},[39,102268,102270],{"className":102269,"ariaHidden":70},[69],[39,102271,102273,102276],{"className":102272},[74],[39,102274],{"className":102275,"style":28824},[78],[39,102277,96325],{"className":102278,"style":438},[83,84]," 是自然参数，",[39,102281,102283,102297],{"className":102282,"translate":43},[42],[39,102284,102286],{"className":102285},[47],[49,102287,102288],{"xmlns":51},[53,102289,102290,102294],{},[56,102291,102292],{},[59,102293,80653],{},[63,102295,102296],{"encoding":65},"\\psi",[39,102298,102300],{"className":102299,"ariaHidden":70},[69],[39,102301,102303,102306],{"className":102302},[74],[39,102304],{"className":102305,"style":51280},[78],[39,102307,80653],{"className":102308,"style":2114},[83,84]," 是对数配分函数，负责把总质量归一化成 1。抛硬币就是 ",[39,102311,102313,102337],{"className":102312,"translate":43},[42],[39,102314,102316],{"className":102315},[47],[49,102317,102318],{"xmlns":51},[53,102319,102320,102334],{},[56,102321,102322,102324,102326,102328,102330,102332],{},[59,102323,11772],{},[133,102325,136],{"stretchy":135},[59,102327,1755],{},[133,102329,161],{"stretchy":135},[133,102331,793],{},[59,102333,1755],{},[63,102335,102336],{"encoding":65},"T(x)=x",[39,102338,102340,102367],{"className":102339,"ariaHidden":70},[69],[39,102341,102343,102346,102349,102352,102355,102358,102361,102364],{"className":102342},[74],[39,102344],{"className":102345,"style":245},[78],[39,102347,11772],{"className":102348,"style":178},[83,84],[39,102350,136],{"className":102351},[182],[39,102353,1755],{"className":102354},[83,84],[39,102356,161],{"className":102357},[263],[39,102359],{"className":102360,"style":231},[230],[39,102362,793],{"className":102363},[235],[39,102365],{"className":102366,"style":231},[230],[39,102368,102370,102373],{"className":102369},[74],[39,102371],{"className":102372,"style":290},[78],[39,102374,1755],{"className":102375},[83,84],[39,102377,102379,102411],{"className":102378,"translate":43},[42],[39,102380,102382],{"className":102381},[47],[49,102383,102384],{"xmlns":51},[53,102385,102386,102408],{},[56,102387,102388,102390,102392,102394,102396],{},[59,102389,96325],{},[133,102391,793],{},[59,102393,92483],{},[133,102395,1690],{},[5362,102397,102398,102400],{},[59,102399,11],{},[56,102401,102402,102404,102406],{},[539,102403,546],{},[133,102405,5378],{},[59,102407,11],{},[63,102409,102410],{"encoding":65},"\\theta=\\log\\frac{p}{1-p}",[39,102412,102414,102432],{"className":102413,"ariaHidden":70},[69],[39,102415,102417,102420,102423,102426,102429],{"className":102416},[74],[39,102418],{"className":102419,"style":28824},[78],[39,102421,96325],{"className":102422,"style":438},[83,84],[39,102424],{"className":102425,"style":231},[230],[39,102427,793],{"className":102428},[235],[39,102430],{"className":102431,"style":231},[230],[39,102433,102435,102439,102444,102447],{"className":102434},[74],[39,102436],{"className":102437,"style":102438},[78],"height:1.2286em;vertical-align:-0.4811em;",[39,102440,92721,102442],{"className":102441},[1744],[39,102443,39209],{"style":35078},[39,102445],{"className":102446,"style":256},[230],[39,102448,102450,102453,102521],{"className":102449},[83],[39,102451],{"className":102452},[182,5463],[39,102454,102456],{"className":102455},[5362],[39,102457,102459,102512],{"className":102458},[196,649],[39,102460,102462,102509],{"className":102461},[200],[39,102463,102466,102486,102494],{"className":102464,"style":102465},[204],"height:0.7475em;",[39,102467,102468,102471],{"style":74816},[39,102469],{"className":102470,"style":935},[212],[39,102472,102474],{"className":102473},[217,218,219,220],[39,102475,102477,102480,102483],{"className":102476},[83,220],[39,102478,546],{"className":102479},[83,220],[39,102481,5378],{"className":102482},[465,220],[39,102484,11],{"className":102485},[83,84,220],[39,102487,102488,102491],{"style":5503},[39,102489],{"className":102490,"style":935},[212],[39,102492],{"className":102493,"style":5511},[5510],[39,102495,102497,102500],{"style":102496},"top:-3.4461em;",[39,102498],{"className":102499,"style":935},[212],[39,102501,102503],{"className":102502},[217,218,219,220],[39,102504,102506],{"className":102505},[83,220],[39,102507,11],{"className":102508},[83,84,220],[39,102510,673],{"className":102511},[672],[39,102513,102515],{"className":102514},[200],[39,102516,102519],{"className":102517,"style":102518},[204],"height:0.4811em;",[39,102520],{},[39,102522],{"className":102523},[263,5463],[39,102525,102527,102568],{"className":102526,"translate":43},[42],[39,102528,102530],{"className":102529},[47],[49,102531,102532],{"xmlns":51},[53,102533,102534,102565],{},[56,102535,102536,102538,102540,102542,102544,102546,102548,102550,102552,102554,102556,102563],{},[59,102537,80653],{},[133,102539,136],{"stretchy":135},[59,102541,96325],{},[133,102543,161],{"stretchy":135},[133,102545,793],{},[59,102547,92483],{},[133,102549,1690],{},[133,102551,136],{"stretchy":135},[539,102553,546],{},[133,102555,788],{},[138,102557,102558,102561],{},[59,102559,102560],{},"e",[59,102562,96325],{},[133,102564,161],{"stretchy":135},[63,102566,102567],{"encoding":65},"\\psi(\\theta)=\\log(1+e^\\theta)",[39,102569,102571,102598,102624],{"className":102570,"ariaHidden":70},[69],[39,102572,102574,102577,102580,102583,102586,102589,102592,102595],{"className":102573},[74],[39,102575],{"className":102576,"style":245},[78],[39,102578,80653],{"className":102579,"style":2114},[83,84],[39,102581,136],{"className":102582},[182],[39,102584,96325],{"className":102585,"style":438},[83,84],[39,102587,161],{"className":102588},[263],[39,102590],{"className":102591,"style":231},[230],[39,102593,793],{"className":102594},[235],[39,102596],{"className":102597,"style":231},[230],[39,102599,102601,102604,102609,102612,102615,102618,102621],{"className":102600},[74],[39,102602],{"className":102603,"style":245},[78],[39,102605,92721,102607],{"className":102606},[1744],[39,102608,39209],{"style":35078},[39,102610,136],{"className":102611},[182],[39,102613,546],{"className":102614},[83],[39,102616],{"className":102617,"style":461},[230],[39,102619,788],{"className":102620},[465],[39,102622],{"className":102623,"style":461},[230],[39,102625,102627,102630,102659],{"className":102626},[74],[39,102628],{"className":102629,"style":15555},[78],[39,102631,102633,102636],{"className":102632},[83],[39,102634,102560],{"className":102635},[83,84],[39,102637,102639],{"className":102638},[192],[39,102640,102642],{"className":102641},[196],[39,102643,102645],{"className":102644},[200],[39,102646,102648],{"className":102647,"style":15577},[204],[39,102649,102650,102653],{"style":208},[39,102651],{"className":102652,"style":213},[212],[39,102654,102656],{"className":102655},[217,218,219,220],[39,102657,96325],{"className":102658,"style":438},[83,84,220],[39,102660,161],{"className":102661},[263],[11,102663,102664,102692,102693,742,102833,103012,103013,103041],{},[39,102665,102667,102680],{"className":102666,"translate":43},[42],[39,102668,102670],{"className":102669},[47],[49,102671,102672],{"xmlns":51},[53,102673,102674,102678],{},[56,102675,102676],{},[59,102677,80653],{},[63,102679,102296],{"encoding":65},[39,102681,102683],{"className":102682,"ariaHidden":70},[69],[39,102684,102686,102689],{"className":102685},[74],[39,102687],{"className":102688,"style":51280},[78],[39,102690,80653],{"className":102691,"style":2114},[83,84]," 的性质很好：",[39,102694,102696,102738],{"className":102695,"translate":43},[42],[39,102697,102699],{"className":102698},[47],[49,102700,102701],{"xmlns":51},[53,102702,102703,102735],{},[56,102704,102705,102707,102709,102711,102713,102715,102717,102723,102725,102727,102729,102731,102733],{},[59,102706,84725],{"mathvariant":145},[59,102708,80653],{},[133,102710,136],{"stretchy":135},[59,102712,96325],{},[133,102714,161],{"stretchy":135},[133,102716,793],{},[613,102718,102719,102721],{},[59,102720,4253],{"mathvariant":421},[59,102722,96325],{},[133,102724,537],{"stretchy":135},[59,102726,11772],{},[133,102728,136],{"stretchy":135},[59,102730,29420],{},[133,102732,161],{"stretchy":135},[133,102734,4280],{"stretchy":135},[63,102736,102737],{"encoding":65},"\\nabla\\psi(\\theta)=\\mathbb E_\\theta[T(X)]",[39,102739,102741,102771],{"className":102740,"ariaHidden":70},[69],[39,102742,102744,102747,102750,102753,102756,102759,102762,102765,102768],{"className":102743},[74],[39,102745],{"className":102746,"style":245},[78],[39,102748,84725],{"className":102749},[83],[39,102751,80653],{"className":102752,"style":2114},[83,84],[39,102754,136],{"className":102755},[182],[39,102757,96325],{"className":102758,"style":438},[83,84],[39,102760,161],{"className":102761},[263],[39,102763],{"className":102764,"style":231},[230],[39,102766,793],{"className":102767},[235],[39,102769],{"className":102770,"style":231},[230],[39,102772,102774,102777,102817,102820,102823,102826,102829],{"className":102773},[74],[39,102775],{"className":102776,"style":245},[78],[39,102778,102780,102783],{"className":102779},[83],[39,102781,4253],{"className":102782},[83,515],[39,102784,102786],{"className":102785},[192],[39,102787,102789,102809],{"className":102788},[196,649],[39,102790,102792,102806],{"className":102791},[200],[39,102793,102795],{"className":102794,"style":3419},[204],[39,102796,102797,102800],{"style":966},[39,102798],{"className":102799,"style":213},[212],[39,102801,102803],{"className":102802},[217,218,219,220],[39,102804,96325],{"className":102805,"style":438},[83,84,220],[39,102807,673],{"className":102808},[672],[39,102810,102812],{"className":102811},[200],[39,102813,102815],{"className":102814,"style":680},[204],[39,102816],{},[39,102818,537],{"className":102819},[182],[39,102821,11772],{"className":102822,"style":178},[83,84],[39,102824,136],{"className":102825},[182],[39,102827,29420],{"className":102828,"style":14420},[83,84],[39,102830,102832],{"className":102831},[263],")]",[39,102834,102836,102888],{"className":102835,"translate":43},[42],[39,102837,102839],{"className":102838},[47],[49,102840,102841],{"xmlns":51},[53,102842,102843,102885],{},[56,102844,102845,102851,102853,102855,102857,102859,102861,102873,102875,102877,102879,102881,102883],{},[138,102846,102847,102849],{},[59,102848,84725],{"mathvariant":145},[539,102850,17359],{},[59,102852,80653],{},[133,102854,136],{"stretchy":135},[59,102856,96325],{},[133,102858,161],{"stretchy":135},[133,102860,793],{},[613,102862,102863,102871],{},[56,102864,102865,102867,102869],{},[59,102866,48444],{"mathvariant":145},[59,102868,35010],{"mathvariant":145},[59,102870,77971],{"mathvariant":145},[59,102872,96325],{},[133,102874,537],{"stretchy":135},[59,102876,11772],{},[133,102878,136],{"stretchy":135},[59,102880,29420],{},[133,102882,161],{"stretchy":135},[133,102884,4280],{"stretchy":135},[63,102886,102887],{"encoding":65},"\\nabla^2\\psi(\\theta)=\\mathrm{Cov}_\\theta[T(X)]",[39,102889,102891,102947],{"className":102890,"ariaHidden":70},[69],[39,102892,102894,102897,102926,102929,102932,102935,102938,102941,102944],{"className":102893},[74],[39,102895],{"className":102896,"style":84615},[78],[39,102898,102900,102903],{"className":102899},[83],[39,102901,84725],{"className":102902},[83],[39,102904,102906],{"className":102905},[192],[39,102907,102909],{"className":102908},[196],[39,102910,102912],{"className":102911},[200],[39,102913,102915],{"className":102914,"style":19710},[204],[39,102916,102917,102920],{"style":208},[39,102918],{"className":102919,"style":213},[212],[39,102921,102923],{"className":102922},[217,218,219,220],[39,102924,17359],{"className":102925},[83,220],[39,102927,80653],{"className":102928,"style":2114},[83,84],[39,102930,136],{"className":102931},[182],[39,102933,96325],{"className":102934,"style":438},[83,84],[39,102936,161],{"className":102937},[263],[39,102939],{"className":102940,"style":231},[230],[39,102942,793],{"className":102943},[235],[39,102945],{"className":102946,"style":231},[230],[39,102948,102950,102953,102997,103000,103003,103006,103009],{"className":102949},[74],[39,102951],{"className":102952,"style":245},[78],[39,102954,102956,102963],{"className":102955},[83],[39,102957,102959],{"className":102958},[83],[39,102960,102962],{"className":102961,"style":35078},[83,35077],"Cov",[39,102964,102966],{"className":102965},[192],[39,102967,102969,102989],{"className":102968},[196,649],[39,102970,102972,102986],{"className":102971},[200],[39,102973,102975],{"className":102974,"style":3419},[204],[39,102976,102977,102980],{"style":44021},[39,102978],{"className":102979,"style":213},[212],[39,102981,102983],{"className":102982},[217,218,219,220],[39,102984,96325],{"className":102985,"style":438},[83,84,220],[39,102987,673],{"className":102988},[672],[39,102990,102992],{"className":102991},[200],[39,102993,102995],{"className":102994,"style":680},[204],[39,102996],{},[39,102998,537],{"className":102999},[182],[39,103001,11772],{"className":103002,"style":178},[83,84],[39,103004,136],{"className":103005},[182],[39,103007,29420],{"className":103008,"style":14420},[83,84],[39,103010,102832],{"className":103011},[263],"。第二个等式说明 Fisher 度量就是 ",[39,103014,103016,103029],{"className":103015,"translate":43},[42],[39,103017,103019],{"className":103018},[47],[49,103020,103021],{"xmlns":51},[53,103022,103023,103027],{},[56,103024,103025],{},[59,103026,80653],{},[63,103028,102296],{"encoding":65},[39,103030,103032],{"className":103031,"ariaHidden":70},[69],[39,103033,103035,103038],{"className":103034},[74],[39,103036],{"className":103037,"style":51280},[78],[39,103039,80653],{"className":103040,"style":2114},[83,84]," 的 Hessian，上一节里“度量由凸函数给出”的说法在这里变成了具体的东西。",[11,103043,60385,103044,103116,103117,103145,103146,103372,103373,103474,103475,103503,103504,103533],{},[39,103045,103047,103074],{"className":103046,"translate":43},[42],[39,103048,103050],{"className":103049},[47],[49,103051,103052],{"xmlns":51},[53,103053,103054,103071],{},[56,103055,103056,103059,103061,103063,103065,103067,103069],{},[59,103057,103058],{},"η",[133,103060,793],{},[59,103062,84725],{"mathvariant":145},[59,103064,80653],{},[133,103066,136],{"stretchy":135},[59,103068,96325],{},[133,103070,161],{"stretchy":135},[63,103072,103073],{"encoding":65},"\\eta=\\nabla\\psi(\\theta)",[39,103075,103077,103095],{"className":103076,"ariaHidden":70},[69],[39,103078,103080,103083,103086,103089,103092],{"className":103079},[74],[39,103081],{"className":103082,"style":2654},[78],[39,103084,103058],{"className":103085,"style":2114},[83,84],[39,103087],{"className":103088,"style":231},[230],[39,103090,793],{"className":103091},[235],[39,103093],{"className":103094,"style":231},[230],[39,103096,103098,103101,103104,103107,103110,103113],{"className":103097},[74],[39,103099],{"className":103100,"style":245},[78],[39,103102,84725],{"className":103103},[83],[39,103105,80653],{"className":103106,"style":2114},[83,84],[39,103108,136],{"className":103109},[182],[39,103111,96325],{"className":103112,"style":438},[83,84],[39,103114,161],{"className":103115},[263]," 为期望参数，它与 ",[39,103118,103120,103133],{"className":103119,"translate":43},[42],[39,103121,103123],{"className":103122},[47],[49,103124,103125],{"xmlns":51},[53,103126,103127,103131],{},[56,103128,103129],{},[59,103130,96325],{},[63,103132,97273],{"encoding":65},[39,103134,103136],{"className":103135,"ariaHidden":70},[69],[39,103137,103139,103142],{"className":103138},[74],[39,103140],{"className":103141,"style":28824},[78],[39,103143,96325],{"className":103144,"style":438},[83,84]," 之间是 Legendre 变换的关系：",[39,103147,103149,103207],{"className":103148,"translate":43},[42],[39,103150,103152],{"className":103151},[47],[49,103153,103154],{"xmlns":51},[53,103155,103156,103204],{},[56,103157,103158,103164,103166,103168,103170,103172,103182,103184,103190,103192,103194,103196,103198,103200,103202],{},[138,103159,103160,103162],{},[59,103161,80653],{},[133,103163,21884],{},[133,103165,136],{"stretchy":135},[59,103167,103058],{},[133,103169,161],{"stretchy":135},[133,103171,793],{},[613,103173,103174,103180],{},[56,103175,103176,103178],{},[59,103177,21465],{},[133,103179,1690],{},[59,103181,96325],{},[133,103183,136],{"stretchy":135},[138,103185,103186,103188],{},[59,103187,96325],{},[59,103189,93341],{"mathvariant":145},[59,103191,103058],{},[133,103193,5378],{},[59,103195,80653],{},[133,103197,136],{"stretchy":135},[59,103199,96325],{},[133,103201,161],{"stretchy":135},[133,103203,161],{"stretchy":135},[63,103205,103206],{"encoding":65},"\\psi^*(\\eta)=\\sup_\\theta(\\theta^\\top\\eta-\\psi(\\theta))",[39,103208,103210,103263,103354],{"className":103209,"ariaHidden":70},[69],[39,103211,103213,103216,103245,103248,103251,103254,103257,103260],{"className":103212},[74],[39,103214],{"className":103215,"style":245},[78],[39,103217,103219,103222],{"className":103218},[83],[39,103220,80653],{"className":103221,"style":2114},[83,84],[39,103223,103225],{"className":103224},[192],[39,103226,103228],{"className":103227},[196],[39,103229,103231],{"className":103230},[200],[39,103232,103234],{"className":103233,"style":21897},[204],[39,103235,103236,103239],{"style":208},[39,103237],{"className":103238,"style":213},[212],[39,103240,103242],{"className":103241},[217,218,219,220],[39,103243,21884],{"className":103244},[465,220],[39,103246,136],{"className":103247},[182],[39,103249,103058],{"className":103250,"style":2114},[83,84],[39,103252,161],{"className":103253},[263],[39,103255],{"className":103256,"style":231},[230],[39,103258,793],{"className":103259},[235],[39,103261],{"className":103262,"style":231},[230],[39,103264,103266,103269,103310,103313,103342,103345,103348,103351],{"className":103265},[74],[39,103267],{"className":103268,"style":15555},[78],[39,103270,103272,103275],{"className":103271},[1744],[39,103273,21465],{"className":103274},[1744],[39,103276,103278],{"className":103277},[192],[39,103279,103281,103302],{"className":103280},[196,649],[39,103282,103284,103299],{"className":103283},[200],[39,103285,103288],{"className":103286,"style":103287},[204],"height:0.242em;",[39,103289,103290,103293],{"style":63928},[39,103291],{"className":103292,"style":213},[212],[39,103294,103296],{"className":103295},[217,218,219,220],[39,103297,96325],{"className":103298,"style":438},[83,84,220],[39,103300,673],{"className":103301},[672],[39,103303,103305],{"className":103304},[200],[39,103306,103308],{"className":103307,"style":63947},[204],[39,103309],{},[39,103311,136],{"className":103312},[182],[39,103314,103316,103319],{"className":103315},[83],[39,103317,96325],{"className":103318,"style":438},[83,84],[39,103320,103322],{"className":103321},[192],[39,103323,103325],{"className":103324},[196],[39,103326,103328],{"className":103327},[200],[39,103329,103331],{"className":103330,"style":15577},[204],[39,103332,103333,103336],{"style":208},[39,103334],{"className":103335,"style":213},[212],[39,103337,103339],{"className":103338},[217,218,219,220],[39,103340,93341],{"className":103341},[83,220],[39,103343,103058],{"className":103344,"style":2114},[83,84],[39,103346],{"className":103347,"style":461},[230],[39,103349,5378],{"className":103350},[465],[39,103352],{"className":103353,"style":461},[230],[39,103355,103357,103360,103363,103366,103369],{"className":103356},[74],[39,103358],{"className":103359,"style":245},[78],[39,103361,80653],{"className":103362,"style":2114},[83,84],[39,103364,136],{"className":103365},[182],[39,103367,96325],{"className":103368,"style":438},[83,84],[39,103370,74175],{"className":103371},[263],"，且 ",[39,103374,103376,103406],{"className":103375,"translate":43},[42],[39,103377,103379],{"className":103378},[47],[49,103380,103381],{"xmlns":51},[53,103382,103383,103403],{},[56,103384,103385,103387,103393,103395,103397,103399,103401],{},[59,103386,84725],{"mathvariant":145},[138,103388,103389,103391],{},[59,103390,80653],{},[133,103392,21884],{},[133,103394,136],{"stretchy":135},[59,103396,103058],{},[133,103398,161],{"stretchy":135},[133,103400,793],{},[59,103402,96325],{},[63,103404,103405],{"encoding":65},"\\nabla\\psi^*(\\eta)=\\theta",[39,103407,103409,103465],{"className":103408,"ariaHidden":70},[69],[39,103410,103412,103415,103418,103447,103450,103453,103456,103459,103462],{"className":103411},[74],[39,103413],{"className":103414,"style":245},[78],[39,103416,84725],{"className":103417},[83],[39,103419,103421,103424],{"className":103420},[83],[39,103422,80653],{"className":103423,"style":2114},[83,84],[39,103425,103427],{"className":103426},[192],[39,103428,103430],{"className":103429},[196],[39,103431,103433],{"className":103432},[200],[39,103434,103436],{"className":103435,"style":21897},[204],[39,103437,103438,103441],{"style":208},[39,103439],{"className":103440,"style":213},[212],[39,103442,103444],{"className":103443},[217,218,219,220],[39,103445,21884],{"className":103446},[465,220],[39,103448,136],{"className":103449},[182],[39,103451,103058],{"className":103452,"style":2114},[83,84],[39,103454,161],{"className":103455},[263],[39,103457],{"className":103458,"style":231},[230],[39,103460,793],{"className":103461},[235],[39,103463],{"className":103464,"style":231},[230],[39,103466,103468,103471],{"className":103467},[74],[39,103469],{"className":103470,"style":28824},[78],[39,103472,96325],{"className":103473,"style":438},[83,84],"。所以同一个分布有两套坐标，自然参数 ",[39,103476,103478,103491],{"className":103477,"translate":43},[42],[39,103479,103481],{"className":103480},[47],[49,103482,103483],{"xmlns":51},[53,103484,103485,103489],{},[56,103486,103487],{},[59,103488,96325],{},[63,103490,97273],{"encoding":65},[39,103492,103494],{"className":103493,"ariaHidden":70},[69],[39,103495,103497,103500],{"className":103496},[74],[39,103498],{"className":103499,"style":28824},[78],[39,103501,96325],{"className":103502,"style":438},[83,84]," 和期望参数 ",[39,103505,103507,103521],{"className":103506,"translate":43},[42],[39,103508,103510],{"className":103509},[47],[49,103511,103512],{"xmlns":51},[53,103513,103514,103518],{},[56,103515,103516],{},[59,103517,103058],{},[63,103519,103520],{"encoding":65},"\\eta",[39,103522,103524],{"className":103523,"ariaHidden":70},[69],[39,103525,103527,103530],{"className":103526},[74],[39,103528],{"className":103529,"style":2654},[78],[39,103531,103058],{"className":103532,"style":2114},[83,84],"，互为对偶。",[11,103535,103536],{},"现在算两个成员之间的 KL：",[39,103538,103540],{"className":103539,"translate":43},[746],[39,103541,103543,103855],{"className":103542,"translate":43},[42],[39,103544,103546],{"className":103545},[47],[49,103547,103548],{"xmlns":51,"display":755},[53,103549,103550,103852],{},[759,103551,103552,103634,103710,103780],{"rowspacing":761,"columnalign":762,"columnspacing":146},[764,103553,103554,103588],{},[767,103555,103556],{},[770,103557,103558],{"scriptlevel":541,"displaystyle":70},[56,103559,103560,103566,103568,103574,103576,103586],{},[56,103561,103562,103564],{},[59,103563,77187],{"mathvariant":145},[59,103565,77190],{"mathvariant":145},[133,103567,136],{"stretchy":135},[613,103569,103570,103572],{},[59,103571,11],{},[59,103573,96325],{},[59,103575,15389],{"mathvariant":145},[613,103577,103578,103580],{},[59,103579,11],{},[138,103581,103582,103584],{},[59,103583,96325],{},[133,103585,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,103587,161],{"stretchy":135},[767,103589,103590],{},[770,103591,103592],{"scriptlevel":541,"displaystyle":70},[56,103593,103594,103596,103598,103604,103606,103608,103610,103616,103618,103620,103622,103632],{},[56,103595],{},[133,103597,793],{},[613,103599,103600,103602],{},[59,103601,4253],{"mathvariant":421},[59,103603,96325],{},[133,103605,537],{"fence":70,"stretchy":70,"minsize":73875,"maxsize":73875},[59,103607,92483],{},[133,103609,1690],{},[613,103611,103612,103614],{},[59,103613,11],{},[59,103615,96325],{},[133,103617,5378],{},[59,103619,92483],{},[133,103621,1690],{},[613,103623,103624,103626],{},[59,103625,11],{},[138,103627,103628,103630],{},[59,103629,96325],{},[133,103631,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,103633,4280],{"fence":70,"stretchy":70,"minsize":73875,"maxsize":73875},[764,103635,103636,103642],{},[767,103637,103638],{},[770,103639,103640],{"scriptlevel":541,"displaystyle":70},[56,103641],{},[767,103643,103644],{},[770,103645,103646],{"scriptlevel":541,"displaystyle":70},[56,103647,103648,103650,103652,103658,103660,103662,103664,103666,103672,103678,103680,103682,103684,103686,103688,103690,103692,103694,103696,103698,103700,103706,103708],{},[56,103649],{},[133,103651,793],{},[613,103653,103654,103656],{},[59,103655,4253],{"mathvariant":421},[59,103657,96325],{},[133,103659,537],{"fence":70,"stretchy":70,"minsize":73875,"maxsize":73875},[133,103661,136],{"stretchy":135},[59,103663,96325],{},[133,103665,5378],{},[138,103667,103668,103670],{},[59,103669,96325],{},[133,103671,147],{"mathvariant":145,"lspace":146,"rspace":146},[138,103673,103674,103676],{},[133,103675,161],{"stretchy":135},[59,103677,93341],{"mathvariant":145},[59,103679,11772],{},[133,103681,4280],{"fence":70,"stretchy":70,"minsize":73875,"maxsize":73875},[133,103683,5378],{},[133,103685,136],{"fence":70,"stretchy":70,"minsize":73875,"maxsize":73875},[59,103687,80653],{},[133,103689,136],{"stretchy":135},[59,103691,96325],{},[133,103693,161],{"stretchy":135},[133,103695,5378],{},[59,103697,80653],{},[133,103699,136],{"stretchy":135},[138,103701,103702,103704],{},[59,103703,96325],{},[133,103705,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,103707,161],{"stretchy":135},[133,103709,161],{"fence":70,"stretchy":70,"minsize":73875,"maxsize":73875},[764,103711,103712,103718],{},[767,103713,103714],{},[770,103715,103716],{"scriptlevel":541,"displaystyle":70},[56,103717],{},[767,103719,103720],{},[770,103721,103722],{"scriptlevel":541,"displaystyle":70},[56,103723,103724,103726,103728,103730,103732,103734,103740,103746,103748,103750,103752,103754,103756,103758,103760,103762,103764,103766,103768,103770,103772,103778],{},[56,103725],{},[133,103727,793],{},[133,103729,136],{"stretchy":135},[59,103731,96325],{},[133,103733,5378],{},[138,103735,103736,103738],{},[59,103737,96325],{},[133,103739,147],{"mathvariant":145,"lspace":146,"rspace":146},[138,103741,103742,103744],{},[133,103743,161],{"stretchy":135},[59,103745,93341],{"mathvariant":145},[59,103747,84725],{"mathvariant":145},[59,103749,80653],{},[133,103751,136],{"stretchy":135},[59,103753,96325],{},[133,103755,161],{"stretchy":135},[133,103757,5378],{},[59,103759,80653],{},[133,103761,136],{"stretchy":135},[59,103763,96325],{},[133,103765,161],{"stretchy":135},[133,103767,788],{},[59,103769,80653],{},[133,103771,136],{"stretchy":135},[138,103773,103774,103776],{},[59,103775,96325],{},[133,103777,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,103779,161],{"stretchy":135},[764,103781,103782,103788],{},[767,103783,103784],{},[770,103785,103786],{"scriptlevel":541,"displaystyle":70},[56,103787],{},[767,103789,103790],{},[770,103791,103792],{"scriptlevel":541,"displaystyle":70},[56,103793,103794,103796,103798,103800,103802,103808,103810,103812,103814,103816,103818,103820,103822,103824,103826,103828,103830,103836,103838,103844,103846,103848,103850],{},[56,103795],{},[133,103797,793],{},[59,103799,80653],{},[133,103801,136],{"stretchy":135},[138,103803,103804,103806],{},[59,103805,96325],{},[133,103807,147],{"mathvariant":145,"ls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E_\\theta\\bigl[\\log p_\\theta-\\log p_{\\theta'}\\bigr]\\\\\n&=\\mathbb E_\\theta\\bigl[(\\theta-\\theta')^\\top T\\bigr]-\\bigl(\\psi(\\theta)-\\psi(\\theta')\\bigr)\\\\\n&=(\\theta-\\theta')^\\top\\nabla\\psi(\\theta)-\\psi(\\theta)+\\psi(\\theta')\\\\\n&=\\psi(\\theta')-\\psi(\\theta)-\\nabla\\psi(\\theta)^\\top(\\theta'-\\theta).\n\\end{aligned}",[39,103856,103858],{"className":103857,"ariaHidden":70},[69],[39,103859,103861,103865],{"className":103860},[74],[39,103862],{"className":103863,"style":103864},[78],"height:5.8873em;vertical-align:-2.6937em;",[39,103866,103868],{"className":103867},[83],[39,103869,103871,104059],{"className":103870},[759],[39,103872,103874],{"className":103873},[918],[39,103875,103877,104050],{"className":103876},[196,649],[39,103878,103880,104047],{"className":103879},[200],[39,103881,103884,104020,104029,104038],{"className":103882,"style":103883},[204],"height:3.1937em;",[39,103885,103887,103890],{"style":103886},"top:-5.3437em;",[39,103888],{"className":103889,"style":935},[212],[39,103891,103893,103899,103902,103942,103945,104017],{"className":103892},[83],[39,103894,103896],{"className":103895},[83],[39,103897,77233],{"className":103898},[83,35077],[39,103900,136],{"className":103901},[182],[39,103903,103905,103908],{"className":103904},[83],[39,103906,11],{"className":103907},[83,84],[39,103909,103911],{"className":103910},[192],[39,103912,103914,103934],{"className":103913},[196,649],[39,103915,103917,103931],{"className":103916},[200],[39,103918,103920],{"className":103919,"style":3419},[204],[39,103921,103922,103925],{"style":966},[39,103923],{"className":103924,"style":213},[212],[39,103926,103928],{"className":103927},[217,218,219,220],[39,103929,96325],{"className":103930,"style":438},[83,84,220],[39,103932,673],{"className":103933},[672],[39,103935,103937],{"className":103936},[200],[39,103938,103940],{"className":103939,"style":680},[204],[39,103941],{},[39,103943,15389],{"className":103944},[83],[39,103946,103948,103951],{"className":103947},[83],[39,103949,11],{"className":103950},[83,84],[39,103952,103954],{"className":103953},[192],[39,103955,103957,104009],{"className":103956},[196,649],[39,103958,103960,104006],{"className":103959},[200],[39,103961,103963],{"className":103962,"style":3419},[204],[39,103964,103965,103968],{"style":966},[39,103966],{"className":103967,"style":213},[212],[39,103969,103971],{"className":103970},[217,218,219,220],[39,103972,103974],{"className":103973},[83,220],[39,103975,103977,103980],{"className":103976},[83,220],[39,103978,96325],{"className":103979,"style":438},[83,84,220],[39,103981,103983],{"className":103982},[192],[39,103984,103986],{"className":103985},[196],[39,103987,103989],{"className":103988},[200],[39,103990,103992],{"className":103991,"style":3997},[204],[39,103993,103994,103997],{"style":4000},[39,103995],{"className":103996,"style":4004},[212],[39,103998,104000],{"className":103999},[217,4008,4009,220],[39,104001,104003],{"className":104002},[83,220],[39,104004,147],{"className":104005},[83,220],[39,104007,673],{"className":104008},[672],[39,104010,104012],{"className":104011},[200],[39,104013,104015],{"className":104014,"style":680},[204],[39,104016],{},[39,104018,161],{"className":104019},[263],[39,104021,104023,104026],{"style":104022},"top:-3.7846em;",[39,104024],{"className":104025,"style":935},[212],[39,104027],{"className":104028},[83],[39,104030,104032,104035],{"style":104031},"top:-2.2254em;",[39,104033],{"className":104034,"style":935},[212],[39,104036],{"className":104037},[83],[39,104039,104041,104044],{"style":104040},"top:-0.6663em;",[39,104042],{"className":104043,"style":935},[212],[39,104045],{"className":104046},[83],[39,104048,673],{"className":104049},[672],[39,104051,104053],{"className":104052},[200],[39,104054,104057],{"className":104055,"style":104056},[204],"height:2.6937em;",[39,104058],{},[39,104060,104062],{"className":104061},[1223],[39,104063,104065,104884],{"className":104064},[196,649],[39,104066,104068,104881],{"className":104067},[200],[39,104069,104071,104280,104514,104696],{"className":104070,"style":103883},[204],[39,104072,104073,104076],{"style":103886},[39,104074],{"className":104075,"style":935},[212],[39,104077,104079,104082,104085,104088,104091,104131,104137,104142,104145,104185,104188,104191,104194,104199,104202,104274],{"className":104078},[83],[39,104080],{"className":104081},[83],[39,104083],{"className":104084,"style":231},[230],[39,104086,793],{"className":104087},[235],[39,104089],{"className":104090,"style":231},[230],[39,104092,104094,104097],{"className":104093},[83],[39,104095,4253],{"className":104096},[83,515],[39,104098,104100],{"className":104099},[192],[39,104101,104103,104123],{"className":104102},[196,649],[39,104104,104106,104120],{"className":104105},[200],[39,104107,104109],{"className":104108,"style":3419},[204],[39,104110,104111,104114],{"style":966},[39,104112],{"className":104113,"style":213},[212],[39,104115,104117],{"className":104116},[217,218,219,220],[39,104118,96325],{"className":104119,"style":438},[83,84,220],[39,104121,673],{"className":104122}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",[39,104896,104898,104911],{"className":104897,"translate":43},[42],[39,104899,104901],{"className":104900},[47],[49,104902,104903],{"xmlns":51},[53,104904,104905,104909],{},[56,104906,104907],{},[59,104908,80653],{},[63,104910,102296],{"encoding":65},[39,104912,104914],{"className":104913,"ariaHidden":70},[69],[39,104915,104917,104920],{"className":104916},[74],[39,104918],{"className":104919,"style":51280},[78],[39,104921,80653],{"className":104922,"style":2114},[83,84],[39,104924,104926,104939],{"className":104925,"translate":43},[42],[39,104927,104929],{"className":104928},[47],[49,104930,104931],{"xmlns":51},[53,104932,104933,104937],{},[56,104934,104935],{},[59,104936,96325],{},[63,104938,97273],{"encoding":65},[39,104940,104942],{"className":104941,"ariaHidden":70},[69],[39,104943,104945,104948],{"className":104944},[74],[39,104946],{"className":104947,"style":28824},[78],[39,104949,96325],{"className":104950,"style":438},[83,84]," 处的切平面在 ",[39,104953,104955,104973],{"className":104954,"translate":43},[42],[39,104956,104958],{"className":104957},[47],[49,104959,104960],{"xmlns":51},[53,104961,104962,104970],{},[56,104963,104964],{},[138,104965,104966,104968],{},[59,104967,96325],{},[133,104969,147],{"mathvariant":145,"lspace":146,"rspace":146},[63,104971,104972],{"encoding":65},"\\theta'",[39,104974,104976],{"className":104975,"ariaHidden":70},[69],[39,104977,104979,104982],{"className":104978},[74],[39,104980],{"className":104981,"style":205},[78],[39,104983,104985,104988],{"className":104984},[83],[39,104986,96325],{"className":104987,"style":438},[83,84],[39,104989,104991],{"className":104990},[192],[39,104992,104994],{"className":104993},[196],[39,104995,104997],{"className":104996},[200],[39,104998,105000],{"className":104999,"style":205},[204],[39,105001,105002,105005],{"style":208},[39,105003],{"className":105004,"style":213},[212],[39,105006,105008],{"className":105007},[217,218,219,220],[39,105009,105011],{"className":105010},[83,220],[39,105012,147],{"className":105013},[83,220]," 处高出的那部分。它有名字：Bregman 散度。",[11,105016,105017,105018,742,105069,105172],{},"拿方差固定为 1 的高斯族验一下。此时 ",[39,105019,105021,105039],{"className":105020,"translate":43},[42],[39,105022,105024],{"className":105023},[47],[49,105025,105026],{"xmlns":51},[53,105027,105028,105036],{},[56,105029,105030,105032,105034],{},[59,105031,96325],{},[133,105033,793],{},[59,105035,73578],{},[63,105037,105038],{"encoding":65},"\\theta=\\mu",[39,105040,105042,105060],{"className":105041,"ariaHidden":70},[69],[39,105043,105045,105048,105051,105054,105057],{"className":105044},[74],[39,105046],{"className":105047,"style":28824},[78],[39,105049,96325],{"className":105050,"style":438},[83,84],[39,105052],{"className":105053,"style":231},[230],[39,105055,793],{"className":105056},[235],[39,105058],{"className":105059,"style":231},[230],[39,105061,105063,105066],{"className":105062},[74],[39,105064],{"className":105065,"style":2654},[78],[39,105067,73578],{"className":105068},[83,84],[39,105070,105072,105104],{"className":105071,"translate":43},[42],[39,105073,105075],{"className":105074},[47],[49,105076,105077],{"xmlns":51},[53,105078,105079,105101],{},[56,105080,105081,105083,105085,105087,105089,105091,105097,105099],{},[59,105082,80653],{},[133,105084,136],{"stretchy":135},[59,105086,96325],{},[133,105088,161],{"stretchy":135},[133,105090,793],{},[138,105092,105093,105095],{},[59,105094,96325],{},[539,105096,17359],{},[59,105098,45691],{"mathvariant":145},[539,105100,17359],{},[63,105102,105103],{"encoding":65},"\\psi(\\theta)=\\theta^2\u002F2",[39,105105,105107,105134],{"className":105106,"ariaHidden":70},[69],[39,105108,105110,105113,105116,105119,105122,105125,105128,105131],{"className":105109},[74],[39,105111],{"className":105112,"style":245},[78],[39,105114,80653],{"className":105115,"style":2114},[83,84],[39,105117,136],{"className":105118},[182],[39,105120,96325],{"className":105121,"style":438},[83,84],[39,105123,161],{"className":105124},[263],[39,105126],{"className":105127,"style":231},[230],[39,105129,793],{"className":105130},[235],[39,105132],{"className":105133,"style":231},[230],[39,105135,105137,105140,105169],{"className":105136},[74],[39,105138],{"className":105139,"style":84615},[78],[39,105141,105143,105146],{"className":105142},[83],[39,105144,96325],{"className":105145,"style":438},[83,84],[39,105147,105149],{"className":105148},[192],[39,105150,105152],{"className":105151},[196],[39,105153,105155],{"className":105154},[200],[39,105156,105158],{"className":105157,"style":19710},[204],[39,105159,105160,105163],{"style":208},[39,105161],{"className":105162,"style":213},[212],[39,105164,105166],{"className":105165},[217,218,219,220],[39,105167,17359],{"className":105168},[83,220],[39,105170,87914],{"className":105171},[83],"，手算",[39,105174,105176],{"className":105175,"translate":43},[746],[39,105177,105179,105267],{"className":105178,"translate":43},[42],[39,105180,105182],{"className":105181},[47],[49,105183,105184],{"xmlns":51,"display":755},[53,105185,105186,105264],{},[56,105187,105188,105194,105196,105198,105200,105206,105208,105210,105212,105214,105216,105218,105224,105226,105228,105230,105232,105234,105262],{},[56,105189,105190,105192],{},[59,105191,77187],{"mathvariant":145},[59,105193,77190],{"mathvariant":145},[133,105195,136],{"fence":70,"stretchy":70,"minsize":73875,"maxsize":73875},[59,105197,77559],{"mathvariant":617},[133,105199,136],{"stretchy":135},[613,105201,105202,105204],{},[59,105203,96325],{},[539,105205,546],{},[133,105207,155],{"separator":70},[539,105209,546],{},[133,105211,161],{"stretchy":135},[133,105213,15389],{"fence":135,"stretchy":70,"minsize":73875,"maxsize":73875},[59,105215,77559],{"mathvariant":617},[133,105217,136],{"stretchy":135},[613,105219,105220,105222],{},[59,105221,96325],{},[539,105223,17359],{},[133,105225,155],{"separator":70},[539,105227,546],{},[133,105229,161],{"stretchy":135},[133,105231,161],{"fence":70,"stretchy":70,"minsize":73875,"maxsize":73875},[133,105233,793],{},[5362,105235,105236,105260],{},[56,105237,105238,105240,105246,105248,105254],{},[133,105239,136],{"stretchy":135},[613,105241,105242,105244],{},[59,105243,96325],{},[539,105245,546],{},[133,105247,5378],{},[613,105249,105250,105252],{},[59,105251,96325],{},[539,105253,17359],{},[138,105255,105256,105258],{},[133,105257,161],{"stretchy":135},[539,105259,17359],{},[539,105261,17359],{},[133,105263,155],{"separator":70},[63,105265,105266],{"encoding":65},"\\mathrm{KL}\\bigl(\\mathcal 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6254,106255,106258],{"style":5503},[39,106256],{"className":106257,"style":935},[212],[39,106259],{"className":106260,"style":5511},[5510],[39,106262,106263,106266],{"style":74839},[39,106264],{"className":106265,"style":935},[212],[39,106267,106269],{"className":106268},[217,218,219,220],[39,106270,106272],{"className":106271},[83,220],[39,106273,546],{"className":106274},[83,220],[39,106276,673],{"className":106277},[672],[39,106279,106281],{"className":106280},[200],[39,106282,106284],{"className":106283,"style":74861},[204],[39,106285],{},[39,106287],{"className":106288},[263,5463],[39,106290,136],{"className":106291},[182],[39,106293,106295,106298],{"className":106294},[83],[39,106296,96325],{"className":106297,"style":438},[83,84],[39,106299,106301],{"className":106300},[192],[39,106302,106304,106324],{"className":106303},[196,649],[39,106305,106307,106321],{"className":106306},[200],[39,106308,106310],{"className":106309,"style":963},[204],[39,106311,106312,106315],{"style":2937},[39,106313],{"className":106314,"style":213},[212],[39,106316,106318],{"className":106317},[217,218,219,220],[39,106319,17359],{"className":106320},[83,220],[39,106322,673],{"className":106323},[672],[39,106325,106327],{"className":106326},[200],[39,106328,106330],{"className":106329,"style":680},[204],[39,106331],{},[39,106333],{"className":106334,"style":461},[230],[39,106336,5378],{"className":106337},[465],[39,106339],{"className":106340,"style":461},[230],[39,106342,106344,106347,106387],{"className":106343},[74],[39,106345],{"className":106346,"style":84615},[78],[39,106348,106350,106353],{"className":106349},[83],[39,106351,96325],{"className":106352,"style":438},[83,84],[39,106354,106356],{"className":106355},[192],[39,106357,106359,106379],{"className":106358},[196,649],[39,106360,106362,106376],{"className":106361},[200],[39,106363,106365],{"className":106364,"style":963},[204],[39,106366,106367,106370],{"style":2937},[39,106368],{"className":106369,"style":213},[212],[39,106371,106373],{"className":106372},[217,218,219,220],[39,106374,546],{"className":106375},[83,220],[39,106377,673],{"className":106378},[672],[39,106380,106382],{"className":106381},[200],[39,106383,106385],{"className":106384,"style":680},[204],[39,106386],{},[39,106388,106390,106393],{"className":106389},[263],[39,106391,161],{"className":106392},[263],[39,106394,106396],{"className":106395},[192],[39,106397,106399],{"className":106398},[196],[39,106400,106402],{"className":106401},[200],[39,106403,106405],{"className":106404,"style":19710},[204],[39,106406,106407,106410],{"style":208},[39,106408],{"className":106409,"style":213},[212],[39,106411,106413],{"className":106412},[217,218,219,220],[39,106414,17359],{"className":106415},[83,220],"。同一个数。",[11,106418,106419],{},"所以在这类模型里，KL 不是某种抽象的“信息量”，它就是 Bregman 散度；而它的 Hessian 就是 Fisher 度量。散度和度量是同一个凸函数的两个侧面。这是我写这篇博客最想讲清楚的一件事。",[21,106421,106423],{"id":106422},"bregman","Bregman",[11,106425,106426,106427,106501,106502,106576],{},"先看一个会让人不舒服的例子。",[39,106428,106430,106456],{"className":106429,"translate":43},[42],[39,106431,106433],{"className":106432},[47],[49,106434,106435],{"xmlns":51},[53,106436,106437,106453],{},[56,106438,106439,106441,106443,106445,106447,106449,106451],{},[59,106440,11],{},[133,106442,793],{},[133,106444,136],{"stretchy":135},[539,106446,546],{},[133,106448,155],{"separator":70},[539,106450,541],{},[133,106452,161],{"stretchy":135},[63,106454,106455],{"encoding":65},"p=(1,0)",[39,106457,106459,106477],{"className":106458,"ariaHidden":70},[69],[39,106460,106462,106465,106468,106471,106474],{"className":106461},[74],[39,106463],{"className":106464,"style":2654},[78],[39,106466,11],{"className":106467},[83,84],[39,106469],{"className":106470,"style":231},[230],[39,106472,793],{"className":106473},[235],[39,106475],{"className":106476,"style":231},[230],[39,106478,106480,106483,106486,106489,106492,106495,106498],{"className":106479},[74],[39,106481],{"className":106482,"style":245},[78],[39,106484,136],{"className":106485},[182],[39,106487,546],{"className":106488},[83],[39,106490,155],{"className":106491},[252],[39,106493],{"className":106494,"style":256},[230],[39,106496,541],{"className":106497},[83],[39,106499,161],{"className":106500},[263]," 是“真相”，",[39,106503,106505,106531],{"className":106504,"translate":43},[42],[39,106506,106508],{"className":106507},[47],[49,106509,106510],{"xmlns":51},[53,106511,106512,106528],{},[56,106513,106514,106516,106518,106520,106522,106524,106526],{},[59,106515,100187],{},[133,106517,793],{},[133,106519,136],{"stretchy":135},[539,106521,95695],{},[133,106523,155],{"separator":70},[539,106525,95695],{},[133,106527,161],{"stretchy":135},[63,106529,106530],{"encoding":65},"q=(0.5,0.5)",[39,106532,106534,106552],{"className":106533,"ariaHidden":70},[69],[39,106535,106537,106540,106543,106546,106549],{"className":106536},[74],[39,106538],{"className":106539,"style":2654},[78],[39,106541,100187],{"className":106542,"style":2114},[83,84],[39,106544],{"className":106545,"style":231},[230],[39,106547,793],{"className":106548},[235],[39,106550],{"className":106551,"style":231},[230],[39,106553,106555,106558,106561,106564,106567,106570,106573],{"className":106554},[74],[39,106556],{"className":106557,"style":245},[78],[39,106559,136],{"className":106560},[182],[39,106562,95695],{"className":106563},[83],[39,106565,155],{"className":106566},[252],[39,106568],{"className":106569,"style":256},[230],[39,106571,95695],{"className":106572},[83],[39,106574,161],{"className":106575},[263]," 是“模型的预测”。",[39,106578,106580],{"className":106579,"translate":43},[746],[39,106581,106583,106652],{"className":106582,"translate":43},[42],[39,106584,106586],{"className":106585},[47],[49,106587,106588],{"xmlns":51,"display":755},[53,106589,106590,106649],{},[56,106591,106592,106598,106600,106602,106604,106606,106608,106610,106612,106614,106616,106618,106621,106623,106625,106631,106633,106635,106637,106639,106641,106643,106645,106647],{},[56,106593,106594,106596],{},[59,106595,77187],{"mathvariant":145},[59,106597,77190],{"mathvariant":145},[133,106599,136],{"stretchy":135},[59,106601,11],{},[59,106603,15389],{"mathvariant":145},[59,106605,100187],{},[133,106607,161],{"stretchy":135},[133,106609,793],{},[59,106611,92483],{},[133,106613,1690],{},[539,106615,17359],{},[133,106617,99744],{},[539,106619,106620],{},"0.693",[133,106622,155],{"separator":70},[230,106624],{"width":10251},[56,106626,106627,106629],{},[59,106628,77187],{"mathvariant":145},[59,106630,77190],{"mathvariant":145},[133,106632,136],{"stretchy":135},[59,106634,100187],{},[59,106636,15389],{"mathvariant":145},[59,106638,11],{},[133,106640,161],{"stretchy":135},[133,106642,793],{},[133,106644,788],{},[59,106646,1696],{"mathvariant":145},[59,106648,2903],{"mathvariant":145},[63,106650,106651],{"encoding":65},"\\mathrm{KL}(p\\|q)=\\log2\\approx0.693,\n\\qquad\n\\mathrm{KL}(q\\|p)=+\\infty .",[39,106653,106655,106691,106717,106765],{"className":106654,"ariaHidden":70},[69],[39,106656,106658,106661,106667,106670,106673,106676,106679,106682,106685,106688],{"className":106657},[74],[39,106659],{"className":106660,"style":245},[78],[39,106662,106664],{"className":106663},[83],[39,106665,77233],{"className":106666},[83,35077],[39,106668,136],{"className":106669},[182],[39,106671,11],{"className":106672},[83,84],[39,106674,15389],{"className":106675},[83],[39,106677,100187],{"className":106678,"style":2114},[83,84],[39,106680,161],{"className":106681},[263],[39,106683],{"className":106684,"style":231},[230],[39,106686,793],{"className":106687},[235],[39,106689],{"className":106690,"style":231},[230],[39,106692,106694,106697,106702,106705,106708,106711,106714],{"className":106693},[74],[39,106695],{"className":106696,"style":51280},[78],[39,106698,92721,106700],{"className":106699},[1744],[39,106701,39209],{"style":35078},[39,106703],{"className":106704,"style":256},[230],[39,106706,17359],{"className":106707},[83],[39,106709],{"className":106710,"style":231},[230],[39,106712,99744],{"className":106713},[235],[39,106715],{"className":106716,"style":231},[230],[39,106718,106720,106723,106726,106729,106732,106735,106741,106744,106747,106750,106753,106756,106759,106762],{"className":106719},[74],[39,106721],{"className":106722,"style":245},[78],[39,106724,106620],{"className":106725},[83],[39,106727,155],{"className":106728},[252],[39,106730],{"className":106731,"style":10787},[230],[39,106733],{"className":106734,"style":256},[230],[39,106736,106738],{"className":106737},[83],[39,106739,77233],{"className":106740},[83,35077],[39,106742,136],{"className":106743},[182],[39,106745,100187],{"className":106746,"style":2114},[83,84],[39,106748,15389],{"className":106749},[83],[39,106751,11],{"className":106752},[83,84],[39,106754,161],{"className":106755},[263],[39,106757],{"className":106758,"style":231},[230],[39,106760,793],{"className":106761},[235],[39,106763],{"className":106764,"style":231},[230],[39,106766,106768,106771,106774],{"className":106767},[74],[39,106769],{"className":106770,"style":73043},[78],[39,106772,788],{"className":106773},[83],[39,106775,106777],{"className":106776},[83],"∞.",[11,106779,106780,106781,106809,106810,106838,106839,106867,106868,106896,106897,106867,106925,106953,106954,106982],{},"同一个“差”，从左往右量是 0.69，从右往左量是无穷。原因不难想：",[39,106782,106784,106797],{"className":106783,"translate":43},[42],[39,106785,106787],{"className":106786},[47],[49,106788,106789],{"xmlns":51},[53,106790,106791,106795],{},[56,106792,106793],{},[59,106794,11],{},[63,106796,11],{"encoding":65},[39,106798,106800],{"className":106799,"ariaHidden":70},[69],[39,106801,106803,106806],{"className":106802},[74],[39,106804],{"className":106805,"style":2654},[78],[39,106807,11],{"className":106808},[83,84]," 在第二位上放了 0，而 ",[39,106811,106813,106826],{"className":106812,"translate":43},[42],[39,106814,106816],{"className":106815},[47],[49,106817,106818],{"xmlns":51},[53,106819,106820,106824],{},[56,106821,106822],{},[59,106823,100187],{},[63,106825,100187],{"encoding":65},[39,106827,106829],{"className":106828,"ariaHidden":70},[69],[39,106830,106832,106835],{"className":106831},[74],[39,106833],{"className":106834,"style":2654},[78],[39,106836,100187],{"className":106837,"style":2114},[83,84]," 在那里放了 0.5，用 ",[39,106840,106842,106855],{"className":106841,"translate":43},[42],[39,106843,106845],{"className":106844},[47],[49,106846,106847],{"xmlns":51},[53,106848,106849,106853],{},[56,106850,106851],{},[59,106852,11],{},[63,106854,11],{"encoding":65},[39,106856,106858],{"className":106857,"ariaHidden":70},[69],[39,106859,106861,106864],{"className":106860},[74],[39,106862],{"className":106863,"style":2654},[78],[39,106865,11],{"className":106866},[83,84]," 去看 ",[39,106869,106871,106884],{"className":106870,"translate":43},[42],[39,106872,106874],{"className":106873},[47],[49,106875,106876],{"xmlns":51},[53,106877,106878,106882],{},[56,106879,106880],{},[59,106881,100187],{},[63,106883,100187],{"encoding":65},[39,106885,106887],{"className":106886,"ariaHidden":70},[69],[39,106888,106890,106893],{"className":106889},[74],[39,106891],{"className":106892,"style":2654},[78],[39,106894,100187],{"className":106895,"style":2114},[83,84],"，只是多花了 0.69 nat；用 ",[39,106898,106900,106913],{"className":106899,"translate":43},[42],[39,106901,106903],{"className":106902},[47],[49,106904,106905],{"xmlns":51},[53,106906,106907,106911],{},[56,106908,106909],{},[59,106910,100187],{},[63,106912,100187],{"encoding":65},[39,106914,106916],{"className":106915,"ariaHidden":70},[69],[39,106917,106919,106922],{"className":106918},[74],[39,106920],{"className":106921,"style":2654},[78],[39,106923,100187],{"className":106924,"style":2114},[83,84],[39,106926,106928,106941],{"className":106927,"translate":43},[42],[39,106929,106931],{"className":106930},[47],[49,106932,106933],{"xmlns":51},[53,106934,106935,106939],{},[56,106936,106937],{},[59,106938,11],{},[63,106940,11],{"encoding":65},[39,106942,106944],{"className":106943,"ariaHidden":70},[69],[39,106945,106947,106950],{"className":106946},[74],[39,106948],{"className":106949,"style":2654},[78],[39,106951,11],{"className":106952},[83,84],"，等于说“我给了一个 ",[39,106955,106957,106970],{"className":106956,"translate":43},[42],[39,106958,106960],{"className":106959},[47],[49,106961,106962],{"xmlns":51},[53,106963,106964,106968],{},[56,106965,106966],{},[59,106967,11],{},[63,106969,11],{"encoding":65},[39,106971,106973],{"className":106972,"ariaHidden":70},[69],[39,106974,106976,106979],{"className":106975},[74],[39,106977],{"className":106978,"style":2654},[78],[39,106980,11],{"className":106981},[83,84]," 认为绝不可能的东西分配了概率”，这种错误在 KL 里罚到无穷。",[11,106984,106985,106986,107047,107048,107076,107077,107105],{},"这正是交叉熵写成 ",[39,106987,106989,107017],{"className":106988,"translate":43},[42],[39,106990,106992],{"className":106991},[47],[49,106993,106994],{"xmlns":51},[53,106995,106996,107014],{},[56,106997,106998,107004,107006,107008,107010,107012],{},[56,106999,107000,107002],{},[59,107001,77187],{"mathvariant":145},[59,107003,77190],{"mathvariant":145},[133,107005,136],{"stretchy":135},[59,107007,29625],{},[59,107009,15389],{"mathvariant":145},[59,107011,100187],{},[133,107013,161],{"stretchy":135},[63,107015,107016],{"encoding":65},"\\mathrm{KL}(y\\|q)",[39,107018,107020],{"className":107019,"ariaHidden":70},[69],[39,107021,107023,107026,107032,107035,107038,107041,107044],{"className":107022},[74],[39,107024],{"className":107025,"style":245},[78],[39,107027,107029],{"className":107028},[83],[39,107030,77233],{"className":107031},[83,35077],[39,107033,136],{"className":107034},[182],[39,107036,29625],{"className":107037,"style":2114},[83,84],[39,107039,15389],{"className":107040},[83],[39,107042,100187],{"className":107043,"style":2114},[83,84],[39,107045,161],{"className":107046},[263]," 而不是反过来的原因：",[39,107049,107051,107064],{"className":107050,"translate":43},[42],[39,107052,107054],{"className":107053},[47],[49,107055,107056],{"xmlns":51},[53,107057,107058,107062],{},[56,107059,107060],{},[59,107061,29625],{},[63,107063,29625],{"encoding":65},[39,107065,107067],{"className":107066,"ariaHidden":70},[69],[39,107068,107070,107073],{"className":107069},[74],[39,107071],{"className":107072,"style":2654},[78],[39,107074,29625],{"className":107075,"style":2114},[83,84]," 是真相，模型 ",[39,107078,107080,107093],{"className":107079,"translate":43},[42],[39,107081,107083],{"className":107082},[47],[49,107084,107085],{"xmlns":51},[53,107086,107087,107091],{},[56,107088,107089],{},[59,107090,100187],{},[63,107092,100187],{"encoding":65},[39,107094,107096],{"className":107095,"ariaHidden":70},[69],[39,107097,107099,107102],{"className":107098},[74],[39,107100],{"className":107101,"style":2654},[78],[39,107103,100187],{"className":107104,"style":2114},[83,84]," 不该在真答案上给 0 概率。",[11,107107,107108,107109,107320,107321,107502,107503,107583,107584,107612,107613,75103,107642,107670],{},"再看一个尺度上的例子。Itakura–Saito 散度 ",[39,107110,107112,107172],{"className":107111,"translate":43},[42],[39,107113,107115],{"className":107114},[47],[49,107116,107117],{"xmlns":51},[53,107118,107119,107169],{},[56,107120,107121,107131,107133,107135,107137,107139,107141,107143,107145,107147,107149,107151,107153,107155,107157,107159,107161,107163,107165,107167],{},[613,107122,107123,107125],{},[59,107124,48449],{},[56,107126,107127,107129],{},[59,107128,14373],{"mathvariant":145},[59,107130,61],{"mathvariant":145},[133,107132,136],{"stretchy":135},[59,107134,1755],{},[59,107136,15389],{"mathvariant":145},[59,107138,29625],{},[133,107140,161],{"stretchy":135},[133,107142,793],{},[59,107144,1755],{},[59,107146,45691],{"mathvariant":145},[59,107148,29625],{},[133,107150,5378],{},[59,107152,92483],{},[133,107154,1690],{},[133,107156,136],{"stretchy":135},[59,107158,1755],{},[59,107160,45691],{"mathvariant":145},[59,107162,29625],{},[133,107164,161],{"stretchy":135},[133,107166,5378],{},[539,107168,546],{},[63,107170,107171],{"encoding":65},"D_{\\mathrm{IS}}(x\\|y)=x\u002Fy-\\log(x\u002Fy)-1",[39,107173,107175,107252,107276,107311],{"className":107174,"ariaHidden":70},[69],[39,107176,107178,107181,107228,107231,107234,107237,107240,107243,107246,107249],{"className":107177},[74],[39,107179],{"className":107180,"style":245},[78],[39,107182,107184,107187],{"className":107183},[83],[39,107185,48449],{"className":107186,"style":438},[83,84],[39,107188,107190],{"className":107189},[192],[39,107191,107193,107220],{"className":107192},[196,649],[39,107194,107196,107217],{"className":107195},[200],[39,107197,107199],{"className":107198,"style":62921},[204],[39,107200,107201,107204],{"style":2937},[39,107202],{"className":107203,"style":213},[212],[39,107205,107207],{"className":107206},[217,218,219,220],[39,107208,107210],{"className":107209},[83,220],[39,107211,107213],{"className":107212},[83,220],[39,107214,107216],{"className":107215},[83,35077,220],"IS",[39,107218,673],{"className":107219},[672],[39,107221,107223],{"className":107222},[200],[39,107224,107226],{"className":107225,"style":680},[204],[39,107227],{},[39,107229,136],{"className":107230},[182],[39,107232,1755],{"className":107233},[83,84],[39,107235,15389],{"className":107236},[83],[39,107238,29625],{"className":107239,"style":2114},[83,84],[39,107241,161],{"className":107242},[263],[39,107244],{"className":107245,"style":231},[230],[39,107247,793],{"className":107248},[235],[39,107250],{"className":107251,"style":231},[230],[39,107253,107255,107258,107261,107264,107267,107270,107273],{"className":107254},[74],[39,107256],{"className":107257,"style":245},[78],[39,107259,1755],{"className":107260},[83,84],[39,107262,45691],{"className":107263},[83],[39,107265,29625],{"className":107266,"style":2114},[83,84],[39,107268],{"className":107269,"style":461},[230],[39,107271,5378],{"className":107272},[465],[39,107274],{"className":107275,"style":461},[230],[39,107277,107279,107282,107287,107290,107293,107296,107299,107302,107305,107308],{"className":107278},[74],[39,107280],{"className":107281,"style":245},[78],[39,107283,92721,107285],{"className":107284},[1744],[39,107286,39209],{"style":35078},[39,107288,136],{"className":107289},[182],[39,107291,1755],{"className":107292},[83,84],[39,107294,45691],{"className":107295},[83],[39,107297,29625],{"className":107298,"style":2114},[83,84],[39,107300,161],{"className":107301},[263],[39,107303],{"className":107304,"style":461},[230],[39,107306,5378],{"className":107307},[465],[39,107309],{"className":107310,"style":461},[230],[39,107312,107314,107317],{"className":107313},[74],[39,107315],{"className":107316,"style":2473},[78],[39,107318,546],{"className":107319},[83],"：把音量从 1 减半到 0.5，",[39,107322,107324,107367],{"className":107323,"translate":43},[42],[39,107325,107327],{"className":107326},[47],[49,107328,107329],{"xmlns":51},[53,107330,107331,107364],{},[56,107332,107333,107343,107345,107347,107349,107351,107353,107355,107357,107359,107361],{},[613,107334,107335,107337],{},[59,107336,48449],{},[56,107338,107339,107341],{},[59,107340,14373],{"mathvariant":145},[59,107342,61],{"mathvariant":145},[133,107344,793],{},[539,107346,17359],{},[133,107348,5378],{},[59,107350,92483],{},[133,107352,1690],{},[539,107354,17359],{},[133,107356,5378],{},[539,107358,546],{},[133,107360,793],{},[539,107362,107363],{},"0.307",[63,107365,107366],{"encoding":65},"D_{\\mathrm{IS}}=2-\\log2-1=0.307",[39,107368,107370,107431,107449,107475,107493],{"className":107369,"ariaHidden":70},[69],[39,107371,107373,107376,107422,107425,107428],{"className":107372},[74],[39,107374],{"className":107375,"style":634},[78],[39,107377,107379,107382],{"className":107378},[83],[39,107380,48449],{"className":107381,"style":438},[83,84],[39,107383,107385],{"className":107384},[192],[39,107386,107388,107414],{"className":107387},[196,649],[39,107389,107391,107411],{"className":107390},[200],[39,107392,107394],{"className":107393,"style":62921},[204],[39,107395,107396,107399],{"style":2937},[39,107397],{"className":107398,"style":213},[212],[39,107400,107402],{"className":107401},[217,218,219,220],[39,107403,107405],{"className":107404},[83,220],[39,107406,107408],{"className":107407},[83,220],[39,107409,107216],{"className":107410},[83,35077,220],[39,107412,673],{"className":107413},[672],[39,107415,107417],{"className":107416},[200],[39,107418,107420],{"className":107419,"style":680},[204],[39,107421],{},[39,107423],{"className":107424,"style":231},[230],[39,107426,793],{"className":107427},[235],[39,107429],{"className":107430,"style":231},[230],[39,107432,107434,107437,107440,107443,107446],{"className":107433},[74],[39,107435],{"className":107436,"style":17882},[78],[39,107438,17359],{"className":107439},[83],[39,107441],{"className":107442,"style":461},[230],[39,107444,5378],{"className":107445},[465],[39,107447],{"className":107448,"style":461},[230],[39,107450,107452,107455,107460,107463,107466,107469,107472],{"className":107451},[74],[39,107453],{"className":107454,"style":51280},[78],[39,107456,92721,107458],{"className":107457},[1744],[39,107459,39209],{"style":35078},[39,107461],{"className":107462,"style":256},[230],[39,107464,17359],{"className":107465},[83],[39,107467],{"className":107468,"style":461},[230],[39,107470,5378],{"className":107471},[465],[39,107473],{"className":107474,"style":461},[230],[39,107476,107478,107481,107484,107487,107490],{"className":107477},[74],[39,107479],{"className":107480,"style":2473},[78],[39,107482,546],{"className":107483},[83],[39,107485],{"className":107486,"style":231},[230],[39,107488,793],{"className":107489},[235],[39,107491],{"className":107492,"style":231},[230],[39,107494,107496,107499],{"className":107495},[74],[39,107497],{"className":107498,"style":2473},[78],[39,107500,107363],{"className":107501},[83],"；从 2 减半到 1，",[39,107504,107506,107528],{"className":107505,"translate":43},[42],[39,107507,107509],{"className":107508},[47],[49,107510,107511],{"xmlns":51},[53,107512,107513,107525],{},[56,107514,107515],{},[613,107516,107517,107519],{},[59,107518,48449],{},[56,107520,107521,107523],{},[59,107522,14373],{"mathvariant":145},[59,107524,61],{"mathvariant":145},[63,107526,107527],{"encoding":65},"D_{\\mathrm{IS}}",[39,107529,107531],{"className":107530,"ariaHidden":70},[69],[39,107532,107534,107537],{"className":107533},[74],[39,107535],{"className":107536,"style":634},[78],[39,107538,107540,107543],{"className":107539},[83],[39,107541,48449],{"className":107542,"style":438},[83,84],[39,107544,107546],{"className":107545},[192],[39,107547,107549,107575],{"className":107548},[196,649],[39,107550,107552,107572],{"className":107551},[200],[39,107553,107555],{"className":107554,"style":62921},[204],[39,107556,107557,107560],{"style":2937},[39,107558],{"className":107559,"style":213},[212],[39,107561,107563],{"className":107562},[217,218,219,220],[39,107564,107566],{"className":107565},[83,220],[39,107567,107569],{"className":107568},[83,220],[39,107570,107216],{"className":107571},[83,35077,220],[39,107573,673],{"className":107574},[672],[39,107576,107578],{"className":107577},[200],[39,107579,107581],{"className":107580,"style":680},[204],[39,107582],{}," 还是 ",[39,107585,107587,107600],{"className":107586,"translate":43},[42],[39,107588,107590],{"className":107589},[47],[49,107591,107592],{"xmlns":51},[53,107593,107594,107598],{},[56,107595,107596],{},[539,107597,107363],{},[63,107599,107363],{"encoding":65},[39,107601,107603],{"className":107602,"ariaHidden":70},[69],[39,107604,107606,107609],{"className":107605},[74],[39,107607],{"className":107608,"style":2473},[78],[39,107610,107363],{"className":107611},[83],"。同样的比例变化，距离一样。而欧氏距离给的是 ",[39,107614,107616,107630],{"className":107615,"translate":43},[42],[39,107617,107619],{"className":107618},[47],[49,107620,107621],{"xmlns":51},[53,107622,107623,107628],{},[56,107624,107625],{},[539,107626,107627],{},"0.25",[63,107629,107627],{"encoding":65},[39,107631,107633],{"className":107632,"ariaHidden":70},[69],[39,107634,107636,107639],{"className":107635},[74],[39,107637],{"className":107638,"style":2473},[78],[39,107640,107627],{"className":107641},[83],[39,107643,107645,107658],{"className":107644,"translate":43},[42],[39,107646,107648],{"className":107647},[47],[49,107649,107650],{"xmlns":51},[53,107651,107652,107656],{},[56,107653,107654],{},[539,107655,546],{},[63,107657,546],{"encoding":65},[39,107659,107661],{"className":107660,"ariaHidden":70},[69],[39,107662,107664,107667],{"className":107663},[74],[39,107665],{"className":107666,"style":2473},[78],[39,107668,546],{"className":107669},[83],"，差了四倍——因为欧氏觉得“从 2 到 1 变化更大”，可听感上这明明是同一件事。",[11,107672,107673],{},"这两个例子指向同一件事：距离可以不对称，也可以不按绝对差来量。",[11,107675,107676,107677,107706,107707,742],{},"一般地，给定凸函数 ",[39,107678,107680,107693],{"className":107679,"translate":43},[42],[39,107681,107683],{"className":107682},[47],[49,107684,107685],{"xmlns":51},[53,107686,107687,107691],{},[56,107688,107689],{},[59,107690,74335],{},[63,107692,74335],{"encoding":65},[39,107694,107696],{"className":107695,"ariaHidden":70},[69],[39,107697,107699,107702],{"className":107698},[74],[39,107700],{"className":107701,"style":51280},[78],[39,107703,74335],{"className":107704,"style":107705},[83,84],"margin-right:0.1076em;"," 和可微点 ",[39,107708,107710,107723],{"className":107709,"translate":43},[42],[39,107711,107713],{"className":107712},[47],[49,107714,107715],{"xmlns":51},[53,107716,107717,107721],{},[56,107718,107719],{},[59,107720,29625],{},[63,107722,29625],{"encoding":65},[39,107724,107726],{"className":107725,"ariaHidden":70},[69],[39,107727,107729,107732],{"className":107728},[74],[39,107730],{"className":107731,"style":2654},[78],[39,107733,29625],{"className":107734,"style":2114},[83,84],[39,107736,107738],{"className":107737,"translate":43},[746],[39,107739,107741,107817],{"className":107740,"translate":43},[42],[39,107742,107744],{"className":107743},[47],[49,107745,107746],{"xmlns":51,"display":755},[53,107747,107748,107814],{},[56,107749,107750,107756,107758,107760,107762,107764,107766,107768,107770,107772,107774,107776,107778,107780,107782,107784,107786,107788,107790,107792,107794,107796,107802,107804,107806,107808,107810,107812],{},[613,107751,107752,107754],{},[59,107753,48449],{},[59,107755,74335],{},[133,107757,136],{"stretchy":135},[59,107759,1755],{},[59,107761,15389],{"mathvariant":145},[59,107763,29625],{},[133,107765,161],{"stretchy":135},[133,107767,793],{},[59,107769,74335],{},[133,107771,136],{"stretchy":135},[59,107773,1755],{},[133,107775,161],{"stretchy":135},[133,107777,5378],{},[59,107779,74335],{},[133,107781,136],{"stretchy":135},[59,107783,29625],{},[133,107785,161],{"stretchy":135},[133,107787,5378],{},[59,107789,84725],{"mathvariant":145},[59,107791,74335],{},[133,107793,136],{"stretchy":135},[59,107795,29625],{},[138,107797,107798,107800],{},[133,107799,161],{"stretchy":135},[59,107801,93341],{"mathvariant":145},[133,107803,136],{"stretchy":135},[59,107805,1755],{},[133,107807,5378],{},[59,107809,29625],{},[133,107811,161],{"stretchy":135},[59,107813,2903],{"mathvariant":145},[63,107815,107816],{"encoding":65},"D_f(x\\|y)=f(x)-f(y)-\\nabla f(y)^\\top(x-y).",[39,107818,107820,107890,107917,107944,108006],{"className":107819,"ariaHidden":70},[69],[39,107821,107823,107826,107866,107869,107872,107875,107878,107881,107884,107887],{"className":107822},[74],[39,107824],{"className":107825,"style":41560},[78],[39,107827,107829,107832],{"className":107828},[83],[39,107830,48449],{"className":107831,"style":438},[83,84],[39,107833,107835],{"className":107834},[192],[39,107836,107838,107858],{"className":107837},[196,649],[39,107839,107841,107855],{"className":107840},[200],[39,107842,107844],{"className":107843,"style":3419},[204],[39,107845,107846,107849],{"style":2937},[39,107847],{"className":107848,"style":213},[212],[39,107850,107852],{"className":107851},[217,218,219,220],[39,107853,74335],{"className":107854,"style":107705},[83,84,220],[39,107856,673],{"className":107857},[672],[39,107859,107861],{"className":107860},[200],[39,107862,107864],{"className":107863,"style":35396},[204],[39,107865],{},[39,107867,136],{"className":107868},[182],[39,107870,1755],{"className":107871},[83,84],[39,107873,15389],{"className":107874},[83],[39,107876,29625],{"className":107877,"style":2114},[83,84],[39,107879,161],{"className":107880},[263],[39,107882],{"className":107883,"style":231},[230],[39,107885,793],{"className":107886},[235],[39,107888],{"className":107889,"style":231},[230],[39,107891,107893,107896,107899,107902,107905,107908,107911,107914],{"className":107892},[74],[39,107894],{"className":107895,"style":245},[78],[39,107897,74335],{"className":107898,"style":107705},[83,84],[39,107900,136],{"className":107901},[182],[39,107903,1755],{"className":107904},[83,84],[39,107906,161],{"className":107907},[263],[39,107909],{"className":107910,"style":461},[230],[39,107912,5378],{"className":107913},[465],[39,107915],{"className":107916,"style":461},[230],[39,107918,107920,107923,107926,107929,107932,107935,107938,107941],{"className":107919},[74],[39,107921],{"className":107922,"style":245},[78],[39,107924,74335],{"className":107925,"style":107705},[83,84],[39,107927,136],{"className":107928},[182],[39,107930,29625],{"className":107931,"style":2114},[83,84],[39,107933,161],{"className":107934},[263],[39,107936],{"className":107937,"style":461},[230],[39,107939,5378],{"className":107940},[465],[39,107942],{"className":107943,"style":461},[230],[39,107945,107947,107950,107953,107956,107959,107962,107991,107994,107997,108000,108003],{"className":107946},[74],[39,107948],{"className":107949,"style":30346},[78],[39,107951,84725],{"className":107952},[83],[39,107954,74335],{"className":107955,"style":107705},[83,84],[39,107957,136],{"className":107958},[182],[39,107960,29625],{"className":107961,"style":2114},[83,84],[39,107963,107965,107968],{"className":107964},[263],[39,107966,161],{"className":107967},[263],[39,107969,107971],{"className":107970},[192],[39,107972,107974],{"className":107973},[196],[39,107975,107977],{"className":107976},[200],[39,107978,107980],{"className":107979,"style":3389},[204],[39,107981,107982,107985],{"style":1027},[39,107983],{"className":107984,"style":213},[212],[39,107986,107988],{"className":107987},[217,218,219,220],[39,107989,93341],{"className":107990},[83,220],[39,107992,136],{"className":107993},[182],[39,107995,1755],{"className":107996},[83,84],[39,107998],{"className":107999,"style":461},[230],[39,108001,5378],{"className":108002},[465],[39,108004],{"className":108005,"style":461},[230],[39,108007,108009,108012,108015,108018],{"className":108008},[74],[39,108010],{"className":108011,"style":245},[78],[39,108013,29625],{"className":108014,"style":2114},[83,84],[39,108016,161],{"className":108017},[263],[39,108019,2903],{"className":108020},[83],[11,108022,108023,108024,108052,108053,108081,108082,108110,108111,108139,108140,108191],{},"几何上，这是 ",[39,108025,108027,108040],{"className":108026,"translate":43},[42],[39,108028,108030],{"className":108029},[47],[49,108031,108032],{"xmlns":51},[53,108033,108034,108038],{},[56,108035,108036],{},[59,108037,74335],{},[63,108039,74335],{"encoding":65},[39,108041,108043],{"className":108042,"ariaHidden":70},[69],[39,108044,108046,108049],{"className":108045},[74],[39,108047],{"className":108048,"style":51280},[78],[39,108050,74335],{"className":108051,"style":107705},[83,84]," 的图像在 ",[39,108054,108056,108069],{"className":108055,"translate":43},[42],[39,108057,108059],{"className":108058},[47],[49,108060,108061],{"xmlns":51},[53,108062,108063,108067],{},[56,108064,108065],{},[59,108066,1755],{},[63,108068,1755],{"encoding":65},[39,108070,108072],{"className":108071,"ariaHidden":70},[69],[39,108073,108075,108078],{"className":108074},[74],[39,108076],{"className":108077,"style":290},[78],[39,108079,1755],{"className":108080},[83,84]," 处比 ",[39,108083,108085,108098],{"className":108084,"translate":43},[42],[39,108086,108088],{"className":108087},[47],[49,108089,108090],{"xmlns":51},[53,108091,108092,108096],{},[56,108093,108094],{},[59,108095,29625],{},[63,108097,29625],{"encoding":65},[39,108099,108101],{"className":108100,"ariaHidden":70},[69],[39,108102,108104,108107],{"className":108103},[74],[39,108105],{"className":108106,"style":2654},[78],[39,108108,29625],{"className":108109,"style":2114},[83,84]," 处的切平面高出多少。",[39,108112,108114,108127],{"className":108113,"translate":43},[42],[39,108115,108117],{"className":108116},[47],[49,108118,108119],{"xmlns":51},[53,108120,108121,108125],{},[56,108122,108123],{},[59,108124,74335],{},[63,108126,74335],{"encoding":65},[39,108128,108130],{"className":108129,"ariaHidden":70},[69],[39,108131,108133,108136],{"className":108132},[74],[39,108134],{"className":108135,"style":51280},[78],[39,108137,74335],{"className":108138,"style":107705},[83,84]," 凸所以非负，严格凸所以只在 ",[39,108141,108143,108161],{"className":108142,"translate":43},[42],[39,108144,108146],{"className":108145},[47],[49,108147,108148],{"xmlns":51},[53,108149,108150,108158],{},[56,108151,108152,108154,108156],{},[59,108153,1755],{},[133,108155,793],{},[59,108157,29625],{},[63,108159,108160],{"encoding":65},"x=y",[39,108162,108164,108182],{"className":108163,"ariaHidden":70},[69],[39,108165,108167,108170,108173,108176,108179],{"className":108166},[74],[39,108168],{"className":108169,"style":290},[78],[39,108171,1755],{"className":108172},[83,84],[39,108174],{"className":108175,"style":231},[230],[39,108177,793],{"className":108178},[235],[39,108180],{"className":108181,"style":231},[230],[39,108183,108185,108188],{"className":108184},[74],[39,108186],{"className":108187,"style":2654},[78],[39,108189,29625],{"className":108190,"style":2114},[83,84]," 时为零。它不对称，也不满足三角不等式，所以我不叫它距离，叫散度。",[11,108193,108194,108195,108377,108378,108545,108546,108763,108764,108545,108825,109001,109002,109185],{},"几个我常用的：",[39,108196,108198,108238],{"className":108197,"translate":43},[42],[39,108199,108201],{"className":108200},[47],[49,108202,108203],{"xmlns":51},[53,108204,108205,108235],{},[56,108206,108207,108209,108211,108213,108215,108217,108225,108227,108229],{},[59,108208,74335],{},[133,108210,136],{"stretchy":135},[59,108212,1755],{},[133,108214,161],{"stretchy":135},[133,108216,793],{},[770,108218,108219],{"scriptlevel":541,"displaystyle":135},[5362,108220,108221,108223],{},[539,108222,546],{},[539,108224,17359],{},[59,108226,15389],{"mathvariant":145},[59,108228,1755],{},[138,108230,108231,108233],{},[59,108232,15389],{"mathvariant":145},[539,108234,17359],{},[63,108236,108237],{"encoding":65},"f(x)=\\tfrac12\\|x\\|^2",[39,108239,108241,108268],{"className":108240,"ariaHidden":70},[69],[39,108242,108244,108247,108250,108253,108256,108259,108262,108265],{"className":108243},[74],[39,108245],{"className":108246,"style":245},[78],[39,108248,74335],{"className":108249,"style":107705},[83,84],[39,108251,136],{"className":108252},[182],[39,108254,1755],{"className":108255},[83,84],[39,108257,161],{"className":108258},[263],[39,108260],{"className":108261,"style":231},[230],[39,108263,793],{"className":108264},[235],[39,108266],{"className":108267,"style":231},[230],[39,108269,108271,108274,108342,108345,108348],{"className":108270},[74],[39,108272],{"className":108273,"style":74794},[78],[39,108275,108277,108280,108339],{"className":108276},[83],[39,108278],{"className":108279},[182,5463],[39,108281,108283],{"className":108282},[5362],[39,108284,108286,108331],{"className":108285},[196,649],[39,108287,108289,108328],{"className":108288},[200],[39,108290,108292,108306,108314],{"className":108291,"style":74813},[204],[39,108293,108294,108297],{"style":74816},[39,108295],{"className":108296,"style":935},[212],[39,108298,108300],{"className":108299},[217,218,219,220],[39,108301,108303],{"className":108302},[83,220],[39,108304,17359],{"className":108305},[83,220],[39,108307,108308,108311],{"style":5503},[39,108309],{"className":108310,"style":935},[212],[39,108312],{"className":108313,"style":5511},[5510],[39,108315,108316,108319],{"style":74839},[39,108317],{"className":108318,"style":935},[212],[39,108320,108322],{"className":108321},[217,218,219,220],[39,108323,108325],{"className":108324},[83,220],[39,108326,546],{"className":108327},[83,220],[39,108329,673],{"className":108330},[672],[39,108332,108334],{"className":108333},[200],[39,108335,108337],{"className":108336,"style":74861},[204],[39,108338],{},[39,108340],{"className":108341},[263,5463],[39,108343,15389],{"className":108344},[83],[39,108346,1755],{"className":108347},[83,84],[39,108349,108351,108354],{"className":108350},[83],[39,108352,15389],{"className":108353},[83],[39,108355,108357],{"className":108356},[192],[39,108358,108360],{"className":108359},[196],[39,108361,108363],{"className":108362},[200],[39,108364,108366],{"className":108365,"style":19710},[204],[39,108367,108368,108371],{"style":208},[39,108369],{"className":108370,"style":213},[212],[39,108372,108374],{"className":108373},[217,218,219,220],[39,108375,17359],{"className":108376},[83,220]," 给 ",[39,108379,108381,108415],{"className":108380,"translate":43},[42],[39,108382,108384],{"className":108383},[47],[49,108385,108386],{"xmlns":51},[53,108387,108388,108412],{},[56,108389,108390,108398,108400,108402,108404,108406],{},[770,108391,108392],{"scriptlevel":541,"displaystyle":135},[5362,108393,108394,108396],{},[539,108395,546],{},[539,108397,17359],{},[59,108399,15389],{"mathvariant":145},[59,108401,1755],{},[133,108403,5378],{},[59,108405,29625],{},[138,108407,108408,108410],{},[59,108409,15389],{"mathvariant":145},[539,108411,17359],{},[63,108413,108414],{"encoding":65},"\\tfrac12\\|x-y\\|^2",[39,108416,108418,108507],{"className":108417,"ariaHidden":70},[69],[39,108419,108421,108424,108492,108495,108498,108501,108504],{"className":108420},[74],[39,108422],{"className":108423,"style":74794},[78],[39,108425,108427,108430,108489],{"className":108426},[83],[39,108428],{"className":108429},[182,5463],[39,108431,108433],{"className":108432},[5362],[39,108434,108436,108481],{"className":108435},[196,649],[39,108437,108439,108478],{"className":108438},[200],[39,108440,108442,108456,108464],{"className":108441,"style":74813},[204],[39,108443,108444,108447],{"style":74816},[39,108445],{"className":108446,"style":935},[212],[39,108448,108450],{"className":108449},[217,218,219,220],[39,108451,108453],{"className":108452},[83,220],[39,108454,17359],{"className":108455},[83,220],[39,108457,108458,108461],{"style":5503},[39,108459],{"className":108460,"style":935},[212],[39,108462],{"className":108463,"style":5511},[5510],[39,108465,108466,108469],{"style":74839},[39,108467],{"className":108468,"style":935},[212],[39,108470,108472],{"className":108471},[217,218,219,220],[39,108473,108475],{"className":108474},[83,220],[39,108476,546],{"className":108477},[83,220],[39,108479,673],{"className":108480},[672],[39,108482,108484],{"className":108483},[200],[39,108485,108487],{"className":108486,"style":74861},[204],[39,108488],{},[39,108490],{"className":108491},[263,5463],[39,108493,15389],{"className":108494},[83],[39,108496,1755],{"className":108497},[83,84],[39,108499],{"className":108500,"style":461},[230],[39,108502,5378],{"className":108503},[465],[39,108505],{"className":108506,"style":461},[230],[39,108508,108510,108513,108516],{"className":108509},[74],[39,108511],{"className":108512,"style":84615},[78],[39,108514,29625],{"className":108515,"style":2114},[83,84],[39,108517,108519,108522],{"className":108518},[83],[39,108520,15389],{"className":108521},[83],[39,108523,108525],{"className":108524},[192],[39,108526,108528],{"className":108527},[196],[39,108529,108531],{"className":108530},[200],[39,108532,108534],{"className":108533,"style":19710},[204],[39,108535,108536,108539],{"style":208},[39,108537],{"className":108538,"style":213},[212],[39,108540,108542],{"className":108541},[217,218,219,220],[39,108543,17359],{"className":108544},[83,220],"；",[39,108547,108549,108593],{"className":108548,"translate":43},[42],[39,108550,108552],{"className":108551},[47],[49,108553,108554],{"xmlns":51},[53,108555,108556,108590],{},[56,108557,108558,108560,108562,108564,108566,108568,108574,108580,108582,108584],{},[59,108559,74335],{},[133,108561,136],{"stretchy":135},[59,108563,11],{},[133,108565,161],{"stretchy":135},[133,108567,793],{},[613,108569,108570,108572],{},[133,108571,3191],{},[59,108573,13377],{},[613,108575,108576,108578],{},[59,108577,11],{},[59,108579,13377],{},[59,108581,92483],{},[133,108583,1690],{},[613,108585,108586,108588],{},[59,108587,11],{},[59,108589,13377],{},[63,108591,108592],{"encoding":65},"f(p)=\\sum_ip_i\\log p_i",[39,108594,108596,108623],{"className":108595,"ariaHidden":70},[69],[39,108597,108599,108602,108605,108608,108611,108614,108617,108620],{"className":108598},[74],[39,108600],{"className":108601,"style":245},[78],[39,108603,74335],{"className":108604,"style":107705},[83,84],[39,108606,136],{"className":108607},[182],[39,108609,11],{"className":108610},[83,84],[39,108612,161],{"className":108613},[263],[39,108615],{"className":108616,"style":231},[230],[39,108618,793],{"className":108619},[235],[39,108621],{"className":108622,"style":231},[230],[39,108624,108626,108629,108669,108672,108712,108715,108720,108723],{"className":108625},[74],[39,108627],{"className":108628,"style":93897},[78],[39,108630,108632,108635],{"className":108631},[1744],[39,108633,3191],{"className":108634,"style":3953},[1744,3339,3952],[39,108636,108638],{"className":108637},[192],[39,108639,108641,108661],{"className":108640},[196,649],[39,108642,108644,108658],{"className":108643},[200],[39,108645,108647],{"className":108646,"style":93922},[204],[39,108648,108649,108652],{"style":3969},[39,108650],{"className":108651,"style":213},[212],[39,108653,108655],{"className":108654},[217,218,219,220],[39,108656,13377],{"className":108657},[83,84,220],[39,108659,673],{"className":108660},[672],[39,108662,108664],{"className":108663},[200],[39,108665,108667],{"className":108666,"style":4025},[204],[39,108668],{},[39,108670],{"className":108671,"style":256},[230],[39,108673,108675,108678],{"className":108674},[83],[39,108676,11],{"className":108677},[83,84],[39,108679,108681],{"className":108680},[192],[39,108682,108684,108704],{"className":108683},[196,649],[39,108685,108687,108701],{"className":108686},[200],[39,108688,108690],{"className":108689,"style":13604},[204],[39,108691,108692,108695],{"style":966},[39,108693],{"className":108694,"style":213},[212],[39,108696,108698],{"className":108697},[217,218,219,220],[39,108699,13377],{"className":108700},[83,84,220],[39,108702,673],{"className":108703},[672],[39,108705,108707],{"className":108706},[200],[39,108708,108710],{"className":108709,"style":680},[204],[39,108711],{},[39,108713],{"className":108714,"style":256},[230],[39,108716,92721,108718],{"className":108717},[1744],[39,108719,39209],{"style":35078},[39,108721],{"className":108722,"style":256},[230],[39,108724,108726,108729],{"className":108725},[83],[39,108727,11],{"className":108728},[83,84],[39,108730,108732],{"className":108731},[192],[39,108733,108735,108755],{"className":108734},[196,649],[39,108736,108738,108752],{"className":108737},[200],[39,108739,108741],{"className":108740,"style":13604},[204],[39,108742,108743,108746],{"style":966},[39,108744],{"className":108745,"style":213},[212],[39,108747,108749],{"className":108748},[217,218,219,220],[39,108750,13377],{"className":108751},[83,84,220],[39,108753,673],{"className":108754},[672],[39,108756,108758],{"className":108757},[200],[39,108759,108761],{"className":108760,"style":680},[204],[39,108762],{},"（负熵）给 ",[39,108765,108767,108795],{"className":108766,"translate":43},[42],[39,108768,108770],{"className":108769},[47],[49,108771,108772],{"xmlns":51},[53,108773,108774,108792],{},[56,108775,108776,108782,108784,108786,108788,108790],{},[56,108777,108778,108780],{},[59,108779,77187],{"mathvariant":145},[59,108781,77190],{"mathvariant":145},[133,108783,136],{"stretchy":135},[59,108785,11],{},[59,108787,15389],{"mathvariant":145},[59,108789,100187],{},[133,108791,161],{"stretchy":135},[63,108793,108794],{"encoding":65},"\\mathrm{KL}(p\\|q)",[39,108796,108798],{"className":108797,"ariaHidden":70},[69],[39,108799,108801,108804,108810,108813,108816,108819,108822],{"className":108800},[74],[39,108802],{"className":108803,"style":245},[78],[39,108805,108807],{"className":108806},[83],[39,108808,77233],{"className":108809},[83,35077],[39,108811,136],{"className":108812},[182],[39,108814,11],{"className":108815},[83,84],[39,108817,15389],{"className":108818},[83],[39,108820,100187],{"className":108821,"style":2114},[83,84],[39,108823,161],{"className":108824},[263],[39,108826,108828,108868],{"className":108827,"translate":43},[42],[39,108829,108831],{"className":108830},[47],[49,108832,108833],{"xmlns":51},[53,108834,108835,108865],{},[56,108836,108837,108839,108841,108843,108845,108847,108849,108855,108857,108859],{},[59,108838,74335],{},[133,108840,136],{"stretchy":135},[59,108842,11],{},[133,108844,161],{"stretchy":135},[133,108846,793],{},[133,108848,5378],{},[613,108850,108851,108853],{},[133,108852,3191],{},[59,108854,13377],{},[59,108856,92483],{},[133,108858,1690],{},[613,108860,108861,108863],{},[59,108862,11],{},[59,108864,13377],{},[63,108866,108867],{"encoding":65},"f(p)=-\\sum_i\\log p_i",[39,108869,108871,108898],{"className":108870,"ariaHidden":70},[69],[39,108872,108874,108877,108880,108883,108886,108889,108892,108895],{"className":108873},[74],[39,108875],{"className":108876,"style":245},[78],[39,108878,74335],{"className":108879,"style":107705},[83,84],[39,108881,136],{"className":108882},[182],[39,108884,11],{"className":108885},[83,84],[39,108887,161],{"className":108888},[263],[39,108890],{"className":108891,"style":231},[230],[39,108893,793],{"className":108894},[235],[39,108896],{"className":108897,"style":231},[230],[39,108899,108901,108904,108907,108910,108950,108953,108958,108961],{"className":108900},[74],[39,108902],{"className":108903,"style":93897},[78],[39,108905,5378],{"className":108906},[83],[39,108908],{"className":108909,"style":256},[230],[39,108911,108913,108916],{"className":108912},[1744],[39,108914,3191],{"className":108915,"style":3953},[1744,3339,3952],[39,108917,108919],{"className":108918},[192],[39,108920,108922,108942],{"className":108921},[196,649],[39,108923,108925,108939],{"className":108924},[200],[39,108926,108928],{"className":108927,"style":93922},[204],[39,108929,108930,108933],{"style":3969},[39,108931],{"className":108932,"style":213},[212],[39,108934,108936],{"className":108935},[217,218,219,220],[39,108937,13377],{"className":108938},[83,84,220],[39,108940,673],{"className":108941},[672],[39,108943,108945],{"className":108944},[200],[39,108946,108948],{"className":108947,"style":4025},[204],[39,108949],{},[39,108951],{"className":108952,"style":256},[230],[39,108954,92721,108956],{"className":108955},[1744],[39,108957,39209],{"style":35078},[39,108959],{"className":108960,"style":256},[230],[39,108962,108964,108967],{"className":108963},[83],[39,108965,11],{"className":108966},[83,84],[39,108968,108970],{"className":108969},[192],[39,108971,108973,108993],{"className":108972},[196,649],[39,108974,108976,108990],{"className":108975},[200],[39,108977,108979],{"className":108978,"style":13604},[204],[39,108980,108981,108984],{"style":966},[39,108982],{"className":108983,"style":213},[212],[39,108985,108987],{"className":108986},[217,218,219,220],[39,108988,13377],{"className":108989},[83,84,220],[39,108991,673],{"className":108992},[672],[39,108994,108996],{"className":108995},[200],[39,108997,108999],{"className":108998,"style":680},[204],[39,109000],{}," 给 Itakura–Saito；",[39,109003,109005,109045],{"className":109004,"translate":43},[42],[39,109006,109008],{"className":109007},[47],[49,109009,109010],{"xmlns":51},[53,109011,109012,109042],{},[56,109013,109014,109016,109018,109020,109022,109024,109032,109038,109040],{},[59,109015,74335],{},[133,109017,136],{"stretchy":135},[59,109019,1755],{},[133,109021,161],{"stretchy":135},[133,109023,793],{},[770,109025,109026],{"scriptlevel":541,"displaystyle":135},[5362,109027,109028,109030],{},[539,109029,546],{},[539,109031,17359],{},[138,109033,109034,109036],{},[59,109035,1755],{},[59,109037,93341],{"mathvariant":145},[59,109039,101],{},[59,109041,1755],{},[63,109043,109044],{"encoding":65},"f(x)=\\tfrac12x^\\top 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给 Mahalanobis 距离。",[11,109187,109188],{},"它在机器学习里出现的频率比多数人以为的高。交叉熵",[39,109190,109192],{"className":109191,"translate":43},[746],[39,109193,109195,109312],{"className":109194,"translate":43},[42],[39,109196,109198],{"className":109197},[47],[49,109199,109200],{"xmlns":51,"display":755},[53,109201,109202,109309],{},[56,109203,109204,109210,109212,109214,109216,109218,109220,109222,109224,109230,109236,109238,109240,109246,109248,109254,109256,109258,109260,109262,109264,109266,109269,109271,109273,109275,109277,109287,109289,109291,109293,109295,109297,109299,109301,109303,109305,109307],{},[56,109205,109206,109208],{},[59,109207,48444],{"mathvariant":145},[59,109209,4253],{"mathvariant":145},[133,109211,136],{"stretchy":135},[59,109213,29625],{},[133,109215,155],{"separator":70},[59,109217,100187],{},[133,109219,161],{"stretchy":135},[133,109221,793],{},[133,109223,5378],{},[6556,109225,109226,109228],{},[133,109227,3191],{},[59,109229,13377],{},[613,109231,109232,109234],{},[59,109233,29625],{},[59,109235,13377],{},[59,109237,92483],{},[133,109239,1690],{},[613,109241,109242,109244],{},[59,109243,100187],{},[59,109245,13377],{},[133,109247,793],{},[56,109249,109250,109252],{},[59,109251,77187],{"mathvariant":145},[59,109253,77190],{"mathvariant":145},[133,109255,136],{"stretchy":135},[59,109257,29625],{},[59,109259,15389],{"mathvariant":145},[59,109261,100187],{},[133,109263,161],{"stretchy":135},[133,109265,788],{},[59,109267,109268],{},"H",[133,109270,136],{"stretchy":135},[59,109272,29625],{},[133,109274,161],{"stretchy":135},[133,109276,793],{},[613,109278,109279,109281],{},[59,109280,48449],{},[56,109282,109283,109285],{},[133,109284,5378],{},[59,109286,109268],{},[133,109288,136],{"stretchy":135},[59,109290,29625],{},[59,109292,15389],{"mathvariant":145},[59,109294,100187],{},[133,109296,161],{"stretchy":135},[133,109298,788],{},[59,109300,109268],{},[133,109302,136],{"stretchy":135},[59,109304,29625],{},[133,109306,161],{"stretchy":135},[133,109308,155],{"separator":70},[63,109310,109311],{"encoding":65},"\\mathrm{CE}(y,q)=-\\sum_iy_i\\log q_i=\\mathrm{KL}(y\\|q)+H(y)=D_{-H}(y\\|q)+H(y),",[39,109313,109315,109355,109514,109550,109578,109654],{"className":109314,"ariaHidden":70},[69],[39,109316,109318,109321,109328,109331,109334,109337,109340,109343,109346,109349,109352],{"className":109317},[74],[39,109319],{"className":109320,"style":245},[78],[39,109322,109324],{"className":109323},[83],[39,109325,109327],{"className":109326},[83,35077],"CE",[39,109329,136],{"className":109330},[182],[39,109332,29625],{"className":109333,"style":2114},[83,84],[39,109335,155],{"className":109336},[252],[39,109338],{"className":109339,"style":256},[230],[39,109341,100187],{"className":109342,"style":2114},[83,84],[39,109344,161],{"className":109345},[263],[39,109347],{"className":109348,"style":231},[230],[39,109350,793],{"className":109351},[235],[39,109353],{"className":109354,"style":231},[230],[39,109356,109358,109361,109364,109367,109411,109414,109454,109457,109462,109465,109505,109508,109511],{"className":109357},[74],[39,109359],{"className":109360,"style":82834},[78],[39,109362,5378],{"className":109363},[83],[39,109365],{"className":109366,"style":256},[230],[39,109368,109370],{"className":109369},[1744,3294],[39,109371,109373,109403],{"className":109372},[196,649],[39,109374,109376,109400],{"className":109375},[200],[39,109377,109379,109390],{"className":109378,"style":6670},[204],[39,109380,109381,109384],{"style":13926},[39,109382],{"className":109383,"style":3311},[212],[39,109385,109387],{"className":109386},[217,218,219,220],[39,109388,13377],{"className":109389},[83,84,220],[39,109391,109392,109395],{"style":3330},[39,109393],{"className":109394,"style":3311},[212],[39,109396,109397],{},[39,109398,3191],{"className":109399},[1744,3339,3340],[39,109401,673],{"className":109402},[672],[39,109404,109406],{"className":109405},[200],[39,109407,109409],{"className":109408,"style":78935},[204],[39,109410],{},[39,109412],{"className":109413,"style":256},[230],[39,109415,109417,109420],{"className":109416},[83],[39,109418,29625],{"className":109419,"style":2114},[83,84],[39,109421,109423],{"className":109422},[192],[39,109424,109426,109446],{"className":109425},[196,649],[39,109427,109429,109443],{"className":109428},[200],[39,109430,109432],{"className":109431,"style":13604},[204],[39,109433,109434,109437],{"style":41316},[39,109435],{"className":109436,"style":213},[212],[39,109438,109440],{"className":109439},[217,218,219,220],[39,109441,13377],{"className":109442},[83,84,220],[39,109444,673],{"className":109445},[672],[39,109447,109449],{"className":109448},[200],[39,109450,109452],{"className":109451,"style":680},[204],[39,109453],{},[39,109455],{"className":109456,"style":256},[230],[39,109458,92721,109460],{"className":109459},[1744],[39,109461,39209],{"style":35078},[39,109463],{"className":109464,"style":256},[230],[39,109466,109468,109471],{"className":109467},[83],[39,109469,100187],{"className":109470,"style":2114},[83,84],[39,109472,109474],{"className":109473},[192],[39,109475,109477,109497],{"className":109476},[196,649],[39,109478,109480,109494],{"className":109479},[200],[39,109481,109483],{"className":109482,"style":13604},[204],[39,109484,109485,109488],{"style":41316},[39,109486],{"className":109487,"style":213},[212],[39,109489,109491],{"className":109490},[217,218,219,220],[39,109492,13377],{"className":109493},[83,84,220],[39,109495,673],{"className":109496},[672],[39,109498,109500],{"className":109499},[200],[39,109501,109503],{"className":109502,"style":680},[204],[39,109504],{},[39,109506],{"className":109507,"style":231},[230],[39,109509,793],{"className":109510},[235],[39,109512],{"className":109513,"style":231},[230],[39,109515,109517,109520,109526,109529,109532,109535,109538,109541,109544,109547],{"className":109516},[74],[39,109518],{"className":109519,"style":245},[78],[39,109521,109523],{"className":109522},[83],[39,109524,77233],{"className":109525},[83,35077],[39,109527,136],{"className":109528},[182],[39,109530,29625],{"className":109531,"style":2114},[83,84],[39,109533,15389],{"className":109534},[83],[39,109536,100187],{"className":109537,"style":2114},[83,84],[39,109539,161],{"className":109540},[263],[39,109542],{"className":109543,"style":461},[230],[39,109545,788],{"className":109546},[465],[39,109548],{"className":109549,"style":461},[230],[39,109551,109553,109556,109560,109563,109566,109569,109572,109575],{"className":109552},[74],[39,109554],{"className":109555,"style":245},[78],[39,109557,109268],{"className":109558,"style":109559},[83,84],"margin-right:0.0813em;",[39,109561,136],{"className":109562},[182],[39,109564,29625],{"className":109565,"style":2114},[83,84],[39,109567,161],{"className":109568},[263],[39,109570],{"className":109571,"style":231},[230],[39,109573,793],{"className":109574},[235],[39,109576],{"className":109577,"style":231},[230],[39,109579,109581,109584,109630,109633,109636,109639,109642,109645,109648,109651],{"className":109580},[74],[39,109582],{"className":109583,"style":245},[78],[3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散度，",[39,109755,109757],{"className":109756,"translate":43},[746],[39,109758,109760,109836],{"className":109759,"translate":43},[42],[39,109761,109763],{"className":109762},[47],[49,109764,109765],{"xmlns":51,"display":755},[53,109766,109767,109833],{},[56,109768,109769,109771,109773,109783,109785,109791,109797,109799,109805,109807,109809,109811,109813,109819,109825,109831],{},[59,109770,39047],{},[133,109772,1690],{},[6556,109774,109775,109781],{},[56,109776,109777,109779],{},[59,109778,92422],{},[133,109780,1690],{},[59,109782,24077],{},[35841,109784,74223],{},[6556,109786,109787,109789],{},[133,109788,3191],{},[59,109790,13377],{},[613,109792,109793,109795],{},[59,109794,48449],{},[59,109796,74335],{},[133,109798,136],{"stretchy":135},[613,109800,109801,109803],{},[59,109802,1755],{},[59,109804,13377],{},[59,109806,15389],{"mathvariant":145},[59,109808,24077],{},[133,109810,161],{"stretchy":135},[133,109812,793],{},[5362,109814,109815,109817],{},[539,109816,546],{},[59,109818,12962],{},[6556,109820,109821,109823],{},[133,109822,3191],{},[59,109824,13377],{},[613,109826,109827,109829],{},[59,109828,1755],{},[59,109830,13377],{},[59,109832,2903],{"mathvariant":145},[63,109834,109835],{"encoding":65},"\\arg\\min_c\\ \\sum_iD_f(x_i\\|c)=\\frac1n\\sum_ix_i 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ame":110119},[263,5463],[39,110121],{"className":110122,"style":256},[230],[39,110124,110126],{"className":110125},[1744,3294],[39,110127,110129,110159],{"className":110128},[196,649],[39,110130,110132,110156],{"className":110131},[200],[39,110133,110135,110146],{"className":110134,"style":6670},[204],[39,110136,110137,110140],{"style":13926},[39,110138],{"className":110139,"style":3311},[212],[39,110141,110143],{"className":110142},[217,218,219,220],[39,110144,13377],{"className":110145},[83,84,220],[39,110147,110148,110151],{"style":3330},[39,110149],{"className":110150,"style":3311},[212],[39,110152,110153],{},[39,110154,3191],{"className":110155},[1744,3339,3340],[39,110157,673],{"className":110158},[672],[39,110160,110162],{"className":110161},[200],[39,110163,110165],{"className":110164,"style":78935},[204],[39,110166],{},[39,110168],{"className":110169,"style":256},[230],[39,110171,110173,110176],{"className":110172},[83],[39,110174,1755],{"className":110175},[83,84],[39,110177,110179],{"className":110178},[192],[39,110180,110182,110202],{"className":110181},[196,649],[39,110183,110185,110199],{"className":110184},[200],[39,110186,110188],{"className":110187,"style":13604},[204],[39,110189,110190,110193],{"style":966},[39,110191],{"className":110192,"style":213},[212],[39,110194,110196],{"className":110195},[217,218,219,220],[39,110197,13377],{"className":110198},[83,84,220],[39,110200,673],{"className":110201},[672],[39,110203,110205],{"className":110204},[200],[39,110206,110208],{"className":110207,"style":680},[204],[39,110209],{},[39,110211,2903],{"className":110212},[83],[11,110214,110215,110216,110445],{},"求导得到 ",[39,110217,110219,110269],{"className":110218,"translate":43},[42],[39,110220,110222],{"className":110221},[47],[49,110223,110224],{"xmlns":51},[53,110225,110226,110266],{},[56,110227,110228,110234,110240,110242,110244,110246,110248,110250,110256,110258,110260,110262,110264],{},[613,110229,110230,110232],{},[133,110231,3191],{},[59,110233,13377],{},[138,110235,110236,110238],{},[59,110237,84725],{"mathvariant":145},[539,110239,17359],{},[59,110241,74335],{},[133,110243,136],{"stretchy":135},[59,110245,24077],{},[133,110247,161],{"stretchy":135},[133,110249,136],{"stretchy":135},[613,110251,110252,110254],{},[59,110253,1755],{},[59,110255,13377],{},[133,110257,5378],{},[59,110259,24077],{},[133,110261,161],{"stretchy":135},[133,110263,793],{},[539,110265,541],{},[63,110267,110268],{"encoding":65},"\\sum_i\\nabla^2f(c)(x_i-c)=0",[39,110270,110272,110415,110436],{"className":110271,"ariaHidden":70},[69],[39,110273,110275,110279,110319,110322,110351,110354,110357,110360,110363,110366,110406,110409,110412],{"className":110274},[74],[39,110276],{"className":110277,"style":110278},[78],"height:1.1138em;vertical-align:-0.2997em;",[39,110280,110282,110285],{"className":110281},[1744],[39,110283,3191],{"className":110284,"style":3953},[1744,3339,3952],[39,110286,110288],{"className":110287},[192],[39,110289,110291,110311],{"className":110290},[196,649],[39,110292,110294,110308],{"className":110293},[200],[39,110295,110297],{"className":110296,"style":93922},[204],[39,110298,110299,110302],{"style":3969},[39,110300],{"className":110301,"style":213},[212],[39,110303,110305],{"className":110304},[217,218,219,220],[39,110306,13377],{"className":110307},[83,84,220],[39,110309,673],{"className":110310},[672],[39,110312,110314],{"className":110313},[200],[39,110315,110317],{"className":110316,"style":4025},[204],[39,110318],{},[39,110320],{"className":110321,"style":256},[230],[39,110323,110325,110328],{"className":110324},[83],[39,110326,84725],{"className":110327},[83],[39,110329,110331],{"className":110330},[192],[39,110332,110334],{"className":110333},[196],[39,110335,110337],{"className":110336},[200],[39,110338,110340],{"className":110339,"style":19710},[204],[39,110341,110342,110345],{"style":208},[39,110343],{"className":110344,"style":213},[212],[39,110346,110348],{"className":110347},[217,218,219,220],[39,110349,17359],{"className":110350},[83,220],[39,110352,74335],{"className":110353,"style":107705},[83,84],[39,110355,136],{"className":110356},[182],[39,110358,24077],{"className":110359},[83,84],[39,110361,161],{"className":110362},[263],[39,110364,136],{"className":110365},[182],[39,110367,110369,110372],{"className":110368},[83],[39,110370,1755],{"className":110371},[83,84],[39,110373,110375],{"className":110374},[192],[39,110376,110378,110398],{"className":110377},[196,649],[39,110379,110381,110395],{"className":110380},[200],[39,110382,110384],{"className":110383,"style":13604},[204],[39,110385,110386,110389],{"style":966},[39,110387],{"className":110388,"style":213},[212],[39,110390,110392],{"className":110391},[217,218,219,220],[39,110393,13377],{"className":110394},[83,84,220],[39,110396,673],{"className":110397},[672],[39,110399,110401],{"className":110400},[200],[39,110402,110404],{"className":110403,"style":680},[204],[39,110405],{},[39,110407],{"className":110408,"style":461},[230],[39,110410,5378],{"className":110411},[465],[39,110413],{"className":110414,"style":461},[230],[39,110416,110418,110421,110424,110427,110430,110433],{"className":110417},[74],[39,110419],{"className":110420,"style":245},[78],[39,110422,24077],{"className":110423},[83,84],[39,110425,161],{"className":110426},[263],[39,110428],{"className":110429,"style":231},[230],[39,110431,793],{"className":110432},[235],[39,110434],{"className":110435,"style":231},[230],[39,110437,110439,110442],{"className":110438},[74],[39,110440],{"className":110441,"style":2473},[78],[39,110443,541],{"className":110444},[83],"，结论就是这一行。所以换散度不会改变“平均”这件事，但会改变“谁离谁近”这件事。",[11,110447,110448,110449,110573,110574,1639,110710,110846],{},"这一点值得展开，因为它关系到分类边界。设两个类原型 ",[39,110450,110452,110478],{"className":110451,"translate":43},[42],[39,110453,110455],{"className":110454},[47],[49,110456,110457],{"xmlns":51},[53,110458,110459,110475],{},[56,110460,110461,110467,110469],{},[613,110462,110463,110465],{},[59,110464,24077],{},[539,110466,546],{},[133,110468,155],{"separator":70},[613,110470,110471,110473],{},[59,110472,24077],{},[539,110474,17359],{},[63,110476,110477],{"encoding":65},"c_1,c_2",[39,110479,110481],{"className":110480,"ariaHidden":70},[69],[39,110482,110484,110487,110527,110530,110533],{"className":110483},[74],[39,110485],{"className":110486,"style":2654},[78],[39,110488,110490,110493],{"className":110489},[83],[39,110491,24077],{"className":110492},[83,84],[39,110494,110496],{"className":110495},[192],[39,110497,110499,110519],{"className":110498},[196,649],[39,110500,110502,110516],{"className":110501},[200],[39,110503,110505],{"className":110504,"style":963},[204],[39,110506,110507,110510],{"style":966},[39,110508],{"className":110509,"style":213},[212],[39,110511,110513],{"className":110512},[217,218,219,220],[39,110514,546],{"className":110515},[83,220],[39,110517,673],{"className":110518},[672],[39,110520,110522],{"className":110521},[200],[39,110523,110525],{"className":110524,"style":680},[204],[39,110526],{},[39,110528,155],{"className":110529},[252],[39,110531],{"className":110532,"style":256},[230],[39,110534,110536,110539],{"className":110535},[83],[39,110537,24077],{"className":110538},[83,84],[39,110540,110542],{"className":110541},[192],[39,110543,110545,110565],{"className":110544},[196,649],[39,110546,110548,110562],{"className":110547},[200],[39,110549,110551],{"className":110550,"style":963},[204],[39,110552,110553,110556],{"style":966},[39,110554],{"className":110555,"style":213},[212],[39,110557,110559],{"className":110558},[217,218,219,220],[39,110560,17359],{"className":110561},[83,220],[39,110563,673],{"className":110564},[672],[39,110566,110568],{"className":110567},[200],[39,110569,110571],{"className":110570,"style":680},[204],[39,110572],{},"，最近原型判类就是比较 ",[39,110575,110577,110609],{"className":110576,"translate":43},[42],[39,110578,110580],{"className":110579},[47],[49,110581,110582],{"xmlns":51},[53,110583,110584,110606],{},[56,110585,110586,110592,110594,110596,110598,110604],{},[613,110587,110588,110590],{},[59,110589,48449],{},[59,110591,74335],{},[133,110593,136],{"stretchy":135},[59,110595,1755],{},[59,110597,15389],{"mathvariant":145},[613,110599,110600,110602],{},[59,110601,24077],{},[539,110603,546],{},[133,110605,161],{"stretchy":135},[63,110607,110608],{"encoding":65},"D_f(x\\|c_1)",[39,110610,110612],{"className":110611,"ariaHidden":70},[69],[39,110613,110615,110618,110658,110661,110664,110667,110707],{"className":110614},[74],[39,110616],{"className":110617,"style":41560},[78],[39,110619,110621,110624],{"className":110620},[83],[39,110622,48449],{"className":110623,"style":438},[83,84],[39,110625,110627],{"className":110626},[192],[39,110628,110630,110650],{"className":110629},[196,649],[39,110631,110633,110647],{"className":110632},[200],[39,110634,110636],{"className":110635,"style":3419},[204],[39,110637,110638,110641],{"style":2937},[39,110639],{"className":110640,"style":213},[212],[39,110642,110644],{"className":110643},[217,218,219,220],[39,110645,74335],{"className":110646,"style":107705},[83,84,220],[39,110648,673],{"className":110649},[672],[39,110651,110653],{"className":110652},[200],[39,110654,110656],{"className":110655,"style":35396},[204],[39,110657],{},[39,110659,136],{"className":110660},[182],[39,110662,1755],{"className":110663},[83,84],[39,110665,15389],{"className":110666},[83],[39,110668,110670,110673],{"className":110669},[83],[39,110671,24077],{"className":110672},[83,84],[39,110674,110676],{"className":110675},[192],[39,110677,110679,110699],{"className":110678},[196,649],[39,110680,110682,110696],{"className":110681},[200],[39,110683,110685],{"className":110684,"style":963},[204],[39,110686,110687,110690],{"style":966},[39,110688],{"className":110689,"style":213},[212],[39,110691,110693],{"className":110692},[217,218,219,220],[39,110694,546],{"className":110695},[83,220],[39,110697,673],{"className":110698},[672],[39,110700,110702],{"className":110701},[200],[39,110703,110705],{"className":110704,"style":680},[204],[39,110706],{},[39,110708,161],{"className":110709},[263],[39,110711,110713,110745],{"className":110712,"translate":43},[42],[39,110714,110716],{"className":110715},[47],[49,110717,110718],{"xmlns":51},[53,110719,110720,110742],{},[56,110721,110722,110728,110730,110732,110734,110740],{},[613,110723,110724,110726],{},[59,110725,48449],{},[59,110727,74335],{},[133,110729,136],{"stretchy":135},[59,110731,1755],{},[59,110733,15389],{"mathvariant":145},[613,110735,110736,110738],{},[59,110737,24077],{},[539,110739,17359],{},[133,110741,161],{"stretchy":135},[63,110743,110744],{"encoding":65},"D_f(x\\|c_2)",[39,110746,110748],{"className":110747,"ariaHidden":70},[69],[39,110749,110751,110754,110794,110797,110800,110803,110843],{"className":110750},[74],[39,110752],{"className":110753,"style":41560},[78],[39,110755,110757,110760],{"className":110756},[83],[39,110758,48449],{"className":110759,"style":438},[83,84],[39,110761,110763],{"className":110762},[192],[39,110764,110766,110786],{"className":110765},[196,649],[39,110767,110769,110783],{"className":110768},[200],[39,110770,110772],{"className":110771,"style":3419},[204],[39,110773,110774,110777],{"style":2937},[39,110775],{"className":110776,"style":213},[212],[39,110778,110780],{"className":110779},[217,218,219,220],[39,110781,74335],{"className":110782,"style":107705},[83,84,220],[39,110784,673],{"className":110785},[672],[39,110787,110789],{"className":110788},[200],[39,110790,110792],{"className":110791,"style":35396},[204],[39,110793],{},[39,110795,136],{"className":110796},[182],[39,110798,1755],{"className":110799},[83,84],[39,110801,15389],{"className":110802},[83],[39,110804,110806,110809],{"className":110805},[83],[39,110807,24077],{"className":110808},[83,84],[39,110810,110812],{"className":110811},[192],[39,110813,110815,110835],{"className":110814},[196,649],[39,110816,110818,110832],{"className":110817},[200],[39,110819,110821],{"className":110820,"style":963},[204],[39,110822,110823,110826],{"style":966},[39,110824],{"className":110825,"style":213},[212],[39,110827,110829],{"className":110828},[217,218,219,220],[39,110830,17359],{"className":110831},[83,220],[39,110833,673],{"className":110834},[672],[39,110836,110838],{"className":110837},[200],[39,110839,110841],{"className":110840,"style":680},[204],[39,110842],{},[39,110844,161],{"className":110845},[263],"。把定义代进去，",[39,110848,110850],{"className":110849,"translate":43},[746],[39,110851,110853,111217],{"className":110852,"translate":43},[42],[39,110854,110856],{"className":110855},[47],[49,110857,110858],{"xmlns":51,"display":755},[53,110859,110860,111214],{},[759,110861,110862,110920,111060],{"rowspacing":761,"columnalign":762,"columnspacing":146},[764,110863,110864,110870],{},[767,110865,110866],{},[770,110867,110868],{"scriptlevel":541,"displaystyle":70},[56,110869],{},[767,110871,110872],{},[770,110873,110874],{"scriptlevel":541,"displaystyle":70},[56,110875,110876,110878,110884,110886,110888,110890,110896,110898,110900,110906,110908,110910,110912,110918],{},[56,110877],{},[613,110879,110880,110882],{},[59,110881,48449],{},[59,110883,74335],{},[133,110885,136],{"stretchy":135},[59,110887,1755],{},[59,110889,15389],{"mathvariant":145},[613,110891,110892,110894],{},[59,110893,24077],{},[539,110895,546],{},[133,110897,161],{"stretchy":135},[133,110899,5378],{},[613,110901,110902,110904],{},[59,110903,48449],{},[59,110905,74335],{},[133,110907,136],{"stretchy":135},[59,110909,1755],{},[59,110911,15389],{"mathvariant":145},[613,110913,110914,110916],{},[59,110915,24077],{},[539,110917,17359],{},[133,110919,161],{"stretchy":135},[764,110921,110922,110928],{},[767,110923,110924],{},[770,110925,110926],{"scriptlevel":541,"displaystyle":70},[56,110927],{},[767,110929,110930],{},[770,110931,110932],{"scriptlevel":541,"displaystyle":70},[56,110933,110934,110936,110938,110940,110942,110944,110946,110948,110950,110952,110954,110960,110962,110964,110966,110968,110970,110976,110982,110984,110986,110988,110994,110996,110998,111000,111002,111004,111006,111008,111010,111012,111014,111016,111022,111024,111026,111028,111030,111032,111038,111044,111046,111048,111050,111056,111058],{},[56,110935],{},[133,110937,793],{},[133,110939,537],{"fence":70,"stretchy":70,"minsize":73875,"maxsize":73875},[59,110941,74335],{},[133,110943,136],{"stretchy":135},[59,110945,1755],{},[133,110947,161],{"stretchy":135},[133,110949,5378],{},[59,110951,74335],{},[133,110953,136],{"stretchy":135},[613,110955,110956,110958],{},[59,110957,24077],{},[539,110959,546],{},[133,110961,161],{"stretchy":135},[133,110963,5378],{},[59,110965,84725],{"mathvariant":145},[59,110967,74335],{},[133,110969,136],{"stretchy":135},[613,110971,110972,110974],{},[59,110973,24077],{},[539,110975,546],{},[138,110977,110978,110980],{},[133,110979,161],{"stretchy":135},[59,110981,93341],{"mathvariant":145},[133,110983,136],{"stretchy":135},[59,110985,1755],{},[133,110987,5378],{},[613,110989,110990,110992],{},[59,110991,24077],{},[539,110993,546],{},[133,110995,161],{"stretchy":135},[133,110997,4280],{"fence":70,"stretchy":70,"minsize":73875,"maxsize":73875},[133,110999,5378],{},[133,111001,537],{"fence":70,"stretchy":70,"minsize":73875,"maxsize":73875},[59,111003,74335],{},[133,111005,136],{"stretchy":135},[59,111007,1755],{},[133,111009,161],{"stretchy":135},[133,111011,5378],{},[59,111013,74335],{},[133,111015,136],{"stretchy":135},[613,111017,111018,111020],{},[59,111019,24077],{},[539,111021,17359],{},[133,111023,161],{"stretchy":135},[133,111025,5378],{},[59,111027,84725],{"mathvariant":145},[59,111029,74335],{},[133,111031,136],{"stretchy":135},[613,111033,111034,111036],{},[59,111035,24077],{},[539,111037,17359],{},[138,111039,111040,111042],{},[133,111041,161],{"stretchy":135},[59,111043,93341],{"mathvariant":145},[133,111045,136],{"stretchy":135},[59,111047,1755],{},[133,111049,5378],{},[613,111051,111052,111054],{},[59,111053,24077],{},[539,111055,17359],{},[133,111057,161],{"stretchy":135},[133,111059,4280],{"fence":70,"stretchy":70,"minsize":73875,"maxsize":73875},[764,111061,111062,111068],{},[767,111063,111064],{},[770,111065,111066],{"scriptlevel":541,"displaystyle":70},[56,111067],{},[767,111069,111070],{},[770,111071,111072],{"scriptlevel":541,"displaystyle":70},[56,111073,111074,111076,111078,111170,111172,111174,111176,111178,111180,111186,111188,111190,111192,111194,111196,111202,111204,111210,111212],{},[56,111075],{},[133,111077,793],{},[6556,111079,111080,111167],{},[6556,111081,111082,111164],{},[56,111083,111084,111086,111088,111090,111096,111098,111100,111102,111104,111110,111112,111114,111116,111118,111120,111126,111132,111138,111140,111142,111144,111146,111152,111158],{},[133,111085,5378],{},[59,111087,74335],{},[133,111089,136],{"stretchy":135},[613,111091,111092,111094],{},[59,111093,24077],{},[539,111095,546],{},[133,111097,161],{"stretchy":135},[133,111099,788],{},[59,111101,74335],{},[133,111103,136],{"stretchy":135},[613,111105,111106,111108],{},[59,111107,24077],{},[539,111109,17359],{},[133,111111,161],{"stretchy":135},[133,111113,788],{},[59,111115,84725],{"mathvariant":145},[59,111117,74335],{},[133,111119,136],{"stretchy":135},[613,111121,111122,111124],{},[59,111123,24077],{},[539,111125,546],{},[138,111127,111128,111130],{},[133,111129,161],{"stretchy":135},[59,111131,93341],{"mathvariant":145},[613,111133,111134,111136],{},[59,111135,24077],{},[539,111137,546],{},[133,111139,5378],{},[59,111141,84725],{"mathvariant":145},[59,111143,74335],{},[133,111145,136],{"stretchy":135},[613,111147,111148,111150],{},[59,111149,24077],{},[539,111151,17359],{},[138,111153,111154,111156],{},[133,111155,161],{"stretchy":135},[59,111157,93341],{"mathvariant":145},[613,111159,111160,111162],{},[59,111161,24077],{},[539,111163,17359],{},[133,111165,111166],{"stretchy":70},"⏟",[35841,111168,111169],{},"常数",[133,111171,788],{},[133,111173,136],{"fence":70,"stretchy":70,"minsize":73875,"maxsize":73875},[59,111175,84725],{"mathvariant":145},[59,111177,74335],{},[133,111179,136],{"stretchy":135},[613,111181,111182,111184],{},[59,111183,24077],{},[539,111185,17359],{},[133,111187,161],{"stretchy":135},[133,111189,5378],{},[59,111191,84725],{"mathvariant":145},[59,111193,74335],{},[133,111195,136],{"stretchy":135},[613,111197,111198,111200],{},[59,111199,24077],{},[539,111201,546],{},[133,111203,161],{"stretchy":135},[138,111205,111206,111208],{},[133,111207,161],{"fence":70,"stretchy":70,"minsize":73875,"maxsize":73875},[59,111209,93341],{"mathvariant":145},[59,111211,1755],{},[59,111213,2903],{"mathvariant":145},[63,111215,111216],{"encoding":65},"\\begin{aligned}\n&D_f(x\\|c_1)-D_f(x\\|c_2)\\\\\n&=\\bigl[f(x)-f(c_1)-\\nabla f(c_1)^\\top(x-c_1)\\bigr]\n -\\bigl[f(x)-f(c_2)-\\nabla f(c_2)^\\top(x-c_2)\\bigr]\\\\\n&=\\underbrace{-f(c_1)+f(c_2)+\\nabla f(c_1)^\\top c_1-\\nabla f(c_2)^\\top c_2}_{\\text{常数}}\n +\\bigl(\\nabla f(c_2)-\\nabla f(c_1)\\bigr)^\\top x .\n\\end{aligned}",[39,111218,111220],{"className":111219,"ariaHidden":70},[69],[39,111221,111223,111227],{"className":111222},[74],[39,111224],{"className":111225,"style":111226},[78],"height:5.7244em;vertical-align:-2.6122em;",[39,111228,111230],{"className":111229},[83],[39,111231,111233,111286],{"className":111232},[759],[39,111234,111236],{"className":111235},[918],[39,111237,111239,111277],{"className":111238},[196,649],[39,111240,111242,111274],{"className":111241},[200],[39,111243,111246,111256,111265],{"className":111244,"style":111245},[204],"height:3.1122em;",[39,111247,111249,111253],{"style":111248},"top:-5.3612em;",[39,111250],{"className":111251,"style":111252},[212],"height:3.089em;",[39,111254],{"className":111255},[83],[39,111257,111259,111262],{"style":111258},"top:-3.8021em;",[39,111260],{"className":111261,"style":111252},[212],[39,111263],{"className":111264},[83],[39,111266,111268,111271],{"style":111267},"top:-2.0531em;",[39,111269],{"className":111270,"style":111252},[212],[39,111272],{"className":111273},[83],[39,111275,673],{"className":111276},[672],[39,111278,111280],{"className":111279},[200],[39,111281,111284],{"className":111282,"style":111283},[204],"height:2.6122em;",[39,111285],{},[39,111287,111289],{"className":111288},[1223],[39,111290,111292,112662],{"className":111291},[196,649],[39,111293,111295,112659],{"className":111294},[200],[39,111296,111298,111502,111985],{"className":111297,"style":111245},[204],[39,111299,111300,111303],{"style":111248},[39,111301],{"className":111302,"style":111252},[212],[39,111304,111306,111309,111349,111352,111355,111358,111398,111401,111404,111407,111410,111450,111453,111456,111459,111499],{"className":111305},[83],[39,111307],{"className":111308},[83],[39,111310,111312,111315],{"className":111311},[83],[39,111313,48449],{"className":111314,"style":438},[83,84],[39,111316,111318],{"className":111317},[192],[39,111319,111321,111341],{"className":111320},[196,649],[39,111322,111324,111338],{"className":111323},[200],[39,111325,111327],{"className":111326,"style":3419},[204],[39,111328,111329,111332],{"style":2937},[39,111330],{"className":111331,"style":213},[212],[39,111333,111335],{"className":111334},[217,218,219,220],[39,111336,74335],{"className":111337,"style":107705},[83,84,220],[39,111339,673],{"className":111340},[672],[39,111342,111344],{"className":111343},[200],[39,111345,111347],{"className":111346,"style":35396},[204],[39,111348],{},[39,111350,136],{"className":111351},[182],[39,111353,1755],{"className":111354},[83,84],[39,111356,15389],{"className":111357},[83],[39,111359,111361,111364],{"className":111360},[83],[39,111362,24077],{"className":111363},[83,84],[39,111365,111367],{"className":111366},[192],[39,111368,111370,111390],{"className":111369},[196,649],[39,111371,111373,111387],{"className":111372},[200],[39,111374,111376],{"className":111375,"style":963},[204],[39,111377,111378,111381],{"style":966},[39,111379],{"className":111380,"style":213},[212],[39,111382,111384],{"className":111383},[217,218,219,220],[39,111385,546],{"className":111386},[83,220],[39,111388,673],{"className":111389},[672],[39,111391,111393],{"className":111392},[200],[39,111394,111396],{"className":111395,"style":680},[204],[39,111397],{},[39,111399,161],{"className":111400},[263],[39,111402],{"className":111403,"style":461},[230],[39,111405,5378],{"className":111406},[465],[39,111408],{"className":111409,"style":461},[230],[39,111411,111413,111416],{"className":111412},[83],[39,111414,48449],{"className":111415,"style":438},[83,84],[39,111417,111419],{"className":111418},[192],[39,111420,111422,111442],{"className":111421},[196,649],[39,111423,111425,111439],{"className":111424},[200],[39,111426,111428],{"className":111427,"style":3419},[204],[39,111429,111430,111433],{"style":2937},[39,111431],{"className":111432,"style":213},[212],[39,111434,111436],{"className":111435},[217,218,219,220],[39,111437,74335],{"className":111438,"style":107705},[83,84,220],[39,111440,673],{"className":111441},[672],[39,111443,111445],{"className":111444},[200],[39,111446,111448],{"className":111447,"style":35396},[204],[39,111449],{},[39,111451,136],{"className":111452},[182],[39,111454,1755],{"className":111455},[83,84],[39,111457,15389],{"className":111458},[83],[39,111460,111462,111465],{"className":111461},[83],[39,111463,24077],{"className":111464},[83,84],[39,111466,111468],{"className":111467},[192],[39,111469,111471,111491],{"className":111470},[196,649],[39,111472,111474,111488],{"className":111473},[200],[39,111475,111477],{"className":111476,"style":963},[204],[39,111478,111479,111482],{"style":966},[39,111480],{"className":111481,"style":213},[212],[39,111483,111485],{"className":111484},[217,218,219,220],[39,111486,17359],{"className":111487},[83,220],[39,111489,673],{"className":111490},[672],[39,111492,111494],{"className":111493},[200],[39,111495,111497],{"className":111496,"style":680},[204],[39,111498],{},[39,111500,161],{"className":111501},[2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20-1h17z",[39,112097,112098,112101],{"style":21599},[39,112099],{"className":112100,"style":935},[212],[39,112102,112104,112107,112110,112113,112153,112156,112159,112162,112165,112168,112171,112211,112214,112217,112220,112223,112226,112229,112232,112272,112301,112341,112344,112347,112350,112353,112356,112359,112399,112428],{"className":112103},[83],[39,112105,5378],{"className":112106},[83],[39,112108,74335],{"className":112109,"style":107705},[83,84],[39,112111,136],{"className":112112},[182],[39,112114,112116,112119],{"className":112115},[83],[39,112117,24077],{"className":112118},[83,84],[39,112120,112122],{"className":112121},[192],[39,112123,112125,112145],{"className":112124},[196,649],[39,112126,112128,112142],{"className":112127},[200],[39,112129,112131],{"className":112130,"style":963},[204],[39,112132,112133,112136],{"style":966},[39,112134],{"className":112135,"style":213},[212],[39,112137,112139],{"className":112138},[217,218,219,220],[39,112140,546],{"className":112141},[83,220],[39,112143,673],{"className":112144},[672],[39,112146,112148],{"className":112147},[200],[39,112149,112151],{"className":112150,"style":680},[204],[39,112152],{},[39,112154,161],{"className":112155},[263],[39,112157],{"className":112158,"style":461},[230],[39,112160,788],{"className":112161},[465],[39,112163],{"className":112164,"style":461},[230],[39,112166,74335],{"className":112167,"style":107705},[83,84],[39,112169,136],{"className":112170},[182],[39,112172,112174,112177],{"className":112173},[83],[39,112175,24077],{"className":112176},[83,84],[39,112178,112180],{"className":112179},[192],[39,112181,112183,112203],{"className":112182},[196,649],[39,112184,112186,112200],{"className":112185},[200],[39,112187,112189],{"className":112188,"style":963},[204],[39,112190,112191,112194],{"style":966},[39,112192],{"className":112193,"style":213},[212],[39,112195,112197],{"className":112196},[217,218,219,220],[39,112198,17359],{"className":112199},[83,220],[39,112201,673],{"className":112202},[672],[39,112204,112206],{"className":112205},[200],[39,112207,112209],{"className":112208,"style":680},[204],[39,112210],{},[39,112212,161],{"className":112213},[263],[39,112215],{"className":112216,"style":461},[230],[39,112218,788],{"className":112219},[465],[39,112221],{"className":112222,"style":461},[230],[39,112224,84725],{"className":112225},[83],[39,112227,74335],{"className":112228,"style":107705},[83,84],[39,112230,136],{"className":112231},[182],[39,112233,112235,112238],{"className":112234},[83],[39,112236,24077],{"className":112237},[83,84],[39,112239,112241],{"className":112240},[192],[39,112242,112244,112264],{"className":112243},[196,649],[39,112245,112247,112261],{"className":112246},[200],[39,112248,112250],{"className":112249,"style":963},[204],[39,112251,112252,112255],{"style":966},[39,112253],{"className":112254,"style":213},[212],[39,112256,112258],{"className":112257},[217,218,219,220],[39,112259,546],{"className":112260},[83,220],[39,112262,673],{"className":112263},[672],[39,112265,112267],{"className":112266},[200],[39,112268,112270],{"className":112269,"style":680},[204],[39,112271],{},[39,112273,112275,112278],{"className":112274},[263],[39,112276,161],{"className":112277},[263],[39,112279,112281],{"className":112280},[192],[39,112282,112284],{"className":112283},[196],[39,112285,112287],{"className":112286},[200],[39,112288,112290],{"className":112289,"style":3389},[204],[39,112291,112292,112295],{"style":1027},[39,112293],{"className":112294,"style":213},[212],[39,112296,112298],{"className":112297},[217,218,219,220],[39,112299,93341],{"className":112300},[83,220],[39,112302,112304,112307],{"className":112303},[83],[39,112305,24077],{"className":112306},[83,84],[39,112308,112310],{"className":112309},[192],[39,112311,112313,112333],{"className":112312},[196,649],[39,112314,112316,112330],{"className":112315},[200],[39,112317,112319],{"className":112318,"style":963},[204],[39,112320,112321,112324],{"style":966},[39,112322],{"className":112323,"style":213},[212],[39,112325,112327],{"className":112326},[217,218,219,220],[39,112328,546],{"className":112329},[83,220],[39,112331,673],{"className":112332},[672],[39,112334,112336],{"className":112335},[200],[39,112337,112339],{"className":112338,"style":680},[204],[39,112340],{},[39,112342],{"className":112343,"style":461},[230],[39,112345,5378],{"className":112346},[465],[39,112348],{"className":112349,"style":461},[230],[39,112351,84725],{"className":112352},[83],[39,112354,74335],{"className":112355,"style":107705},[83,84],[39,112357,136],{"className":112358},[182],[39,112360,112362,112365],{"className":112361},[83],[39,112363,24077],{"className":112364},[83,84],[39,112366,112368],{"className":112367},[192],[39,112369,112371,112391],{"className":112370},[196,649],[39,112372,112374,112388],{"className":112373},[200],[39,112375,112377],{"className":112376,"style":963},[204],[39,112378,112379,112382],{"style":966},[39,112380],{"className":112381,"style":213},[212],[39,112383,112385],{"className":112384},[217,218,219,220],[39,112386,17359],{"className":112387},[83,220],[39,112389,673],{"className":112390},[672],[39,112392,112394],{"className":112393},[200],[39,112395,112397],{"className":112396,"style":680},[204],[39,112398],{},[39,112400,112402,112405],{"className":112401},[263],[39,112403,161],{"className":112404},[263],[39,112406,112408],{"className":112407},[192],[39,112409,112411],{"className":112410},[196],[39,112412,112414],{"className":112413},[200],[39,112415,112417],{"className":112416,"style":3389},[204],[39,112418,112419,112422],{"style":1027},[39,112420],{"className":112421,"style":213},[212],[39,112423,112425],{"className":112424},[217,218,219,220],[39,112426,93341],{"className":112427},[83,220],[39,112429,112431,112434],{"className":112430},[83],[39,112432,24077],{"className":112433},[83,84],[39,112435,112437],{"className":112436},[192],[39,112438,112440,112460],{"className":112439},[196,649],[39,112441,112443,112457],{"className":112442},[200],[39,112444,112446],{"className":112445,"style":963},[204],[39,112447,112448,112451],{"style":966},[39,112449],{"className":112450,"style":213},[212],[39,112452,112454],{"className":112453},[217,218,219,220],[39,112455,17359],{"className":112456},[83,220],[39,112458,673],{"className":112459},[672],[39,112461,112463],{"className":112462},[200],[39,112464,112466],{"className":112465,"style":680},[204],[39,112467],{},[39,112469,673],{"className":112470},[672],[39,112472,112474],{"className":112473},[200],[39,112475,112478],{"className":112476,"style":112477},[204],"height:0.898em;",[39,112479],{},[39,112481,673],{"className":112482},[672],[39,112484,112486],{"className":112485},[200],[39,112487,112490],{"className":112488,"style":112489},[204],"height:1.5763em;",[39,112491],{},[39,112493],{"className":112494,"style":461},[230],[39,112496,788],{"className":112497},[465],[39,112499],{"className":112500,"style":461},[230],[39,112502,112504],{"className":112503},[182],[39,112505,136],{"className":112506},[18520,4009],[39,112508,84725],{"className":112509},[83],[39,112511,74335],{"className":112512,"style":107705},[83,84],[39,112514,136],{"className":112515},[182],[39,112517,112519,112522],{"className":112518},[83],[39,112520,24077],{"className":112521},[83,84],[39,112523,112525],{"className":112524},[192],[39,112526,112528,112548],{"className":112527},[196,649],[39,112529,112531,112545],{"className":112530},[200],[39,112532,112534],{"className":112533,"style":963},[204],[39,112535,112536,112539],{"style":966},[39,112537],{"className":112538,"style":213},[212],[39,112540,112542],{"className":112541},[217,218,219,220],[39,112543,17359],{"className":112544},[83,220],[39,112546,673],{"className":112547},[672],[39,112549,112551],{"className":112550},[200],[39,112552,112554],{"className":112553,"style":680},[204],[39,112555],{},[39,112557,161],{"className":112558},[263],[39,112560],{"className":112561,"style":461},[230],[39,112563,5378],{"className":112564},[465],[39,112566],{"className":112567,"style":461},[230],[39,112569,84725],{"className":112570},[83],[39,112572,74335],{"className":112573,"style":107705},[83,84],[39,112575,136],{"className":112576},[182],[39,112578,112580,112583],{"className":112579},[83],[39,112581,24077],{"className":112582},[83,84],[39,112584,112586],{"className":112585},[192],[39,112587,112589,112609],{"className":112588},[196,649],[39,112590,112592,112606],{"className":112591},[200],[39,112593,112595],{"className":112594,"style":963},[204],[39,112596,112597,112600],{"style":966},[39,112598],{"className":112599,"style":213},[212],[39,112601,112603],{"className":112602},[217,218,219,220],[39,112604,546],{"className":112605},[83,220],[39,112607,673],{"className":112608},[672],[39,112610,112612],{"className":112611},[200],[39,112613,112615],{"className":112614,"style":680},[204],[39,112616],{},[39,112618,161],{"className":112619},[263],[39,112621,112623,112629],{"className":112622},[263],[39,112624,112626],{"className":112625},[263],[39,112627,161],{"className":112628},[18520,4009],[39,112630,112632],{"className":112631},[192],[39,112633,112635],{"className":112634},[196],[39,112636,112638],{"className":112637},[200],[39,112639,112642],{"className":112640,"style":112641},[204],"height:1.089em;",[39,112643,112644,112647],{"style":86136},[39,112645],{"className":112646,"style":213},[212],[39,112648,112650],{"className":112649},[217,218,219,220],[39,112651,93341],{"className":112652},[83,220],[39,112654,1755],{"className":112655},[83,84],[39,112657,2903],{"className":112658},[83],[39,112660,673],{"className":112661},[672],[39,112663,112665],{"className":112664},[200],[39,112666,112668],{"className":112667,"style":111283},[204],[39,112669],{},[11,112671,112672,112716,112717,112745],{},[39,112673,112675,112695],{"className":112674,"translate":43},[42],[39,112676,112678],{"className":112677},[47],[49,112679,112680],{"xmlns":51},[53,112681,112682,112692],{},[56,112683,112684,112686,112688,112690],{},[59,112685,74335],{},[133,112687,136],{"stretchy":135},[59,112689,1755],{},[133,112691,161],{"stretchy":135},[63,112693,112694],{"encoding":65},"f(x)",[39,112696,112698],{"className":112697,"ariaHidden":70},[69],[39,112699,112701,112704,112707,112710,112713],{"className":112700},[74],[39,112702],{"className":112703,"style":245},[78],[39,112705,74335],{"className":112706,"style":107705},[83,84],[39,112708,136],{"className":112709},[182],[39,112711,1755],{"className":112712},[83,84],[39,112714,161],{"className":112715},[263]," 那一项消掉了，剩下的是 ",[39,112718,112720,112733],{"className":112719,"translate":43},[42],[39,112721,112723],{"className":112722},[47],[49,112724,112725],{"xmlns":51},[53,112726,112727,112731],{},[56,112728,112729],{},[59,112730,1755],{},[63,112732,1755],{"encoding":65},[39,112734,112736],{"className":112735,"ariaHidden":70},[69],[39,112737,112739,112742],{"className":112738},[74],[39,112740],{"className":112741,"style":290},[78],[39,112743,1755],{"className":112744},[83,84]," 的仿射函数。所以边界仍然是一个超平面。",[11,112747,112748],{},"这个结论有点扫兴，但我觉得它比“Bregman 给出弯曲边界”更有用：Bregman 改变的不是假设空间，而是“垂直于谁”——这个超平面垂直于 Hessian 度量下的法向，而不是欧氏法向。换句话说，它换了一把尺子，没有换刀。",[11,112750,112751,112752,112803,112804,75103,112833,112862,112863,112916,112917,112946,112947,112976],{},"具体到二维单纯形（其实就是线段 ",[39,112753,112755,112776],{"className":112754,"translate":43},[42],[39,112756,112758],{"className":112757},[47],[49,112759,112760],{"xmlns":51},[53,112761,112762,112774],{},[56,112763,112764,112766,112768,112770,112772],{},[133,112765,537],{"stretchy":135},[539,112767,541],{},[133,112769,155],{"separator":70},[539,112771,546],{},[133,112773,4280],{"stretchy":135},[63,112775,87745],{"encoding":65},[39,112777,112779],{"className":112778,"ariaHidden":70},[69],[39,112780,112782,112785,112788,112791,112794,112797,112800],{"className":112781},[74],[39,112783],{"className":112784,"style":245},[78],[39,112786,537],{"className":112787},[182],[39,112789,541],{"className":112790},[83],[39,112792,155],{"className":112793},[252],[39,112795],{"className":112796,"style":256},[230],[39,112798,546],{"className":112799},[83],[39,112801,4280],{"className":112802},[263],"）上，两个类原型 ",[39,112805,112807,112821],{"className":112806,"translate":43},[42],[39,112808,112810],{"className":112809},[47],[49,112811,112812],{"xmlns":51},[53,112813,112814,112819],{},[56,112815,112816],{},[539,112817,112818],{},"0.2",[63,112820,112818],{"encoding":65},[39,112822,112824],{"className":112823,"ariaHidden":70},[69],[39,112825,112827,112830],{"className":112826},[74],[39,112828],{"className":112829,"style":2473},[78],[39,112831,112818],{"className":112832},[83],[39,112834,112836,112850],{"className":112835,"translate":43},[42],[39,112837,112839],{"className":112838},[47],[49,112840,112841],{"xmlns":51},[53,112842,112843,112848],{},[56,112844,112845],{},[539,112846,112847],{},"0.6",[63,112849,112847],{"encoding":65},[39,112851,112853],{"className":112852,"ariaHidden":70},[69],[39,112854,112856,112859],{"className":112855},[74],[39,112857],{"className":112858,"style":2473},[78],[39,112860,112847],{"className":112861},[83],"，用 KL 判类的分界点在 ",[39,112864,112866,112885],{"className":112865,"translate":43},[42],[39,112867,112869],{"className":112868},[47],[49,112870,112871],{"xmlns":51},[53,112872,112873,112882],{},[56,112874,112875,112877,112879],{},[59,112876,11],{},[133,112878,99744],{},[539,112880,112881],{},"0.387",[63,112883,112884],{"encoding":65},"p\\approx0.387",[39,112886,112888,112907],{"className":112887,"ariaHidden":70},[69],[39,112889,112891,112895,112898,112901,112904],{"className":112890},[74],[39,112892],{"className":112893,"style":112894},[78],"height:0.6776em;vertical-align:-0.1944em;",[39,112896,11],{"className":112897},[83,84],[39,112899],{"className":112900,"style":231},[230],[39,112902,99744],{"className":112903},[235],[39,112905],{"className":112906,"style":231},[230],[39,112908,112910,112913],{"className":112909},[74],[39,112911],{"className":112912,"style":2473},[78],[39,112914,112881],{"className":112915},[83],"；欧氏中点是 ",[39,112918,112920,112934],{"className":112919,"translate":43},[42],[39,112921,112923],{"className":112922},[47],[49,112924,112925],{"xmlns":51},[53,112926,112927,112932],{},[56,112928,112929],{},[539,112930,112931],{},"0.4",[63,112933,112931],{"encoding":65},[39,112935,112937],{"className":112936,"ariaHidden":70},[69],[39,112938,112940,112943],{"className":112939},[74],[39,112941],{"className":112942,"style":2473},[78],[39,112944,112931],{"className":112945},[83],"；而在 logit 坐标里取中点又得到 ",[39,112948,112950,112964],{"className":112949,"translate":43},[42],[39,112951,112953],{"className":112952},[47],[49,112954,112955],{"xmlns":51},[53,112956,112957,112962],{},[56,112958,112959],{},[539,112960,112961],{},"0.380",[63,112963,112961],{"encoding":65},[39,112965,112967],{"className":112966,"ariaHidden":70},[69],[39,112968,112970,112973],{"className":112969},[74],[39,112971],{"className":112972,"style":2473},[78],[39,112974,112961],{"className":112975},[83],"。三个数，三把尺子，没有哪把是“错”的，取决于你认为什么该被同等对待。",[11,112978,112979,112980,113084,113085,29832,113410,113438,113439,113467],{},"真正弯的边界来自两个地方。一是把比较放在约束集上：单纯形加 Fisher–Rao，就是上一节的大圆。二是把它放在非线性映射之后：如果 ",[39,112981,112983,113029],{"className":112982,"translate":43},[42],[39,112984,112986],{"className":112985},[47],[49,112987,112988],{"xmlns":51},[53,112989,112990,113026],{},[56,112991,112992,112994,112996,112998,113000,113002,113018,113020,113022,113024],{},[59,112993,100187],{},[133,112995,136],{"stretchy":135},[59,112997,1755],{},[133,112999,161],{"stretchy":135},[133,113001,793],{},[56,113003,113004,113006,113008,113010,113012,113014,113016],{},[59,113005,142],{"mathvariant":145},[59,113007,35010],{"mathvariant":145},[59,113009,74335],{"mathvariant":145},[59,113011,621],{"mathvariant":145},[59,113013,1748],{"mathvariant":145},[59,113015,158],{"mathvariant":145},[59,113017,1755],{"mathvariant":145},[133,113019,136],{"stretchy":135},[59,113021,23703],{},[59,113023,1755],{},[133,113025,161],{"stretchy":135},[63,113027,113028],{"encoding":65},"q(x)=\\mathrm{softmax}(Wx)",[39,113030,113032,113059],{"className":113031,"ariaHidden":70},[69],[39,113033,113035,113038,113041,113044,113047,113050,113053,113056],{"className":113034},[74],[39,113036],{"className":113037,"style":245},[78],[39,113039,100187],{"className":113040,"style":2114},[83,84],[39,113042,136],{"className":113043},[182],[39,113045,1755],{"className":113046},[83,84],[39,113048,161],{"className":113049},[263],[39,113051],{"className":113052,"style":231},[230],[39,113054,793],{"className":113055},[235],[39,113057],{"className":113058,"style":231},[230],[39,113060,113062,113065,113072,113075,113078,113081],{"className":113061},[74],[39,113063],{"className":113064,"style":245},[78],[39,113066,113068],{"className":113067},[83],[39,113069,113071],{"className":113070},[83,35077],"softmax",[39,113073,136],{"className":113074},[182],[39,113076,23703],{"className":113077,"style":178},[83,84],[39,113079,1755],{"className":113080},[83,84],[39,113082,161],{"className":113083},[263],"，那么 ",[39,113086,113088,113162],{"className":113087,"translate":43},[42],[39,113089,113091],{"className":113090},[47],[49,113092,113093],{"xmlns":51},[53,113094,113095,113159],{},[56,113096,113097,113099,113101,113103,113109,113111,113113,113115,113117,113119,113121,113127,113129,113131,113137,113139,113141,113143,113145,113147,113149,113155,113157],{},[133,113098,12951],{"stretchy":135},[59,113100,1755],{},[133,113102,404],{},[613,113104,113105,113107],{},[59,113106,48449],{},[59,113108,74335],{},[133,113110,136],{"stretchy":135},[59,113112,100187],{},[133,113114,136],{"stretchy":135},[59,113116,1755],{},[133,113118,161],{"stretchy":135},[59,113120,15389],{"mathvariant":145},[613,113122,113123,113125],{},[59,113124,24077],{},[539,113126,546],{},[133,113128,161],{"stretchy":135},[133,113130,793],{},[613,113132,113133,113135],{},[59,113134,48449],{},[59,113136,74335],{},[133,113138,136],{"stretchy":135},[59,113140,100187],{},[133,113142,136],{"stretchy":135},[59,113144,1755],{},[133,113146,161],{"stretchy":135},[59,113148,15389],{"mathvariant":145},[613,113150,113151,113153],{},[59,113152,24077],{},[539,113154,17359],{},[133,113156,161],{"stretchy":135},[133,113158,12965],{"stretchy":135},[63,113160,113161],{"encoding":65},"\\{x:D_f(q(x)\\|c_1)=D_f(q(x)\\|c_2)\\}",[39,113163,113165,113186,113302],{"className":113164,"ariaHidden":70},[69],[39,113166,113168,113171,113174,113177,113180,113183],{"className":113167},[74],[39,113169],{"className":113170,"style":245},[78],[39,113172,12951],{"className":113173},[182],[39,113175,1755],{"className":113176},[83,84],[39,113178],{"className":113179,"style":231},[230],[39,113181,404],{"className":113182},[235],[39,113184],{"className":113185,"style":231},[230],[39,113187,113189,113192,113232,113235,113238,113241,113244,113247,113250,113290,113293,113296,113299],{"className":113188},[74],[39,113190],{"className":113191,"style":41560},[78],[39,113193,113195,113198],{"className":113194},[83],[39,113196,48449],{"className":113197,"style":438},[83,84],[39,113199,113201],{"className":113200},[192],[39,113202,113204,113224],{"className":113203},[196,649],[39,113205,113207,113221],{"className":113206},[200],[39,113208,113210],{"className":113209,"style":3419},[204],[39,113211,113212,113215],{"style":2937},[39,113213],{"className":113214,"style":213},[212],[39,113216,113218],{"className":113217},[217,218,219,220],[39,113219,74335],{"className":113220,"style":107705},[83,84,220],[39,113222,673],{"className":113223},[672],[39,113225,113227],{"className":113226},[200],[39,113228,113230],{"className":113229,"style":35396},[204],[39,113231],{},[39,113233,136],{"className":113234},[182],[39,113236,100187],{"className":113237,"style":2114},[83,84],[39,113239,136],{"className":113240},[182],[39,113242,1755],{"className":113243},[83,84],[39,113245,161],{"className":113246},[263],[39,113248,15389],{"className":113249},[83],[39,113251,113253,113256],{"className":113252},[83],[39,113254,24077],{"className":113255},[83,84],[39,113257,113259],{"className":113258},[192],[39,113260,113262,113282],{"className":113261},[196,649],[39,113263,113265,113279],{"className":113264},[200],[39,113266,113268],{"className":113267,"style":963},[204],[39,113269,113270,113273],{"style":966},[39,113271],{"className":113272,"style":213},[212],[39,113274,113276],{"className":113275},[217,218,219,220],[39,113277,546],{"className":113278},[83,220],[39,113280,673],{"className":113281},[672],[39,113283,113285],{"className":113284},[200],[39,113286,113288],{"className":113287,"style":680},[204],[39,113289],{},[39,113291,161],{"className":113292},[263],[39,113294],{"className":113295,"style":231},[230],[39,113297,793],{"className":113298},[235],[39,113300],{"className":113301,"style":231},[230],[39,113303,113305,113308,113348,113351,113354,113357,113360,113363,113366,113406],{"className":113304},[74],[39,113306],{"className":113307,"style":41560},[78],[39,113309,113311,113314],{"className":113310},[83],[39,113312,48449],{"className":113313,"style":438},[83,84],[39,113315,113317],{"className":113316},[192],[39,113318,113320,113340],{"className":113319},[196,649],[39,113321,113323,113337],{"className":113322},[200],[39,113324,113326],{"className":113325,"style":3419},[204],[39,113327,113328,113331],{"style":2937},[39,113329],{"className":113330,"style":213},[212],[39,113332,113334],{"className":113333},[217,218,219,220],[39,113335,74335],{"className":113336,"style":107705},[83,84,220],[39,113338,673],{"className":113339},[672],[39,113341,113343],{"className":113342},[200],[39,113344,113346],{"className":113345,"style":35396},[204],[39,113347],{},[39,113349,136],{"className":113350},[182],[39,113352,100187],{"className":113353,"style":2114},[83,84],[39,113355,136],{"className":113356},[182],[39,113358,1755],{"className":113359},[83,84],[39,113361,161],{"className":113362},[263],[39,113364,15389],{"className":113365},[83],[39,113367,113369,113372],{"className":113368},[83],[39,113370,24077],{"className":113371},[83,84],[39,11337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空间是超平面，在 ",[39,113440,113442,113455],{"className":113441,"translate":43},[42],[39,113443,113445],{"className":113444},[47],[49,113446,113447],{"xmlns":51},[53,113448,113449,113453],{},[56,113450,113451],{},[59,113452,1755],{},[63,113454,1755],{"encoding":65},[39,113456,113458],{"className":113457,"ariaHidden":70},[69],[39,113459,113461,113464],{"className":113460},[74],[39,113462],{"className":113463,"style":290},[78],[39,113465,1755],{"className":113466},[83,84]," 空间是它的原像，一般是一张弯的曲面。我见到的“非欧氏边界”基本都是后者。",[11,113469,113470],{},"Bregman 还给出两个优化工具。一个是镜像下降：",[39,113472,113474],{"className":113473,"translate":43},[746],[39,113475,113477,113563],{"className":113476,"translate":43},[42],[39,113478,113480],{"className":113479},[47],[49,113481,113482],{"xmlns":51,"display":755},[53,113483,113484,113560],{},[56,113485,113486,113498,113500,113502,113504,113514,113516,113518,113524,113526,113528,113530,113532,113538,113544,113546,113548,113550,113556,113558],{},[613,113487,113488,113490],{},[59,113489,1755],{},[56,113491,113492,113494,113496],{},[59,113493,621],{},[133,113495,788],{},[539,113497,546],{},[133,113499,793],{},[59,113501,39047],{},[133,113503,1690],{},[6556,113505,113506,113512],{},[56,113507,113508,113510],{},[59,113509,92422],{},[133,113511,1690],{},[59,113513,1755],{},[35841,113515,74223],{},[133,113517,92447],{"stretchy":135},[613,113519,113520,113522],{},[59,113521,39209],{},[59,113523,621],{},[133,113525,155],{"separator":70},[59,113527,1755],{},[133,113529,92456],{"stretchy":135},[133,113531,788],{},[5362,113533,113534,113536],{},[539,113535,546],{},[59,113537,103058],{},[613,113539,113540,113542],{},[59,113541,48449],{},[59,113543,74335],{},[133,113545,136],{"stretchy":135},[59,113547,1755],{},[59,113549,15389],{"mathvariant":145},[613,113551,113552,113554],{},[59,113553,1755],{},[59,113555,621],{},[133,113557,161],{"stretchy":135},[133,113559,155],{"separator":70},[63,113561,113562],{"encoding":65},"x_{t+1}=\\arg\\min_x\\ \\langle g_t,x\\rangle+\\frac1\\eta D_f(x\\|x_t),",[39,113564,113566,113630,113756],{"className":113565,"ariaHidden":70},[69],[39,113567,113569,113572,113621,113624,113627],{"className":113568},[74],[39,113570],{"className":113571,"style":3635},[78],[39,113573,113575,113578],{"className":113574},[83],[39,113576,1755],{"className":113577},[83,84],[39,113579,113581],{"className":113580},[192],[39,113582,113584,113613],{"className":113583},[196,649],[39,113585,113587,113610],{"className":113586},[200],[39,113588,113590],{"className":113589,"style":963},[204],[39,113591,113592,113595],{"style":966},[39,113593],{"className":113594,"style":213},[212],[39,113596,113598],{"className":113597},[217,218,219,220],[39,113599,113601,113604,113607],{"className":113600},[83,220],[39,113602,621],{"className":113603},[83,84,220],[39,113605,788],{"className":113606},[465,220],[39,113608,546],{"className":113609},[83,220],[39,113611,673],{"className":113612},[672],[39,113614,113616],{"className":113615},[200],[39,113617,113619],{"className":113618,"style":994},[204],[39,113620],{},[39,113622],{"className":113623,"style":231},[230],[39,113625,793],{"className":113626},[235],[39,113628],{"className":113629,"style":231},[230],[39,113631,113633,113637,113642,113645,113689,113692,113695,113735,113738,113741,113744,113747,113750,113753],{"className":113632},[74],[39,113634],{"className":113635,"style":113636},[78],"height:1.45em;vertical-align:-0.7em;",[39,113638,39206,113640],{"className":113639},[1744],[39,113641,39209],{"style":35078},[39,113643],{"className":113644,"style":256},[230],[39,113646,113648],{"className":113647},[1744,3294],[39,113649,113651,113681],{"className":113650},[196,649],[39,113652,113654,113678],{"className":113653},[200],[39,113655,113657,113668],{"className":113656,"style":92523},[204],[39,113658,113659,113662],{"style":22299},[39,113660],{"className":113661,"style":935},[212],[39,113663,113665],{"className":113664},[217,218,219,220],[39,113666,1755],{"className":113667},[83,84,220],[39,113669,113670,113673],{"style":21599},[39,113671],{"className":113672,"style":935},[212],[39,113674,113675],{},[39,113676,92422],{"className":113677},[1744],[39,113679,673],{"className":113680},[672],[39,113682,113684],{"className":113683},[200],[39,113685,113687],{"className":113686,"style":22331},[204],[39,113688],{},[39,113690,74223],{"className":113691},[230],[39,113693,92447],{"className":113694},[182],[39,113696,113698,113701],{"className":113697},[83],[39,113699,39209],{"className":113700,"style":2114},[83,84],[39,113702,113704],{"className":113703},[192],[39,113705,113707,113727],{"className":113706},[196,649],[39,113708,113710,113724],{"className":113709},[200],[39,113711,113713],{"className":113712,"style":656},[204],[39,113714,113715,113718],{"style":41316},[39,113716],{"className":113717,"style":213},[212],[39,113719,113721],{"className":113720},[217,218,219,220],[39,113722,621],{"className":113723},[83,84,220],[39,113725,673],{"className":113726},[672],[39,113728,113730],{"className":113729},[200],[39,113731,113733],{"className":113732,"style":680},[204],[39,113734],{},[39,113736,155],{"className":113737},[252],[39,113739],{"className":113740,"style":256},[230],[39,113742,1755],{"className":113743},[83,84],[39,113745,92456],{"className":113746},[263],[39,113748],{"className":113749,"style":461},[230],[39,113751,788],{"className":113752},[465],[39,113754],{"className":113755,"style":461},[230],[39,113757,113759,113762,113824,113864,113867,113870,113873,113913,113916],{"className":113758},[74],[39,113760],{"className":113761,"style":72267},[78],[39,113763,113765,113768,113821],{"className":113764},[83],[39,113766],{"className":113767},[182,5463],[39,113769,113771],{"className":113770},[5362],[39,113772,113774,113813],{"className":113773},[196,649],[39,113775,113777,113810],{"className":113776},[200],[39,113778,113780,113791,113799],{"className":113779,"style":18240},[204],[39,113781,113782,113785],{"style":5479},[39,113783],{"className":113784,"style":935},[212],[39,113786,113788],{"className":113787},[83],[39,113789,103058],{"className":113790,"style":2114},[83,84],[39,113792,113793,113796],{"style":5503},[39,113794],{"className":113795,"style":935},[212],[39,113797],{"className":113798,"style":5511},[5510],[39,113800,113801,113804],{"style":5514},[39,113802],{"className":113803,"style":935},[212],[39,113805,113807],{"className":113806},[83],[39,113808,546],{"className":113809},[83],[39,113811,673],{"className":113812},[672],[39,113814,113816],{"className":113815},[200],[39,113817,113819],{"className":113818,"style":5582},[204],[39,113820],{},[39,113822],{"className":113823},[263,5463],[39,113825,113827,113830],{"className":113826},[83],[39,113828,48449],{"className":113829,"style":438},[83,84],[39,113831,113833],{"className":113832},[192],[39,113834,113836,113856],{"className":113835},[196,649],[39,113837,113839,113853],{"className":113838},[200],[39,113840,113842],{"className":113841,"style":3419},[204],[39,113843,113844,113847],{"style":2937},[39,113845],{"className":113846,"style":213},[212],[39,113848,113850],{"className":113849},[217,218,219,220],[39,113851,74335],{"className":113852,"style":107705},[83,84,220],[39,113854,673],{"className":113855},[672],[39,113857,113859],{"className":113858},[200],[39,113860,113862],{"className":113861,"style":35396},[204],[39,113863],{},[39,113865,136],{"className":113866},[182],[39,113868,1755],{"className":113869},[83,84],[39,113871,15389],{"className":113872},[83],[39,113874,113876,113879],{"className":113875},[83],[39,113877,1755],{"className":113878},[83,84],[39,113880,113882],{"className":113881},[192],[39,113883,113885,113905],{"className":113884},[196,649],[39,113886,113888,113902],{"className":113887},[200],[39,113889,113891],{"className":113890,"style":656},[204],[39,113892,113893,113896],{"style":966},[39,113894],{"className":113895,"style":213},[212],[39,113897,113899],{"className":113898},[217,218,219,220],[39,113900,621],{"className":113901},[83,84,220],[39,113903,673],{"className":113904},[672],[39,113906,113908],{"className":113907},[200],[39,113909,113911],{"className":113910,"style":680},[204],[39,113912],{},[39,113914,161],{"className":113915},[263],[39,113917,155],{"className":113918},[252],[11,113920,113921,113922,113950,113951,114118,114119,114147,114148,2183,114393,114421],{},"直觉是“走一步，但别离原来的位置太远”，而“太远”用 ",[39,113923,113925,113938],{"className":113924,"translate":43},[42],[39,113926,113928],{"className":113927},[47],[49,113929,113930],{"xmlns":51},[53,113931,113932,113936],{},[56,113933,113934],{},[59,113935,74335],{},[63,113937,74335],{"encoding":65},[39,113939,113941],{"className":113940,"ariaHidden":70},[69],[39,113942,113944,113947],{"className":113943},[74],[39,113945],{"className":113946,"style":51280},[78],[39,113948,74335],{"className":113949,"style":107705},[83,84]," 来量。取 ",[39,113952,113954,113988],{"className":113953,"translate":43},[42],[39,113955,113957],{"className":113956},[47],[49,113958,113959],{"xmlns":51},[53,113960,113961,113985],{},[56,113962,113963,113965,113967,113975,113977,113979],{},[59,113964,74335],{},[133,113966,793],{},[770,113968,113969],{"scriptlevel":541,"displaystyle":135},[5362,113970,113971,113973],{},[539,113972,546],{},[539,113974,17359],{},[59,113976,15389],{"mathvariant":145},[59,113978,1755],{},[138,113980,113981,113983],{},[59,113982,15389],{"mathvariant":145},[539,113984,17359],{},[63,113986,113987],{"encoding":65},"f=\\tfrac12\\|x\\|^2",[39,113989,113991,114009],{"className":113990,"ariaHidden":70},[69],[39,113992,113994,113997,114000,114003,114006],{"className":113993},[74],[39,113995],{"className":113996,"style":51280},[78],[39,113998,74335],{"className":113999,"style":107705},[83,84],[39,114001],{"className":114002,"style":231},[230],[39,114004,793],{"className":114005},[235],[39,114007],{"className":114008,"style":231},[230],[39,114010,114012,114015,114083,114086,114089],{"className":114011},[74],[39,114013],{"className":114014,"style":74794},[78],[39,114016,114018,114021,114080],{"className":114017},[83],[39,114019],{"className":114020},[182,5463],[39,114022,114024],{"className":114023},[5362],[39,114025,114027,114072],{"className":114026},[196,649],[39,114028,114030,114069],{"className":114029},[200],[39,114031,114033,114047,114055],{"className":114032,"style":74813},[204],[39,114034,114035,114038],{"style":74816},[39,114036],{"className":114037,"style":935},[212],[39,114039,114041],{"className":114040},[217,218,219,220],[39,114042,114044],{"className":114043},[83,220],[39,114045,17359],{"className":114046},[83,220],[39,114048,114049,114052],{"style":5503},[39,114050],{"className":114051,"style":935},[212],[39,114053],{"className":114054,"style":5511},[5510],[39,114056,114057,114060],{"style":74839},[39,114058],{"className":114059,"style":935},[212],[39,114061,114063],{"className":114062},[217,218,219,220],[39,114064,114066],{"className":114065},[83,220],[39,114067,546],{"className":114068},[83,220],[39,114070,673],{"className":114071},[672],[39,114073,114075],{"className":114074},[200],[39,114076,114078],{"className":114077,"style":74861},[204],[39,114079],{},[39,114081],{"className":114082},[263,5463],[39,114084,15389],{"className":114085},[83],[39,114087,1755],{"className":114088},[83,84],[39,114090,114092,114095],{"className":114091},[83],[39,114093,15389],{"className":114094},[83],[39,114096,114098],{"className":114097},[192],[39,114099,114101],{"className":114100},[196],[39,114102,114104],{"className":114103},[200],[39,114105,114107],{"className":114106,"style":19710},[204],[39,114108,114109,114112],{"style":208},[39,114110],{"className":114111,"style":213},[212],[39,114113,114115],{"className":114114},[217,218,219,220],[39,114116,17359],{"className":114117},[83,220]," 就是梯度下降；取 ",[39,114120,114122,114135],{"className":114121,"translate":43},[42],[39,114123,114125],{"className":114124},[47],[49,114126,114127],{"xmlns":51},[53,114128,114129,114133],{},[56,114130,114131],{},[59,114132,74335],{},[63,114134,74335],{"encoding":65},[39,114136,114138],{"className":114137,"ariaHidden":70},[69],[39,114139,114141,114144],{"className":114140},[74],[39,114142],{"className":114143,"style":51280},[78],[39,114145,74335],{"className":114146,"style":107705},[83,84]," 为负熵，在单纯形上得到指数梯度／乘法权重更新，更新式是乘性的，天然保持非负。另一个是自然梯度：",[39,114149,114151,114209],{"className":114150,"translate":43},[42],[39,114152,114154],{"className":114153},[47],[49,114155,114156],{"xmlns":51},[53,114157,114158,114206],{},[56,114159,114160,114172,114174,114180,114182,114184,114186,114188,114190,114192,114202,114204],{},[613,114161,114162,114164],{},[59,114163,96325],{},[56,114165,114166,114168,114170],{},[59,114167,621],{},[133,114169,788],{},[539,114171,546],{},[133,114173,793],{},[613,114175,114176,114178],{},[59,114177,96325],{},[59,114179,621],{},[133,114181,5378],{},[59,114183,103058],{},[35841,114185,35843],{},[59,114187,3181],{},[133,114189,136],{"stretchy":135},[59,114191,96325],{},[138,114193,114194,114196],{},[133,114195,161],{"stretchy":135},[56,114197,114198,114200],{},[133,114199,5378],{},[539,114201,546],{},[59,114203,84725],{"mathvariant":145},[59,114205,77190],{},[63,114207,114208],{"encoding":65},"\\theta_{t+1}=\\theta_t-\\eta\\,G(\\theta)^{-1}\\nabla L",[39,114210,114212,114276,114331],{"className":114211,"ariaHidden":70},[69],[39,114213,114215,114218,114267,114270,114273],{"className":114214},[74],[39,114216],{"className":114217,"style":97549},[78],[39,114219,114221,114224],{"className":114220},[83],[39,114222,96325],{"className":114223,"style":438},[83,84],[39,114225,114227],{"className":114226},[192],[39,114228,114230,114259],{"className":114229},[196,649],[39,114231,114233,114256],{"className":114232},[200],[39,114234,114236],{"className":114235,"style":963},[204],[39,114237,114238,114241],{"style":2937},[39,114239],{"className":114240,"style":213},[212],[39,114242,114244],{"className":114243},[217,218,219,220],[39,114245,114247,114250,114253],{"className":114246},[83,220],[39,114248,621],{"className":114249},[83,84,220],[39,114251,788],{"className":114252},[465,220],[39,114254,546],{"className":114255},[83,220],[39,114257,673],{"className":114258},[672],[39,114260,114262],{"className":114261},[200],[39,114263,114265],{"className":114264,"style":994},[204],[39,114266],{},[39,114268],{"className":114269,"style":231},[230],[39,114271,793],{"className":114272},[235],[39,114274],{"className":114275,"style":231},[230],[39,114277,114279,114282,114322,114325,114328],{"className":114278},[74],[39,114280],{"className":114281,"style":74674},[78],[39,114283,114285,114288],{"className":114284},[83],[39,114286,96325],{"className":114287,"style":438},[83,84],[39,114289,114291],{"className":114290},[192],[39,114292,114294,114314],{"className":114293},[196,649],[39,114295,114297,114311],{"className":114296},[200],[39,114298,114300],{"className":114299,"style":656},[204],[39,114301,114302,114305],{"style":2937},[39,114303],{"className":114304,"style":213},[212],[39,114306,114308],{"className":114307},[217,218,219,220],[39,114309,621],{"className":114310},[83,84,220],[39,114312,673],{"className":114313},[672],[39,114315,114317],{"className":114316},[200],[39,114318,114320],{"className":114319,"style":680},[204],[39,114321],{},[39,114323],{"className":114324,"style":461},[230],[39,114326,5378],{"className":114327},[465],[39,114329],{"className":114330,"style":461},[230],[39,114332,114334,114337,114340,114343,114346,114349,114352,114387,114390],{"className":114333},[74],[39,114335],{"className":114336,"style":84615},[78],[39,114338,103058],{"className":114339,"style":2114},[83,84],[39,114341],{"className":114342,"style":256},[230],[39,114344,3181],{"className":114345},[83,84],[39,114347,136],{"className":114348},[182],[39,114350,96325],{"className":114351,"style":438},[83,84],[39,114353,114355,114358],{"className":114354},[263],[39,114356,161],{"className":114357},[263],[39,114359,114361],{"className":114360},[192],[39,114362,114364],{"className":114363},[196],[39,114365,114367],{"className":114366},[200],[39,114368,114370],{"className":114369,"style":19710},[204],[39,114371,114372,114375],{"style":208},[39,114373],{"className":114374,"style":213},[212],[39,114376,114378],{"className":114377},[217,218,219,220],[39,114379,114381,114384],{"className":114380},[83,220],[39,114382,5378],{"className":114383},[83,220],[39,114385,546],{"className":114386},[83,220],[39,114388,84725],{"className":114389},[83],[39,114391,77190],{"className":114392},[83,84],[39,114394,114396,114409],{"className":114395,"translate":43},[42],[39,114397,114399],{"className":114398},[47],[49,114400,114401],{"xmlns":51},[53,114402,114403,114407],{},[56,114404,114405],{},[59,114406,3181],{},[63,114408,3181],{"encoding":65},[39,114410,114412],{"className":114411,"ariaHidden":70},[69],[39,114413,114415,114418],{"className":114414},[74],[39,114416],{"className":114417,"style":79},[78],[39,114419,3181],{"className":114420},[83,84]," 是 Fisher。它是在 Fisher–Rao 度量下做最速下降，而不是在欧氏度量下。对 softmax 回归，自然梯度会消掉 logits 尺度带来的平台期——因为在 Fisher 度量下，那个尺度根本不算“距离”。",[21,114423,114424],{"id":114424},"点态密度",[11,114426,114427,114428,114533,114534,114562],{},"手上有五个一维样本：",[39,114429,114431,114473],{"className":114430,"translate":43},[42],[39,114432,114434],{"className":114433},[47],[49,114435,114436],{"xmlns":51},[53,114437,114438,114470],{},[56,114439,114440,114442,114444,114446,114449,114451,114453,114456,114458,114460,114463,114465,114467],{},[539,114441,17739],{},[133,114443,155],{"separator":70},[35841,114445,74223],{},[539,114447,114448],{},"1.0",[133,114450,155],{"separator":70},[35841,114452,74223],{},[539,114454,114455],{},"1.1",[133,114457,155],{"separator":70},[35841,114459,74223],{},[539,114461,114462],{},"5.0",[133,114464,155],{"separator":70},[35841,114466,74223],{},[539,114468,114469],{},"5.1",[63,114471,114472],{"encoding":65},"0.9,\\ 1.0,\\ 1.1,\\ 5.0,\\ 5.1",[39,114474,114476],{"className":114475,"ariaHidden":70},[69],[39,114477,114479,114482,114485,114488,114491,114494,114497,114500,114503,114506,114509,114512,114515,114518,114521,114524,114527,114530],{"className":114478},[74],[39,114480],{"className":114481,"style":17578},[78],[39,114483,17739],{"className":114484},[83],[39,114486,155],{"className":114487},[252],[39,114489,74223],{"className":114490},[230],[39,114492],{"className":114493,"style":256},[230],[39,114495,114448],{"className":114496},[83],[39,114498,155],{"className":114499},[252],[39,114501,74223],{"className":114502},[230],[39,114504],{"className":114505,"style":256},[230],[39,114507,114455],{"className":114508},[83],[39,114510,155],{"className":114511},[252],[39,114513,74223],{"className":114514},[230],[39,114516],{"className":114517,"style":256},[230],[39,114519,114462],{"className":114520},[83],[39,114522,155],{"className":114523},[252],[39,114525,74223],{"className":114526},[230],[39,114528],{"className":114529,"style":256},[230],[39,114531,114469],{"className":114532},[83],"。肉眼看出两簇，中间在 ",[39,114535,114537,114550],{"className":114536,"translate":43},[42],[39,114538,114540],{"className":114539},[47],[49,114541,114542],{"xmlns":51},[53,114543,114544,114548],{},[56,114545,114546],{},[539,114547,59187],{},[63,114549,59187],{"encoding":65},[39,114551,114553],{"className":114552,"ariaHidden":70},[69],[39,114554,114556,114559],{"className":114555},[74],[39,114557],{"className":114558,"style":2473},[78],[39,114560,59187],{"className":114561},[83]," 附近是空的。",[11,114564,114565,114566,114595,114596,114624,114625,114676,114677,114706,114707,114758,114759,114788,114789,114792],{},"现在用核密度估计去拟合，只用高斯核、只改带宽。带宽取 ",[39,114567,114569,114583],{"className":114568,"translate":43},[42],[39,114570,114572],{"className":114571},[47],[49,114573,114574],{"xmlns":51},[53,114575,114576,114581],{},[56,114577,114578],{},[539,114579,114580],{},"0.1",[63,114582,114580],{"encoding":65},[39,114584,114586],{"className":114585,"ariaHidden":70},[69],[39,114587,114589,114592],{"className":114588},[74],[39,114590],{"className":114591,"style":2473},[78],[39,114593,114580],{"className":114594},[83],"，每个点都是一个尖峰，曲线在两个簇内部剧烈起伏；带宽取 ",[39,114597,114599,114612],{"className":114598,"translate":43},[42],[39,114600,114602],{"className":114601},[47],[49,114603,114604],{"xmlns":51},[53,114605,114606,114610],{},[56,114607,114608],{},[539,114609,59259],{},[63,114611,59259],{"encoding":65},[39,114613,114615],{"className":114614,"ariaHidden":70},[69],[39,114616,114618,114621],{"className":114617},[74],[39,114619],{"className":114620,"style":2473},[78],[39,114622,59259],{"className":114623},[83],"，曲线被抹平。抹平到什么程度？算一下会发现，",[39,114626,114628,114646],{"className":114627,"translate":43},[42],[39,114629,114631],{"className":114630},[47],[49,114632,114633],{"xmlns":51},[53,114634,114635,114643],{},[56,114636,114637,114639,114641],{},[59,114638,1755],{},[133,114640,793],{},[539,114642,59187],{},[63,114644,114645],{"encoding":65},"x=3",[39,114647,114649,114667],{"className":114648,"ariaHidden":70},[69],[39,114650,114652,114655,114658,114661,114664],{"className":114651},[74],[39,114653],{"className":114654,"style":290},[78],[39,114656,1755],{"className":114657},[83,84],[39,114659],{"className":114660,"style":231},[230],[39,114662,793],{"className":114663},[235],[39,114665],{"className":114666,"style":231},[230],[39,114668,114670,114673],{"className":114669},[74],[39,114671],{"className":114672,"style":2473},[78],[39,114674,59187],{"className":114675},[83]," 那个空洞处的密度大约是 ",[39,114678,114680,114694],{"className":114679,"translate":43},[42],[39,114681,114683],{"className":114682},[47],[49,114684,114685],{"xmlns":51},[53,114686,114687,114692],{},[56,114688,114689],{},[539,114690,114691],{},"0.073",[63,114693,114691],{"encoding":65},[39,114695,114697],{"className":114696,"ariaHidden":70},[69],[39,114698,114700,114703],{"className":114699},[74],[39,114701],{"className":114702,"style":2473},[78],[39,114704,114691],{"className":114705},[83],"，而簇心 ",[39,114708,114710,114728],{"className":114709,"translate":43},[42],[39,114711,114713],{"className":114712},[47],[49,114714,114715],{"xmlns":51},[53,114716,114717,114725],{},[56,114718,114719,114721,114723],{},[59,114720,1755],{},[133,114722,793],{},[539,114724,546],{},[63,114726,114727],{"encoding":65},"x=1",[39,114729,114731,114749],{"className":114730,"ariaHidden":70},[69],[39,114732,114734,114737,114740,114743,114746],{"className":114733},[74],[39,114735],{"className":114736,"style":290},[78],[39,114738,1755],{"className":114739},[83,84],[39,114741],{"className":114742,"style":231},[230],[39,114744,793],{"className":114745},[235],[39,114747],{"className":114748,"style":231},[230],[39,114750,114752,114755],{"className":114751},[74],[39,114753],{"className":114754,"style":2473},[78],[39,114756,546],{"className":114757},[83]," 处是 ",[39,114760,114762,114776],{"className":114761,"translate":43},[42],[39,114763,114765],{"className":114764},[47],[49,114766,114767],{"xmlns":51},[53,114768,114769,114774],{},[56,114770,114771],{},[539,114772,114773],{},"0.071",[63,114775,114773],{"encoding":65},[39,114777,114779],{"className":114778,"ariaHidden":70},[69],[39,114780,114782,114785],{"className":114781},[74],[39,114783],{"className":114784,"style":2473},[78],[39,114786,114773],{"className":114787},[83],"——",[1489,114790,114791],{},"空洞反而比簇心更“密”","。因为核太宽，五个点全都够到了中间，而簇心只够到自己人和远处那两个点的尾巴。",[11,114794,114795,114796,114824,114825,114853,114854,114882],{},"这个例子里，带宽就是分辨率。",[39,114797,114799,114812],{"className":114798,"translate":43},[42],[39,114800,114802],{"className":114801},[47],[49,114803,114804],{"xmlns":51},[53,114805,114806,114810],{},[56,114807,114808],{},[59,114809,69998],{},[63,114811,69998],{"encoding":65},[39,114813,114815],{"className":114814,"ariaHidden":70},[69],[39,114816,114818,114821],{"className":114817},[74],[39,114819],{"className":114820,"style":28824},[78],[39,114822,69998],{"className":114823},[83,84]," 太小，方差爆炸；",[39,114826,114828,114841],{"className":114827,"translate":43},[42],[39,114829,114831],{"className":114830},[47],[49,114832,114833],{"xmlns":51},[53,114834,114835,114839],{},[56,114836,114837],{},[59,114838,69998],{},[63,114840,69998],{"encoding":65},[39,114842,114844],{"className":114843,"ariaHidden":70},[69],[39,114845,114847,114850],{"className":114846},[74],[39,114848],{"className":114849,"style":28824},[78],[39,114851,69998],{"className":114852},[83,84]," 太大，两个类之间的空隙被填平，一个明显偏离的样本反而显得很正常。我用 KDE 做异常检测时最常踩的坑就是这个：调 ",[39,114855,114857,114870],{"className":114856,"translate":43},[42],[39,114858,114860],{"className":114859},[47],[49,114861,114862],{"xmlns":51},[53,114863,114864,114868],{},[56,114865,114866],{},[59,114867,69998],{},[63,114869,69998],{"encoding":65},[39,114871,114873],{"className":114872,"ariaHidden":70},[69],[39,114874,114876,114879],{"className":114875},[74],[39,114877],{"className":114878,"style":28824},[78],[39,114880,69998],{"className":114881},[83,84]," 其实是在调“我允许多大的类间间隙”。",[11,114884,114885],{},"形式上是",[39,114887,114889],{"className":114888,"translate":43},[746],[39,114890,114892,115005],{"className":114891,"translate":43},[42],[39,114893,114895],{"className":114894},[47],[49,114896,114897],{"xmlns":51,"display":755},[53,114898,114899,115002],{},[56,114900,114901,114912,114914,114916,114918,114920,114926,114940,114946,114948,114950,114952,114958,114960,114962,114964,114970,114972,114974,114976,114978,114988,114990,114992,114994,114996,114998,115000],{},[613,114902,114903,114910],{},[39689,114904,114905,114907],{"accent":70},[59,114906,11],{},[133,114908,114909],{},"^",[59,114911,69998],{},[133,114913,136],{"stretchy":135},[59,114915,1755],{},[133,114917,161],{"stretchy":135},[133,114919,793],{},[5362,114921,114922,114924],{},[539,114923,546],{},[59,114925,12962],{},[3187,114927,114928,114930,114938],{},[133,114929,3191],{},[56,114931,114932,114934,114936],{},[59,114933,13377],{},[133,114935,793],{},[539,114937,546],{},[59,114939,12962],{},[613,114941,114942,114944],{},[59,114943,77187],{},[59,114945,69998],{},[133,114947,136],{"stretchy":135},[59,114949,1755],{},[133,114951,5378],{},[613,114953,114954,114956],{},[59,114955,1755],{},[59,114957,13377],{},[133,114959,161],{"stretchy":135},[133,114961,155],{"separator":70},[230,114963],{"width":10251},[613,114965,114966,114968],{},[59,114967,77187],{},[59,114969,69998],{},[133,114971,136],{"stretchy":135},[59,114973,35321],{},[133,114975,161],{"stretchy":135},[133,114977,793],{},[138,114979,114980,114982],{},[59,114981,69998],{},[56,114983,114984,114986],{},[133,114985,5378],{},[59,114987,29425],{},[59,114989,77187],{},[133,114991,136],{"stretchy":135},[59,114993,35321],{},[59,114995,45691],{"mathvariant":145},[59,114997,69998],{},[133,114999,161],{"stretchy":135},[59,115001,2903],{"mathvariant":145},[63,115003,115004],{"encoding":65},"\\hat p_h(x)=\\frac1n\\sum_{i=1}^nK_h(x-x_i),\\qquad K_h(u)=h^{-d}K(u\u002Fh).",[39,115006,115008,115113,115307,115423],{"className":115007,"ariaHidden":70},[69],[39,115009,115011,115014,115095,115098,115101,115104,115107,115110],{"className":115010},[74],[39,115012],{"className":115013,"style":245},[78],[39,115015,115017,115061],{"className":115016},[83],[39,115018,115020],{"className":115019},[83,39712],[39,115021,115023,115052],{"className":115022},[196,649],[39,115024,115026,115049],{"className":115025},[200],[39,115027,115029,115037],{"className":115028,"style":28824},[204],[39,115030,115031,115034],{"style":21599},[39,115032],{"className":115033,"style":935},[212],[39,115035,11],{"className":115036},[83,84],[39,115038,115039,115042],{"style":21599},[39,115040],{"className":115041,"style":935},[212],[39,115043,115046],{"className":115044,"style":115045},[39739],"left:-0.1667em;",[39,115047,114909],{"className":115048},[83],[39,115050,673],{"className":115051},[672],[39,115053,115055],{"className":115054},[200],[39,115056,115059],{"className":115057,"style":115058},[204],"height:0.1944em;",[39,115060],{},[39,115062,115064],{"className":115063},[192],[39,115065,115067,115087],{"className":115066},[196,649],[39,115068,115070,115084],{"className":115069},[200],[39,115071,115073],{"className":115072,"style":3419},[204],[39,115074,115075,115078],{"style":966},[39,115076],{"className":115077,"style":213},[212],[39,115079,115081],{"className":115080},[217,218,219,220],[39,115082,69998],{"className":115083},[83,84,220],[39,115085,673],{"className":115086},[672],[39,115088,115090],{"className":115089},[200],[39,115091,115093],{"className":115092,"style":680},[204],[39,115094],{},[39,115096,136],{"className":115097},[182],[39,115099,1755],{"className":115100},[83,84],[39,115102,161],{"className":115103},[263],[39,115105],{"className":115106,"style":231},[230],[39,115108,793],{"className":115109},[235],[39,115111],{"className":115112,"style":231},[230],[39,115114,115116,115120,115182,115185,115249,115252,115292,115295,115298,115301,115304],{"className":115115},[74],[39,115117],{"className":115118,"style":115119},[78],"height:2.9291em;vertical-align:-1.2777em;",[39,115121,115123,115126,115179],{"className":115122},[83],[39,115124],{"className":115125},[182,5463],[39,115127,115129],{"className":115128},[5362],[39,115130,115132,115171],{"className":115131},[196,649],[39,115133,115135,115168],{"className":115134},[200],[39,115136,115138,115149,115157],{"className":115137,"style":18240},[204],[39,115139,115140,115143],{"style":5479},[39,115141],{"className":115142,"style":935},[212],[39,115144,115146],{"className":115145},[83],[39,115147,12962],{"className":115148},[83,84],[39,115150,115151,115154],{"style":5503},[39,115152],{"className":115153,"style":935},[212],[39,115155],{"className":115156,"style":5511},[5510],[39,115158,115159,115162],{"style":5514},[39,115160],{"className":115161,"style":935},[212],[39,115163,115165],{"className":115164},[83],[39,115166,546],{"className":115167},[83],[39,115169,673],{"className":115170},[672],[39,115172,115174],{"className":115173},[200],[39,115175,115177],{"className":115176,"style":19487},[204],[39,115178],{},[39,115180],{"className":115181},[263,5463],[39,115183],{"className":115184,"style":256},[230],[39,115186,115188],{"className":115187},[1744,3294],[39,115189,115191,115241],{"className":115190},[196,649],[39,115192,115194,115238],{"className":115193},[200],[39,115195,115197,115217,115227],{"className":115196,"style":3304},[204],[39,115198,115199,115202],{"style":13926},[39,115200],{"className":115201,"style":3311},[212],[39,115203,115205],{"className":115204},[217,218,219,220],[39,115206,115208,115211,115214],{"className":115207},[83,220],[39,115209,13377],{"className":115210},[83,84,220],[39,115212,793],{"className":115213},[235,220],[39,115215,546],{"className":115216},[83,220],[39,115218,115219,115222],{"style":3330},[39,115220],{"className":115221,"style":3311},[212],[39,115223,115224],{},[39,115225,3191],{"className":115226},[1744,3339,3340],[39,115228,115229,115232],{"style":3343},[39,115230],{"className":115231,"style":3311},[212],[39,115233,115235],{"className":115234},[217,218,219,220],[39,115236,12962],{"className":115237},[83,84,220],[39,115239,673],{"className":115240},[672],[39,115242,115244],{"className":115243},[200],[39,115245,115247],{"className":115246,"style":78935},[204],[39,115248],{},[39,115250],{"className":115251,"style":256},[230],[39,115253,115255,115258],{"className":115254},[83],[39,115256,77187],{"className":115257,"style":48487},[83,84],[39,115259,115261],{"className":115260},[192],[39,115262,115264,115284],{"className":115263},[196,649],[39,115265,115267,115281],{"className":115266},[200],[39,115268,115270],{"className":115269,"style":3419},[204],[39,115271,115272,115275],{"style":81127},[39,115273],{"className":115274,"style":213},[212],[39,115276,115278],{"className":115277},[217,218,219,220],[39,115279,69998],{"className":115280},[83,84,220],[39,115282,673],{"className":115283},[672],[39,115285,115287],{"className":115286},[200],[39,115288,115290],{"className":115289,"style":680},[204],[39,115291],{},[39,115293,136],{"className":115294},[182],[39,115296,1755],{"className":115297},[83,84],[39,115299],{"className":115300,"style":461},[230],[39,115302,5378],{"className":115303},[465],[39,115305],{"className":115306,"style":461},[230],[39,115308,115310,115313,115353,115356,115359,115362,115365,115405,115408,115411,115414,115417,115420],{"className":115309},[74],[39,115311],{"className":115312,"style":245},[78],[39,115314,115316,115319],{"className":115315},[83],[39,115317,1755],{"className":115318},[83,84],[39,115320,115322],{"className":115321},[192],[39,115323,115325,115345],{"className":115324},[196,649],[39,115326,115328,115342],{"className":115327},[200],[39,115329,115331],{"className":115330,"style":13604},[204],[39,115332,115333,115336],{"style":966},[39,115334],{"className":115335,"style":213},[212],[39,115337,115339],{"className":115338},[217,218,219,220],[39,115340,13377],{"className":115341},[83,84,220],[39,115343,673],{"className":115344},[672],[39,115346,115348],{"className":115347},[200],[39,115349,115351],{"className":115350,"style":680},[204],[39,115352],{},[39,115354,161],{"className":115355},[263],[39,115357,155],{"className":115358},[252],[39,115360],{"className":115361,"style":10787},[230],[39,115363],{"className":115364,"style":256},[230],[39,115366,115368,115371],{"className":115367},[83],[39,115369,77187],{"className":115370,"style":48487},[83,84],[39,115372,115374],{"className":115373},[192],[39,115375,115377,115397],{"className":115376},[196,649],[39,115378,115380,115394],{"className":115379},[200],[39,115381,115383],{"className":115382,"style":3419},[204],[39,115384,115385,115388],{"style":81127},[39,115386],{"className":115387,"style":213},[212],[39,115389,115391],{"className":115390},[217,218,219,220],[39,115392,69998],{"className":115393},[83,84,220],[39,115395,673],{"className":115396},[672],[39,115398,115400],{"className":115399},[200],[39,115401,115403],{"className":115402,"style":680},[204],[39,115404],{},[39,115406,136],{"className":115407},[182],[39,115409,35321],{"className":115410},[83,84],[39,115412,161],{"className":115413},[263],[39,115415],{"className":115416,"style":231},[230],[39,115418,793],{"className":115419},[235],[39,115421],{"className":115422,"style":231},[230],[39,115424,115426,115429,115464,115467,115470,115473,115476,115479,115482],{"className":115425},[74],[39,115427],{"className":115428,"style":30346},[78],[39,115430,115432,115435],{"className":115431},[83],[39,115433,69998],{"className":115434},[83,84],[39,115436,115438],{"className":115437},[192],[39,115439,115441],{"className":115440},[196],[39,115442,115444],{"className":115443},[200],[39,115445,115447],{"className":115446,"style":3389},[204],[39,115448,115449,115452],{"style":1027},[39,115450],{"className":115451,"style":213},[212],[39,115453,115455],{"className":115454},[217,218,219,220],[39,115456,115458,115461],{"className":115457},[83,220],[39,115459,5378],{"className":115460},[83,220],[39,115462,29425],{"className":115463},[83,84,220],[39,115465,77187],{"className":115466,"style":48487},[83,84],[39,115468,136],{"className":115469},[182],[39,115471,35321],{"className":115472},[83,84],[39,115474,45691],{"className":115475},[83],[39,115477,69998],{"className":115478},[83,84],[39,115480,161],{"className":115481},[263],[39,115483,2903],{"className":115484},[83],[11,115486,115487,115488,742,115547,742,115734,115888],{},"它的偏差和方差可以算出来（记 ",[39,115489,115491,115511],{"className":115490,"translate":43},[42],[39,115492,115494],{"className":115493},[47],[49,115495,115496],{"xmlns":51},[53,115497,115498,115508],{},[56,115499,115500,115502,115504,115506],{},[133,115501,69087],{},[59,115503,77187],{},[133,115505,793],{},[539,115507,546],{},[63,115509,115510],{"encoding":65},"\\int K=1",[39,115512,115514,115538],{"className":115513,"ariaHidden":70},[69],[39,115515,115517,115520,115523,115526,115529,115532,115535],{"className":115516},[74],[39,115518],{"className":115519,"style":74140},[78],[39,115521,69087],{"className":115522,"style":74144},[1744,3339,3952],[39,115524],{"className":115525,"style":256},[230],[39,115527,77187],{"className":115528,"style":48487},[83,84],[39,115530],{"className":115531,"style":231},[230],[39,115533,793],{"className":115534},[235],[39,115536],{"className":115537,"style":231},[230],[39,115539,115541,115544],{"className":115540},[74],[39,115542],{"className":115543,"style":2473},[78],[39,115545,546],{"className":115546},[83],[39,115548,115550,115596],{"className":115549,"translate":43},[42],[39,115551,115553],{"className":115552},[47],[49,115554,115555],{"xmlns":51},[53,115556,115557,115593],{},[56,115558,115559,115561,115563,115565,115571,115573,115575,115577,115579,115581,115583,115585],{},[133,115560,69087],{},[59,115562,15389],{"mathvariant":145},[59,115564,35321],{},[138,115566,115567,115569],{},[59,115568,15389],{"mathvariant":145},[539,115570,17359],{},[59,115572,77187],{},[133,115574,136],{"stretchy":135},[59,115576,35321],{},[133,115578,161],{"stretchy":135},[59,115580,29425],{},[59,115582,35321],{},[133,115584,793],{},[9932,115586,115587,115589,115591],{},[59,115588,89893],{},[59,115590,77187],{},[539,115592,17359],{},[63,115594,115595],{"encoding":65},"\\int\\|u\\|^2K(u)du=\\sigma_K^2",[39,115597,115599,115674],{"className":115598,"ariaHidden":70},[69],[39,115600,115602,115606,115609,115612,115615,115618,115647,115650,115653,115656,115659,115662,115665,115668,115671],{"className":115601},[74],[39,115603],{"className":115604,"style":115605},[78],"height:1.1202em;vertical-align:-0.3061em;",[39,115607,69087],{"className":115608,"style":74144},[1744,3339,3952],[39,115610],{"className":115611,"style":256},[230],[39,115613,15389],{"className":115614},[83],[39,115616,35321],{"className":115617},[83,84],[39,115619,115621,115624],{"className":115620},[83],[39,115622,15389],{"className":115623},[83],[39,115625,115627],{"className":115626},[192],[39,115628,115630],{"className":115629},[196],[39,115631,115633],{"className":115632},[200],[39,115634,115636],{"className":115635,"style":19710},[204],[39,115637,115638,115641],{"style":208},[39,115639],{"className":115640,"style":213},[212],[39,115642,115644],{"className":115643},[217,218,219,220],[39,115645,17359],{"className":115646},[83,220],[39,115648,77187],{"className":115649,"style":48487},[83,84],[39,115651,136],{"className":115652},[182],[39,115654,35321],{"className":115655},[83,84],[39,115657,161],{"className":115658},[263],[39,115660,29425],{"className":115661},[83,84],[39,115663,35321],{"className":115664},[83,84],[39,115666],{"className":115667,"style":231},[230],[39,115669,793],{"className":115670},[235],[39,115672],{"className":115673,"style":231},[230],[39,115675,115677,115681],{"className":115676},[74],[39,115678],{"className":115679,"style":115680},[78],"height:1.0894em;vertical-align:-0.2753em;",[39,115682,115684,115687],{"className":115683},[83],[39,115685,89893],{"className":115686,"style":2114},[83,84],[39,115688,115690],{"className":115689},[192],[39,115691,115693,115725],{"className":115692},[196,649],[39,115694,115696,115722],{"className":115695},[200],[39,115697,115699,115711],{"className":115698,"style":19710},[204],[39,115700,115702,115705],{"style":115701},"top:-2.4247em;margin-left:-0.0359em;margin-right:0.05em;",[39,115703],{"className":115704,"style":213},[212],[39,115706,115708],{"className":115707},[217,218,219,220],[39,115709,77187],{"className":115710,"style":48487},[83,84,220],[39,115712,115713,115716],{"style":208},[39,115714],{"className":115715,"style":213},[212],[39,115717,115719],{"className":115718},[217,218,219,220],[39,115720,17359],{"className":115721},[83,220],[39,115723,673],{"className":115724},[672],[39,115726,115728],{"className":115727},[200],[39,115729,115732],{"className":115730,"style":115731},[204],"height:0.2753em;",[39,115733],{},[39,115735,115737,115771],{"className":115736,"translate":43},[42],[39,115738,115740],{"className":115739},[47],[49,115741,115742],{"xmlns":51},[53,115743,115744,115768],{},[56,115745,115746,115748,115750,115758,115760,115762],{},[59,115747,15389],{"mathvariant":145},[59,115749,77187],{},[9932,115751,115752,115754,115756],{},[59,115753,15389],{"mathvariant":145},[539,115755,17359],{},[539,115757,17359],{},[133,115759,793],{},[133,115761,69087],{},[138,115763,115764,115766],{},[59,115765,77187],{},[539,115767,17359],{},[63,115769,115770],{"encoding":65},"\\|K\\|_2^2=\\int K^2",[39,115772,115774,115847],{"className":115773,"ariaHidden":70},[69],[39,115775,115777,115780,115783,115786,115838,115841,115844],{"className":115776},[74],[39,115778],{"className":115779,"style":84615},[78],[39,115781,15389],{"className":115782},[83],[39,115784,77187],{"className":115785,"style":48487},[83,84],[39,115787,115789,115792],{"className":115788},[83],[39,115790,15389],{"className":115791},[83],[39,115793,115795],{"className":115794},[192],[39,115796,115798,115830],{"className":115797},[196,649],[39,115799,115801,115827],{"className":115800},[200],[39,115802,115804,115816],{"className":115803,"style":19710},[204],[39,115805,115807,115810],{"style":115806},"top:-2.4519em;margin-left:0em;margin-right:0.05em;",[39,115808],{"className":115809,"style":213},[212],[39,115811,115813],{"className":115812},[217,218,219,220],[39,115814,17359],{"className":115815},[83,220],[39,115817,115818,115821],{"style":208},[39,115819],{"className":115820,"style":213},[212],[39,115822,115824],{"className":115823},[217,218,219,220],[39,115825,17359],{"className":115826},[83,220],[39,115828,673],{"className":115829},[672],[39,115831,115833],{"className":115832},[200],[39,115834,115836],{"className":115835,"style":90347},[204],[39,115837],{},[39,115839],{"className":115840,"style":231},[230],[39,115842,793],{"className":115843},[235],[39,115845],{"className":115846,"style":231},[230],[39,115848,115850,115853,115856,115859],{"className":115849},[74],[39,115851],{"className":115852,"style":115605},[78],[39,115854,69087],{"className":115855,"style":74144},[1744,3339,3952],[39,115857],{"className":115858,"style":256},[230],[39,115860,115862,115865],{"className":115861},[83],[39,115863,77187],{"className":115864,"style":48487},[83,84],[39,115866,115868],{"className":115867},[192],[39,115869,115871],{"className":115870},[196],[39,115872,115874],{"className":115873},[200],[39,115875,115877],{"className":115876,"style":19710},[204],[39,115878,115879,115882],{"style":208},[39,115880],{"className":115881,"style":213},[212],[39,115883,115885],{"className":115884},[217,218,219,220],[39,115886,17359],{"className":115887},[83,220],"）：",[39,115890,115892],{"className":115891,"translate":43},[746],[39,115893,115895,116037],{"className":115894,"translate":43},[42],[39,115896,115898],{"className":115897},[47],[49,115899,115900],{"xmlns":51,"display":755},[53,115901,115902,116034],{},[56,115903,115904,115906,115908,115918,115920,115922,115924,115926,115928,115930,115932,115934,115936,115956,115958,115960,115962,115964,115966,115968,115970,115978,115980,115990,115992,115994,115996,115998,116032],{},[59,115905,4253],{"mathvariant":421},[35841,115907,35843],{},[613,115909,115910,115916],{},[39689,115911,115912,115914],{"accent":70},[59,115913,11],{},[133,115915,114909],{},[59,115917,69998],{},[133,115919,136],{"stretchy":135},[59,115921,1755],{},[133,115923,161],{"stretchy":135},[133,115925,5378],{},[59,115927,11],{},[133,115929,136],{"stretchy":135},[59,115931,1755],{},[133,115933,161],{"stretchy":135},[133,115935,99744],{},[5362,115937,115938,115954],{},[56,115939,115940,115946],{},[138,115941,115942,115944],{},[59,115943,69998],{},[539,115945,17359],{},[9932,115947,115948,115950,115952],{},[59,115949,89893],{},[59,115951,77187],{},[539,115953,17359],{},[539,115955,17359],{},[59,115957,2134],{"mathvariant":145},[59,115959,11],{},[133,115961,136],{"stretchy":135},[59,115963,1755],{},[133,115965,161],{"stretchy":135},[133,115967,155],{"separator":70},[230,115969],{"width":10251},[56,115971,115972,115974,115976],{},[59,115973,4230],{"mathvariant":145},[59,115975,158],{"mathvariant":145},[59,115977,401],{"mathvariant":145},[35841,115979,35843],{},[613,115981,115982,115988],{},[39689,115983,115984,115986],{"accent":70},[59,115985,11],{},[133,115987,114909],{},[59,115989,69998],{},[133,115991,136],{"stretchy":135},[59,115993,1755],{},[133,115995,161],{"stretchy":135},[133,115997,99744],{},[5362,115999,116000,116022],{},[56,116001,116002,116004,116006,116008,116010,116012,116014],{},[59,116003,11],{},[133,116005,136],{"stretchy":135},[59,116007,1755],{},[133,116009,161],{"stretchy":135},[59,116011,15389],{"mathvariant":145},[59,116013,77187],{},[9932,116015,116016,116018,116020],{},[59,116017,15389],{"mathvariant":145},[539,116019,17359],{},[539,116021,17359],{},[56,116023,116024,116026],{},[59,116025,12962],{},[138,116027,116028,116030],{},[59,116029,69998],{},[59,116031,29425],{},[59,116033,2903],{"mathvariant":145},[63,116035,116036],{"encoding":65},"\\mathbb E\\,\\hat p_h(x)-p(x)\\approx\\frac{h^2\\sigma_K^2}{2}\\Delta p(x),\n\\qquad\n\\mathrm{Var}\\,\\hat p_h(x)\\approx\\frac{p(x)\\|K\\|_2^2}{nh^d}.",[39,116038,116040,116149,116176,116452],{"className":116039,"ariaHidden":70},[69],[39,116041,116043,116046,116049,116052,116131,116134,116137,116140,116143,116146],{"className":116042},[74],[39,116044],{"className":116045,"style":245},[78],[39,116047,4253],{"className":116048},[83,515],[39,116050],{"className":116051,"style":256},[230],[39,116053,116055,116097],{"className":116054},[83],[39,116056,116058],{"className":116057},[83,39712],[39,116059,116061,116089],{"className":116060},[196,649],[39,116062,116064,116086],{"className":116063},[200],[39,116065,116067,116075],{"className":116066,"style":28824},[204],[39,116068,116069,116072],{"style":21599},[39,116070],{"className":116071,"style":935},[212],[39,116073,11],{"className":116074},[83,84],[39,116076,116077,116080],{"style":21599},[39,116078],{"className":116079,"style":935},[212],[39,116081,116083],{"className":116082,"style":115045},[39739],[39,116084,114909],{"className":116085},[83],[39,116087,673],{"className":116088},[672],[39,116090,116092],{"className":116091},[200],[39,116093,116095],{"className":116094,"style":115058},[204],[39,116096],{},[39,116098,116100],{"className":116099},[192],[39,116101,116103,116123],{"className":116102},[196,649],[39,116104,116106,116120],{"className":116105},[200],[39,116107,116109],{"className":116108,"style":3419},[204],[39,116110,116111,116114],{"style":966},[39,116112],{"className":116113,"style":213},[212],[39,116115,116117],{"className":116116},[217,218,219,220],[39,116118,69998],{"className":116119},[83,84,220],[39,116121,673],{"className":116122},[672],[39,116124,116126],{"className":116125},[200],[39,116127,116129],{"className":116128,"style":680},[204],[39,116130],{},[39,116132,136],{"className":116133},[182],[39,116135,1755],{"className":116136},[83,84],[39,116138,161],{"className":116139},[263],[39,116141],{"className":116142,"style":461},[230],[39,116144,5378],{"className":116145},[465],[39,116147],{"className":116148,"style":461},[230],[39,116150,116152,116155,116158,116161,116164,116167,116170,116173],{"className":116151},[74],[39,116153],{"className":116154,"style":245},[78],[39,116156,11],{"className":116157},[83,84],[39,116159,136],{"className":116160},[182],[39,116162,1755],{"className":116163},[83,84],[39,116165,161],{"className":116166},[263],[39,116168],{"className":116169,"style":231},[230],[39,116171,99744],{"className":116172},[235],[39,116174],{"className":116175,"style":231},[230],[39,116177,116179,116182,116321,116324,116327,116330,116333,116336,116339,116342,116345,116352,116355,116434,116437,116440,116443,116446,116449],{"className":116178},[74],[39,116180],{"className":116181,"style":77691},[78],[39,116183,116185,116188,116318],{"className":116184},[83],[39,116186],{"className":116187},[182,5463],[39,116189,116191],{"className":116190},[5362],[39,116192,116194,116310],{"className":116193},[196,649],[39,116195,116197,116307],{"className":116196},[200],[39,116198,116200,116211,116219],{"className":116199,"style":77710},[204],[39,116201,116202,116205],{"style":5479},[39,116203],{"className":116204,"style":935},[212],[39,116206,116208],{"className":116207},[83],[39,116209,17359],{"className":116210},[83],[39,116212,116213,116216],{"style":5503},[39,116214],{"className":116215,"style":935},[212],[39,116217],{"className":116218,"style":5511},[5510],[39,116220,116221,116224],{"style":5514},[39,116222],{"className":116223,"style":935},[212],[39,116225,116227,116256],{"className":116226},[83],[39,116228,116230,116233],{"className":116229},[83],[39,116231,69998],{"className":116232},[83,84],[39,116234,116236],{"className":116235},[192],[39,116237,116239],{"className":116238},[196],[39,116240,116242],{"className":116241},[200],[39,116243,116245],{"className":116244,"style":19710},[204],[39,116246,116247,116250],{"style":208},[39,116248],{"className":116249,"style":213},[212],[39,116251,116253],{"className":116252},[217,218,219,220],[39,116254,17359],{"className":116255},[83,220],[39,116257,116259,116262],{"className":116258},[83],[39,116260,89893],{"className":116261,"style":2114},[83,84],[39,116263,116265],{"className":116264},[192],[39,116266,116268,116299],{"className":116267},[196,649],[39,116269,116271,116296],{"className":116270},[200],[39,116272,116274,116285],{"className":116273,"style":19710},[204],[39,116275,116276,116279],{"style":115701},[39,116277],{"className":116278,"style":213},[212],[39,116280,116282],{"className":116281},[217,218,219,220],[39,116283,77187],{"className":116284,"style":48487},[83,84,220],[39,116286,116287,116290],{"style":208},[39,116288],{"className":116289,"style":213},[212],[39,116291,116293],{"className":116292},[217,218,219,220],[39,116294,17359],{"className":116295},[83,220],[39,116297,673],{"className":116298},[672],[39,116300,116302],{"className":116301},[200],[39,116303,116305],{"className":116304,"style":115731},[204],[39,116306],{},[39,116308,673],{"className":116309},[672],[39,116311,116313],{"className":1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",[39,116623,116625,116638],{"className":116624,"translate":43},[42],[39,116626,116628],{"className":116627},[47],[49,116629,116630],{"xmlns":51},[53,116631,116632,116636],{},[56,116633,116634],{},[59,116635,69998],{},[63,116637,69998],{"encoding":65},[39,116639,116641],{"className":116640,"ariaHidden":70},[69],[39,116642,116644,116647],{"className":116643},[74],[39,116645],{"className":116646,"style":28824},[78],[39,116648,69998],{"className":116649},[83,84]," 增、一个随 ",[39,116652,116654,116667],{"className":116653,"translate":43},[42],[39,116655,116657],{"className":116656},[47],[49,116658,116659],{"xmlns":51},[53,116660,116661,116665],{},[56,116662,116663],{},[59,116664,69998],{},[63,116666,69998],{"encoding":65},[39,116668,116670],{"className":116669,"ariaHidden":70},[69],[39,116671,116673,116676],{"className":116672},[74],[39,116674],{"className":116675,"style":28824},[78],[39,116677,69998],{"className":116678},[83,84]," 减，最优带宽 ",[39,116681,116683,116726],{"className":116682,"translate":43},[42],[39,116684,116686],{"className":116685},[47],[49,116687,116688],{"xmlns":51},[53,116689,116690,116723],{},[56,116691,116692,116698,116701],{},[138,116693,116694,116696],{},[59,116695,69998],{},[133,116697,21884],{},[133,116699,116700],{},"∝",[138,116702,116703,116705],{},[59,116704,12962],{},[56,116706,116707,116709,116711,116713,116715,116717,116719,116721],{},[133,116708,5378],{},[539,116710,546],{},[59,116712,45691],{"mathvariant":145},[133,116714,136],{"stretchy":135},[59,116716,29425],{},[133,116718,788],{},[539,116720,59223],{},[133,116722,161],{"stretchy":135},[63,116724,116725],{"encoding":65},"h^*\\propto n^{-1\u002F(d+4)}",[39,116727,116729,116773],{"className":116728,"ariaHidden":70},[69],[39,116730,116732,116735,116764,116767,116770],{"className":116731},[74],[39,116733],{"className":116734,"style":28824},[78],[39,116736,116738,116741],{"className":116737},[83],[39,116739,69998],{"className":116740},[83,84],[39,116742,116744],{"className":116743},[192],[39,116745,116747],{"className":116746},[196],[39,116748,116750],{"className":116749},[200],[39,116751,116753],{"className":116752,"style":21897},[204],[39,116754,116755,116758],{"style":208},[39,116756],{"className":116757,"style":213},[212],[39,116759,116761],{"className":116760},[217,218,219,220],[39,116762,21884],{"className":116763},[465,220],[39,116765],{"className":116766,"style":231},[230],[39,116768,116700],{"className":116769},[235],[39,116771],{"className":116772,"style":231},[230],[39,116774,116776,116779],{"className":116775},[74],[39,116777],{"className":116778,"style":95385},[78],[39,116780,116782,116785],{"className":116781},[83],[39,116783,12962],{"className":116784},[83,84],[39,116786,116788],{"className":116787},[192],[39,116789,116791],{"className":116790},[196],[39,116792,116794],{"className":116793},[200],[39,116795,116797],{"className":116796,"style":95385},[204],[39,116798,116799,116802],{"style":208},[39,116800],{"className":116801,"style":213},[212],[39,116803,116805],{"className":116804},[217,218,219,220],[39,116806,116808,116811,116814,116817,116820,116823,116826],{"className":116807},[83,220],[39,116809,5378],{"className":116810},[83,220],[39,116812,76672],{"className":116813},[83,220],[39,116815,136],{"className":116816},[182,220],[39,116818,29425],{"className":116819},[83,84,220],[39,116821,788],{"className":116822},[465,220],[39,116824,59223],{"className":116825},[83,220],[39,116827,161],{"className":116828},[263,220],"，此时 ",[39,116831,116833,116877],{"className":116832,"translate":43},[42],[39,116834,116836],{"className":116835},[47],[49,116837,116838],{"xmlns":51},[53,116839,116840,116874],{},[56,116841,116842,116850,116852],{},[56,116843,116844,116846,116848],{},[59,116845,1507],{"mathvariant":145},[59,116847,61],{"mathvariant":145},[59,116849,4253],{"mathvariant":145},[133,116851,116700],{},[138,116853,116854,116856],{},[59,116855,12962],{},[56,116857,116858,116860,116862,116864,116866,116868,116870,116872],{},[133,116859,5378],{},[539,116861,59223],{},[59,116863,45691],{"mathvariant":145},[133,116865,136],{"stretchy":135},[59,116867,29425],{},[133,116869,788],{},[539,116871,59223],{},[133,116873,161],{"stretchy":135},[63,116875,116876],{"encoding":65},"\\mathrm{MSE}\\propto n^{-4\u002F(d+4)}",[39,116878,116880,116902],{"className":116879,"ariaHidden":70},[69],[39,116881,116883,116886,116893,116896,116899],{"className":116882},[74],[39,116884],{"className":116885,"style":79},[78],[39,116887,116889],{"className":116888},[83],[39,116890,116892],{"className":116891},[83,35077],"MSE",[39,116894],{"className":116895,"style":231},[230],[39,116897,116700],{"className":116898},[235],[39,116900],{"className":116901,"style":231},[230],[39,116903,116905,116908],{"className":116904},[74],[39,116906],{"className":116907,"style":95385},[78],[39,116909,116911,116914],{"className":116910},[83],[39,116912,12962],{"className":116913},[83,84],[39,116915,116917],{"className":116916},[192],[39,116918,116920],{"className":116919},[196],[39,116921,116923],{"className":116922},[200],[39,116924,116926],{"className":116925,"style":95385},[204],[39,116927,116928,116931],{"style":208},[39,116929],{"className":116930,"style":213},[212],[39,116932,116934],{"className":116933},[217,218,219,220],[39,116935,116937,116940,116944,116947,116950,116953,116956],{"className":116936},[83,220],[39,116938,5378],{"className":116939},[83,220],[39,116941,116943],{"className":116942},[83,220],"4\u002F",[39,116945,136],{"className":116946},[182,220],[39,116948,29425],{"className":116949},[83,84,220],[39,116951,788],{"className":116952},[465,220],[39,116954,59223],{"className":116955},[83,220],[39,116957,161],{"className":116958},[263,220],[11,116960,116961,117012,117013,117087,117088,117012,117139,117214,117215,117300],{},[39,116962,116964,116982],{"className":116963,"translate":43},[42],[39,116965,116967],{"className":116966},[47],[49,116968,116969],{"xmlns":51},[53,116970,116971,116979],{},[56,116972,116973,116975,116977],{},[59,116974,29425],{},[133,116976,793],{},[539,116978,546],{},[63,116980,116981],{"encoding":65},"d=1",[39,116983,116985,117003],{"className":116984,"ariaHidden":70},[69],[39,116986,116988,116991,116994,116997,117000],{"className":116987},[74],[39,116989],{"className":116990,"style":28824},[78],[39,116992,29425],{"className":116993},[83,84],[39,116995],{"className":116996,"style":231},[230],[39,116998,793],{"className":116999},[235],[39,117001],{"className":117002,"style":231},[230],[39,117004,117006,117009],{"className":117005},[74],[39,117007],{"className":117008,"style":2473},[78],[39,117010,546],{"className":117011},[83]," 时是 ",[39,117014,117016,117042],{"className":117015,"translate":43},[42],[39,117017,117019],{"className":117018},[47],[49,117020,117021],{"xmlns":51},[53,117022,117023,117039],{},[56,117024,117025],{},[138,117026,117027,117029],{},[59,117028,12962],{},[56,117030,117031,117033,117035,117037],{},[133,117032,5378],{},[539,117034,59223],{},[59,117036,45691],{"mathvariant":145},[539,117038,59259],{},[63,117040,117041],{"encoding":65},"n^{-4\u002F5}",[39,117043,117045],{"className":117044,"ariaHidden":70},[69],[39,117046,117048,117051],{"className":117047},[74],[39,117049],{"className":117050,"style":95385},[78],[39,117052,117054,117057],{"className":117053},[83],[39,117055,12962],{"className":117056},[83,84],[39,117058,117060],{"className":117059},[192],[39,117061,117063],{"className":117062},[196],[39,117064,117066],{"className":117065},[200],[39,117067,117069],{"className":117068,"style":95385},[204],[39,117070,117071,117074],{"style":208},[39,117072],{"className":117073,"style":213},[212],[39,117075,117077],{"className":117076},[217,218,219,220],[39,117078,117080,117083],{"className":117079},[83,220],[39,117081,5378],{"className":117082},[83,220],[39,117084,117086],{"className":117085},[83,220],"4\u002F5","，很不错。",[39,117089,117091,117109],{"className":117090,"translate":43},[42],[39,117092,117094],{"className":117093},[47],[49,117095,117096],{"xmlns":51},[53,117097,117098,117106],{},[56,117099,117100,117102,117104],{},[59,117101,29425],{},[133,117103,793],{},[539,117105,48919],{},[63,117107,117108],{"encoding":65},"d=10",[39,117110,117112,117130],{"className":117111,"ariaHidden":70},[69],[39,117113,117115,117118,117121,117124,117127],{"className":117114},[74],[39,117116],{"className":117117,"style":28824},[78],[39,117119,29425],{"className":117120},[83,84],[39,117122],{"className":117123,"style":231},[230],[39,117125,793],{"className":117126},[235],[39,117128],{"className":117129,"style":231},[230],[39,117131,117133,117136],{"className":117132},[74],[39,117134],{"className":117135,"style":2473},[78],[39,117137,48919],{"className":117138},[83],[39,117140,117142,117169],{"className":117141,"translate":43},[42],[39,117143,117145],{"className":117144},[47],[49,117146,117147],{"xmlns":51},[53,117148,117149,117166],{},[56,117150,117151],{},[138,117152,117153,117155],{},[59,117154,12962],{},[56,117156,117157,117159,117161,117163],{},[133,117158,5378],{},[539,117160,17359],{},[59,117162,45691],{"mathvariant":145},[539,117164,117165],{},"7",[63,117167,117168],{"encoding":65},"n^{-2\u002F7}",[39,117170,117172],{"className":117171,"ariaHidden":70},[69],[39,117173,117175,117178],{"className":117174},[74],[39,117176],{"className":117177,"style":95385},[78],[39,117179,117181,117184],{"className":117180},[83],[39,117182,12962],{"className":117183},[83,84],[39,117185,117187],{"className":117186},[192],[39,117188,117190],{"className":117189},[196],[39,117191,117193],{"className":117192},[200],[39,117194,117196],{"className":117195,"style":95385},[204],[39,117197,117198,117201],{"style":208},[39,117199],{"className":117200,"style":213},[212],[39,117202,117204],{"className":117203},[217,218,219,220],[39,117205,117207,117210],{"className":117206},[83,220],[39,117208,5378],{"className":117209},[83,220],[39,117211,117213],{"className":117212},[83,220],"2\u002F7","，想把误差减半，样本量要乘 ",[39,117216,117218,117241],{"className":117217,"translate":43},[42],[39,117219,117221],{"className":117220},[47],[49,117222,117223],{"xmlns":51},[53,117224,117225,117238],{},[56,117226,117227,117234,117236],{},[138,117228,117229,117231],{},[539,117230,17359],{},[539,117232,117233],{},"3.5",[133,117235,99744],{},[539,117237,73213],{},[63,117239,117240],{"encoding":65},"2^{3.5}\\approx11",[39,117242,117244,117291],{"className":117243,"ariaHidden":70},[69],[39,117245,117247,117250,117282,117285,117288],{"className":117246},[74],[39,117248],{"className":117249,"style":19710},[78],[39,117251,117253,117256],{"className":117252},[83],[39,117254,17359],{"className":117255},[83],[39,117257,117259],{"className":117258},[192],[39,117260,117262],{"className":117261},[196],[39,117263,117265],{"className":117264},[200],[39,117266,117268],{"className":117267,"style":19710},[204],[39,117269,117270,117273],{"style":208},[39,117271],{"className":117272,"style":213},[212],[39,117274,117276],{"className":117275},[217,218,219,220],[39,117277,117279],{"className":117278},[83,220],[39,117280,117233],{"className":117281},[83,220],[39,117283],{"className":117284,"style":231},[230],[39,117286,99744],{"className":117287},[235],[39,117289],{"className":117290,"style":231},[230],[39,117292,117294,117297],{"className":117293},[74],[39,117295],{"className":117296,"style":2473},[78],[39,117298,73213],{"className":117299},[83],"。这个数字我建议背下来：每加几个维度，KDE 就从“能用”滑向“不能用”。",[11,117302,117303,117304,117403,117404,117496,117497,117569,117570,117671,117672,117815,117816,117887,117888,117012,117938,1607],{},"但 KDE 在高维还有一层跟带宽无关的麻烦。考虑标准正态 ",[39,117305,117307,117335],{"className":117306,"translate":43},[42],[39,117308,117310],{"className":117309},[47],[49,117311,117312],{"xmlns":51},[53,117313,117314,117332],{},[56,117315,117316,117318,117320,117322,117324,117330],{},[59,117317,77559],{"mathvariant":617},[133,117319,136],{"stretchy":135},[539,117321,541],{},[133,117323,155],{"separator":70},[613,117325,117326,117328],{},[59,117327,14373],{},[59,117329,29425],{},[133,117331,161],{"stretchy":135},[63,117333,117334],{"encoding":65},"\\mathcal N(0,I_d)",[39,117336,117338],{"className":117337,"ariaHidden":70},[69],[39,117339,117341,117344,117347,117350,117353,117356,117359,117400],{"className":117340},[74],[39,117342],{"className":117343,"style":245},[78],[39,117345,77559],{"className":117346,"style":77627},[83,641],[39,117348,136],{"className":117349},[182],[39,117351,541],{"className":117352},[83],[39,117354,155],{"className":117355},[252],[39,117357],{"className":117358,"style":256},[230],[39,117360,117362,117365],{"className":117361},[83],[39,117363,14373],{"className":117364,"style":14420},[83,84],[39,117366,117368],{"className":117367},[192],[39,117369,117371,117392],{"className":117370},[196,649],[39,117372,117374,117389],{"className":117373},[200],[39,117375,117377],{"className":117376,"style":3419},[204],[39,117378,117380,117383],{"style":117379},"top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;",[39,117381],{"className":117382,"style":213},[212],[39,117384,117386],{"className":117385},[217,218,219,220],[39,117387,29425],{"className":117388},[83,84,220],[39,117390,673],{"className":117391},[672],[39,117393,117395],{"className":117394},[200],[39,117396,117398],{"className":117397,"style":680},[204],[39,117399],{},[39,117401,161],{"className":117402},[263],"。众数在原点的密度是 ",[39,117405,117407,117439],{"className":117406,"translate":43},[42],[39,117408,117410],{"className":117409},[47],[49,117411,117412],{"xmlns":51},[53,117413,117414,117436],{},[56,117415,117416,117418,117420,117422],{},[133,117417,136],{"stretchy":135},[539,117419,17359],{},[59,117421,2098],{},[138,117423,117424,117426],{},[133,117425,161],{"stretchy":135},[56,117427,117428,117430,117432,117434],{},[133,117429,5378],{},[59,117431,29425],{},[59,117433,45691],{"mathvariant":145},[539,117435,17359],{},[63,117437,117438],{"encoding":65},"(2\\pi)^{-d\u002F2}",[39,117440,117442],{"className":117441,"ariaHidden":70},[69],[39,117443,117445,117449,117452,117455,117458],{"className":117444},[74],[39,117446],{"className":117447,"style":117448},[78],"height:1.138em;vertical-align:-0.25em;",[39,117450,136],{"className":117451},[182],[39,117453,17359],{"className":117454},[83],[39,117456,2098],{"className":117457,"style":2114},[83,84],[39,117459,117461,117464],{"className":117460},[263],[39,117462,161],{"className":117463},[263],[39,117465,117467],{"className":117466},[192],[39,117468,117470],{"className":117469},[196],[39,117471,117473],{"className":117472},[200],[39,117474,117476],{"className":117475,"style":95385},[204],[39,117477,117478,117481],{"style":208},[39,117479],{"className":117480,"style":213},[212],[39,117482,117484],{"className":117483},[217,218,219,220],[39,117485,117487,117490,117493],{"className":117486},[83,220],[39,117488,5378],{"className":117489},[83,220],[39,117491,29425],{"className":117492},[83,84,220],[39,117494,87914],{"className":117495},[83,220],"；而一个“典型”样本落在半径 ",[39,117498,117500,117516],{"className":117499,"translate":43},[42],[39,117501,117503],{"className":117502},[47],[49,117504,117505],{"xmlns":51},[53,117506,117507,117513],{},[56,117508,117509],{},[98996,117510,117511],{},[59,117512,29425],{},[63,117514,117515],{"encoding":65},"\\sqrt d",[39,117517,117519],{"className":117518,"ariaHidden":70},[69],[39,117520,117522,117525],{"className":117521},[74],[39,117523],{"className":117524,"style":100088},[78],[39,117526,117528],{"className":117527},[83,99058],[39,117529,117531,117561],{"className":117530},[196,649],[39,117532,117534,117558],{"className":117533},[200],[39,117535,117537,117546],{"className":117536,"style":100113},[204],[39,117538,117540,117543],{"className":117539,"style":21599},[99072],[39,117541],{"className":117542,"style":935},[212],[39,117544,29425],{"className":117545,"style":100403},[83,84],[39,117547,117548,117551],{"style":100126},[39,117549],{"className":117550,"style":935},[212],[39,117552,117554],{"className":117553,"style":100133},[99114],[85913,117555,117556],{"xmlns":85915,"width":99118,"height":100136,"viewBox":100137,"preserveAspectRatio":99121},[85919,117557],{"d":100140},[39,117559,673],{"className":117560},[672],[39,117562,117564],{"className":117563},[200],[39,117565,117567],{"className":117566,"style":100150},[204],[39,117568],{}," 的球壳上（因为 ",[39,117571,117573,117603],{"className":117572,"translate":43},[42],[39,117574,117576],{"className":117575},[47],[49,117577,117578],{"xmlns":51},[53,117579,117580,117600],{},[56,117581,117582,117584,117586,117592,117594,117596,117598],{},[59,117583,15389],{"mathvariant":145},[59,117585,29420],{},[138,117587,117588,117590],{},[59,117589,15389],{"mathvariant":145},[539,117591,17359],{},[59,117593,45691],{"mathvariant":145},[59,117595,29425],{},[133,117597,418],{},[539,117599,546],{},[63,117601,117602],{"encoding":65},"\\|X\\|^2\u002Fd\\to1",[39,117604,117606,117662],{"className":117605,"ariaHidden":70},[69],[39,117607,117609,117612,117615,117618,117647,117650,117653,117656,117659],{"className":117608},[74],[39,117610],{"className":117611,"style":84615},[78],[39,117613,15389],{"className":117614},[83],[39,117616,29420],{"className":117617,"style":14420},[83,84],[39,117619,117621,117624],{"className":117620},[83],[39,117622,15389],{"className":117623},[83],[39,117625,117627],{"className":117626},[192],[39,117628,117630],{"className":117629},[196],[39,117631,117633],{"className":117632},[200],[39,117634,117636],{"className":117635,"style":19710},[204],[39,117637,117638,117641],{"style":208},[39,117639],{"className":117640,"style":213},[212],[39,117642,117644],{"className":117643},[217,218,219,220],[39,117645,17359],{"className":117646},[83,220],[39,117648,45691],{"className":117649},[83],[39,117651,29425],{"className":117652},[83,84],[39,117654],{"className":117655,"style":231},[230],[39,117657,418],{"className":117658},[235],[39,117660],{"className":117661,"style":231},[230],[39,117663,117665,117668],{"className":117664},[74],[39,117666],{"className":117667,"style":2473},[78],[39,117669,546],{"className":117670},[83],"），那里的密度是 ",[39,117673,117675,117721],{"className":117674,"translate":43},[42],[39,117676,117678],{"className":117677},[47],[49,117679,117680],{"xmlns":51},[53,117681,117682,117718],{},[56,117683,117684,117686,117688,117690,117704],{},[133,117685,136],{"stretchy":135},[539,117687,17359],{},[59,117689,2098],{},[138,117691,117692,117694],{},[133,117693,161],{"stretchy":135},[56,117695,117696,117698,117700,117702],{},[133,117697,5378],{},[59,117699,29425],{},[59,117701,45691],{"mathvariant":145},[539,117703,17359],{},[138,117705,117706,117708],{},[59,117707,102560],{},[56,117709,117710,117712,117714,117716],{},[133,117711,5378],{},[59,117713,29425],{},[59,117715,45691],{"mathvariant":145},[539,117717,17359],{},[63,117719,117720],{"encoding":65},"(2\\pi)^{-d\u002F2}e^{-d\u002F2}",[39,117722,117724],{"className":117723,"ariaHidden":70},[69],[39,117725,117727,117730,117733,117736,117739,117777],{"className":117726},[74],[39,117728],{"className":117729,"style":117448},[78],[39,117731,136],{"className":117732},[182],[39,117734,17359],{"className":117735},[83],[39,117737,2098],{"className":117738,"style":2114},[83,84],[39,117740,117742,117745],{"className":117741},[263],[39,117743,161],{"className":117744},[263],[39,117746,117748],{"className":117747},[192],[39,117749,117751],{"className":117750},[196],[39,117752,117754],{"className":117753},[200],[39,117755,117757],{"className":117756,"style":95385},[204],[39,117758,117759,117762],{"style":208},[39,117760],{"className":117761,"style":213},[212],[39,117763,117765],{"className":117764},[217,218,219,220],[39,117766,117768,117771,117774],{"className":117767},[83,220],[39,117769,5378],{"className":117770},[83,220],[39,117772,29425],{"className":117773},[83,84,220],[39,117775,87914],{"className":117776},[83,220],[39,117778,117780,117783],{"className":117779},[83],[39,117781,102560],{"className":117782},[83,84],[39,117784,117786],{"className":117785},[192],[39,117787,117789],{"className":117788},[196],[39,117790,117792],{"className":117791},[200],[39,117793,117795],{"className":117794,"style":95385},[204],[39,117796,117797,117800],{"style":208},[39,117798],{"className":117799,"style":213},[212],[39,117801,117803],{"className":117802},[217,218,219,220],[39,117804,117806,117809,117812],{"className":117805},[83,220],[39,117807,5378],{"className":117808},[83,220],[39,117810,29425],{"className":117811},[83,84,220],[39,117813,87914],{"className":117814},[83,220],"。两者差 ",[39,117817,117819,117843],{"className":117818,"translate":43},[42],[39,117820,117822],{"className":117821},[47],[49,117823,117824],{"xmlns":51},[53,117825,117826,117840],{},[56,117827,117828],{},[138,117829,117830,117832],{},[59,117831,102560],{},[56,117833,117834,117836,117838],{},[59,117835,29425],{},[59,117837,45691],{"mathvariant":145},[539,117839,17359],{},[63,117841,117842],{"encoding":65},"e^{d\u002F2}",[39,117844,117846],{"className":117845,"ariaHidden":70},[69],[39,117847,117849,117852],{"className":117848},[74],[39,117850],{"className":117851,"style":95385},[78],[39,117853,117855,117858],{"className":117854},[83],[39,117856,102560],{"className":117857},[83,84],[39,117859,117861],{"className":117860},[192],[39,117862,117864],{"className":117863},[196],[39,117865,117867],{"className":117866},[200],[39,117868,117870],{"className":117869,"style":95385},[204],[39,117871,117872,117875],{"style":208},[39,117873],{"className":117874,"style":213},[212],[39,117876,117878],{"className":117877},[217,218,219,220],[39,117879,117881,117884],{"className":117880},[83,220],[39,117882,29425],{"className":117883},[83,84,220],[39,117885,87914],{"className":117886},[83,220]," 倍，",[39,117889,117891,117908],{"className":117890,"translate":43},[42],[39,117892,117894],{"className":117893},[47],[49,117895,117896],{"xmlns":51},[53,117897,117898,117906],{},[56,117899,117900,117902,117904],{},[59,117901,29425],{},[133,117903,793],{},[539,117905,28724],{},[63,117907,95149],{"encoding":65},[39,117909,117911,117929],{"className":117910,"ariaHidden":70},[69],[39,117912,117914,117917,117920,117923,117926],{"className":117913},[74],[39,117915],{"className":117916,"style":28824},[78],[39,117918,29425],{"className":117919},[83,84],[39,117921],{"className":117922,"style":231},[230],[39,117924,793],{"className":117925},[235],[39,117927],{"className":117928,"style":231},[230],[39,117930,117932,117935],{"className":117931},[74],[39,117933],{"className":117934,"style":2473},[78],[39,117936,28724],{"className":117937},[83],[39,117939,117941,117972],{"className":117940,"translate":43},[42],[39,117942,117944],{"className":117943},[47],[49,117945,117946],{"xmlns":51},[53,117947,117948,117969],{},[56,117949,117950,117956,117958,117960,117962],{},[138,117951,117952,117954],{},[59,117953,102560],{},[539,117955,61535],{},[133,117957,99744],{},[539,117959,59259],{},[133,117961,409],{},[138,117963,117964,117966],{},[539,117965,48919],{},[539,117967,117968],{},"21",[63,117970,117971],{"encoding":65},"e^{50}\\approx5\\times10^{21}",[39,117973,117975,118022,118040],{"className":117974,"ariaHidden":70},[69],[39,117976,117978,117981,118013,118016,118019],{"className":117977},[74],[39,117979],{"className":117980,"style":19710},[78],[39,117982,117984,117987],{"className":117983},[83],[39,117985,102560],{"className":117986},[83,84],[39,117988,117990],{"className":117989},[192],[39,117991,117993],{"className":117992},[196],[39,117994,117996],{"className":117995},[200],[39,117997,117999],{"className":117998,"style":19710},[204],[39,118000,118001,118004],{"style":208},[39,118002],{"className":118003,"style":213},[212],[39,118005,118007],{"className":118006},[217,218,219,220],[39,118008,118010],{"className":118009},[83,220],[39,118011,61535],{"className":118012},[83,220],[39,118014],{"className":118015,"style":231},[230],[39,118017,99744],{"className":118018},[235],[39,118020],{"className":118021,"style":231},[230],[39,118023,118025,118028,118031,118034,118037],{"className":118024},[74],[39,118026],{"className":118027,"style":17882},[78],[39,118029,59259],{"className":118030},[83],[39,118032],{"className":118033,"style":461},[230],[39,118035,409],{"className":118036},[465],[39,118038],{"className":118039,"style":461},[230],[39,118041,118043,118046,118049],{"className":118042},[74],[39,118044],{"className":118045,"style":19710},[78],[39,118047,546],{"className":118048},[83],[39,118050,118052,118055],{"className":118051},[83],[39,118053,541],{"className":118054},[83],[39,118056,118058],{"className":118057},[192],[39,118059,118061],{"className":118060},[196],[39,118062,118064],{"className":118063},[200],[39,118065,118067],{"className":118066,"style":19710},[204],[39,118068,118069,118072],{"style":208},[39,118070],{"className":118071,"style":213},[212],[39,118073,118075],{"className":118074},[217,218,219,220],[39,118076,118078],{"className":118077},[83,220],[39,118079,117968],{"className":118080},[83,220],[11,118082,118083],{},"有个很好记的画面：一个 100 维的橙子，果皮比果肉厚得多。球的体积几乎全在表面附近那层薄壳里，维度越高这层壳占的比例越大。所以“最典型的位置”不是圆心，而是那层壳。密度的最大值在圆心，可那里几乎没有质量。",[11,118085,118086],{},"这意味着什么？如果我用“密度低于阈值”判 OOD，而阈值按众数附近校准，那几乎会把所有正常样本都判成异常。这就是典型集悖论。",[11,118088,118089],{},"所以“低密度等于分布外”这个直觉在高维是错的。密度是点态的，回答的是“这个点附近有多挤”；我真正想问的是“这个点离已知类别的概率质量有多远”。后者是分布态的问题，得用前面那些工具。",[21,118091,118092],{"id":118092},"可逆变换",[11,118094,118095,118096,118100,118101,118157],{},"写采样代码时我总会用到同一招：",[118097,118098,118099],"code",{},"random()"," 只能给 ",[39,118102,118104,118127],{"className":118103,"translate":43},[42],[39,118105,118107],{"className":118106},[47],[49,118108,118109],{"xmlns":51},[53,118110,118111,118125],{},[56,118112,118113,118115,118117,118119,118121,118123],{},[59,118114,85394],{},[133,118116,136],{"stretchy":135},[539,118118,541],{},[133,118120,155],{"separator":70},[539,118122,546],{},[133,118124,161],{"stretchy":135},[63,118126,85466],{"encoding":65},[39,118128,118130],{"className":118129,"ariaHidden":70},[69],[39,118131,118133,118136,118139,118142,118145,118148,118151,118154],{"className":118132},[74],[39,118134],{"className":118135,"style":245},[78],[39,118137,85394],{"className":118138,"style":85420},[83,84],[39,118140,136],{"className":118141},[182],[39,118143,541],{"className":118144},[83],[39,118146,155],{"className":118147},[252],[39,118149],{"className":118150,"style":256},[230],[39,118152,546],{"className":118153},[83],[39,118155,161],{"className":118156},[263],"，可我想要指数分布，于是取",[39,118159,118161],{"className":118160,"translate":43},[746],[39,118162,118164,118205],{"className":118163,"translate":43},[42],[39,118165,118167],{"className":118166},[47],[49,118168,118169],{"xmlns":51,"display":755},[53,118170,118171,118202],{},[56,118172,118173,118175,118177,118179,118200],{},[59,118174,1755],{},[133,118176,793],{},[133,118178,5378],{},[5362,118180,118181,118197],{},[56,118182,118183,118185,118187,118189,118191,118193,118195],{},[59,118184,92483],{},[133,118186,1690],{},[133,118188,136],{"stretchy":135},[539,118190,546],{},[133,118192,5378],{},[59,118194,35321],{},[133,118196,161],{"stretchy":135},[59,118198,118199],{},"λ",[59,118201,2903],{"mathvariant":145},[63,118203,118204],{"encoding":65},"x=-\\frac{\\log(1-u)}{\\lambda}.",[39,118206,118208,118226],{"className":118207,"ariaHidden":70},[69],[39,118209,118211,118214,118217,118220,118223],{"className":118210},[74],[39,118212],{"className":118213,"style":290},[78],[39,118215,1755],{"className":118216},[83,84],[39,118218],{"className":118219,"style":231},[230],[39,118221,793],{"className":118222},[235],[39,118224],{"className":118225,"style":231},[230],[39,118227,118229,118233,118236,118321],{"className":118228},[74],[39,118230],{"className":118231,"style":118232},[78],"height:2.113em;vertical-align:-0.686em;",[39,118234,5378],{"className":118235},[83],[39,118237,118239,118242,118318],{"className":118238},[83],[39,118240],{"className":118241},[182,5463],[39,118243,118245],{"className":118244},[5362],[39,118246,118248,118310],{"className":118247},[196,649],[39,118249,118251,118307],{"className":118250},[200],[39,118252,118254,118265,118273],{"className":118253,"style":58292},[204],[39,118255,118256,118259],{"style":5479},[39,118257],{"className":118258,"style":935},[212],[39,118260,118262],{"className":118261},[83],[39,118263,118199],{"className":118264},[83,84],[39,118266,118267,118270],{"style":5503},[39,118268],{"className":118269,"style":935},[212],[39,118271],{"className":118272,"style":5511},[5510],[39,118274,118275,118278],{"style":5514},[39,118276],{"className":118277,"style":935},[212],[39,118279,118281,118286,118289,118292,118295,118298,118301,118304],{"className":118280},[83],[39,118282,92721,118284],{"className":118283},[1744],[39,118285,39209],{"style":35078},[39,118287,136],{"className":118288},[182],[39,118290,546],{"className":118291},[83],[39,118293],{"className":118294,"style":461},[230],[39,118296,5378],{"className":118297},[465],[39,118299],{"className":118300,"style":461},[230],[39,118302,35321],{"className":118303},[83,84],[39,118305,161],{"className":118306},[263],[39,118308,673],{"className":118309},[672],[39,118311,118313],{"className":118312},[200],[39,118314,118316],{"className":118315,"style":19487},[204],[39,118317],{},[39,118319],{"className":118320},[263,5463],[39,118322,2903],{"className":118323},[83],[11,118325,118326,118327,38249,118445,118473],{},"这一个式子就是一个标准化流。反过来看，",[39,118328,118330,118362],{"className":118329,"translate":43},[42],[39,118331,118333],{"className":118332},[47],[49,118334,118335],{"xmlns":51},[53,118336,118337,118359],{},[56,118338,118339,118341,118343,118345,118347],{},[59,118340,35321],{},[133,118342,793],{},[539,118344,546],{},[133,118346,5378],{},[138,118348,118349,118351],{},[59,118350,102560],{},[56,118352,118353,118355,118357],{},[133,118354,5378],{},[59,118356,118199],{},[59,118358,1755],{},[63,118360,118361],{"encoding":65},"u=1-e^{-\\lambda x}",[39,118363,118365,118383,118401],{"className":118364,"ariaHidden":70},[69],[39,118366,118368,118371,118374,118377,118380],{"className":118367},[74],[39,118369],{"className":118370,"style":290},[78],[39,118372,35321],{"className":118373},[83,84],[39,118375],{"className":118376,"style":231},[230],[39,118378,793],{"className":118379},[235],[39,118381],{"className":118382,"style":231},[230],[39,118384,118386,118389,118392,118395,118398],{"className":118385},[74],[39,118387],{"className":118388,"style":17882},[78],[39,118390,546],{"className":118391},[83],[39,118393],{"className":118394,"style":461},[230],[39,118396,5378],{"className":118397},[465],[39,118399],{"className":118400,"style":461},[230],[39,118402,118404,118407],{"className":118403},[74],[39,118405],{"className":118406,"style":15577},[78],[39,118408,118410,118413],{"className":118409},[83],[39,118411,102560],{"className":118412},[83,84],[39,118414,118416],{"className":118415},[192],[39,118417,118419],{"className":118418},[196],[39,118420,118422],{"className":118421},[200],[39,118423,118425],{"className":118424,"style":15577},[204],[39,118426,118427,118430],{"style":208},[39,118428],{"className":118429,"style":213},[212],[39,118431,118433],{"className":118432},[217,218,219,220],[39,118434,118436,118439,118442],{"className":118435},[83,220],[39,118437,5378],{"className":118438},[83,220],[39,118440,118199],{"className":118441},[83,84,220],[39,118443,1755],{"className":118444},[83,84,220],[39,118446,118448,118461],{"className":118447,"translate":43},[42],[39,118449,118451],{"className":118450},[47],[49,118452,118453],{"xmlns":51},[53,118454,118455,118459],{},[56,118456,118457],{},[59,118458,1755],{},[63,118460,1755],{"encoding":65},[39,118462,118464],{"className":118463,"ariaHidden":70},[69],[39,118465,118467,118470],{"className":118466},[74],[39,118468],{"className":118469,"style":290},[78],[39,118471,1755],{"className":118472},[83,84]," 的 CDF，而密度按变量替换是",[39,118475,118477],{"className":118476,"translate":43},[746],[39,118478,118480,118560],{"className":118479,"translate":43},[42],[39,118481,118483],{"className":118482},[47],[49,118484,118485],{"xmlns":51,"display":755},[53,118486,118487,118557],{},[56,118488,118489,118495,118497,118499,118501,118503,118509,118511,118513,118515,118535,118537,118539,118541,118543,118555],{},[613,118490,118491,118493],{},[59,118492,11],{},[59,118494,1755],{},[133,118496,136],{"stretchy":135},[59,118498,1755],{},[133,118500,161],{"stretchy":135},[133,118502,793],{},[613,118504,118505,118507],{},[59,118506,11],{},[59,118508,35321],{},[133,118510,136],{"stretchy":135},[59,118512,35321],{},[133,118514,161],{"stretchy":135},[56,118516,118517,118519,118533],{},[133,118518,150],{"fence":70},[5362,118520,118521,118527],{},[56,118522,118523,118525],{},[59,118524,29425],{},[59,118526,35321],{},[56,118528,118529,118531],{},[59,118530,29425],{},[59,118532,1755],{},[133,118534,150],{"fence":70},[133,118536,793],{},[539,118538,546],{},[133,118540,8885],{},[59,118542,118199],{},[138,118544,118545,118547],{},[59,118546,102560],{},[56,118548,118549,118551,118553],{},[133,118550,5378],{},[59,118552,118199],{},[59,118554,1755],{},[133,118556,155],{"separator":70},[63,118558,118559],{"encoding":65},"p_x(x)=p_u(u)\\left|\\frac{du}{dx}\\right|=1\\cdot\\lambda e^{-\\lambda 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p_x(x)=\\log p_z(z)-\\log\\bigl|\\det J_f(z)\\bigr|.",[39,119622,119624,119696,119768],{"className":119623,"ariaHidden":70},[69],[39,119625,119627,119630,119635,119638,119678,119681,119684,119687,119690,119693],{"className":119626},[74],[39,119628],{"className":119629,"style":245},[78],[39,119631,92721,119633],{"className":119632},[1744],[39,119634,39209],{"style":35078},[39,119636],{"className":119637,"style":256},[230],[39,119639,119641,119644],{"className":119640},[83],[39,119642,11],{"className":119643},[83,84],[39,119645,119647],{"className":119646},[192],[39,119648,119650,119670],{"className":119649},[196,649],[39,119651,119653,119667],{"className":119652},[200],[39,119654,119656],{"className":119655,"style":1728},[204],[39,119657,119658,119661],{"style":966},[39,119659],{"className":119660,"style":213},[212],[39,119662,119664],{"className":119663},[217,218,219,220],[39,119665,1755],{"className":119666},[83,84,220],[39,119668,673],{"className":119669},[672],[39,119671,119673],{"className":119672},[200],[39,119674,119676],{"className":119675,"style":680},[204],[39,119677],{},[39,119679,136],{"className":119680},[182],[39,119682,1755],{"className":119683},[83,84],[39,119685,161],{"className":119686},[263],[39,119688],{"className":119689,"style":231},[230],[39,119691,793],{"className":119692},[235],[39,119694],{"className":119695,"style":231},[230],[39,119697,119699,119702,119707,119710,119750,119753,119756,119759,119762,119765],{"className":119698},[74],[39,119700],{"className":119701,"style":245},[78],[39,119703,92721,119705],{"className":119704},[1744],[39,119706,39209],{"style":35078},[39,119708],{"className":119709,"style":256},[230],[39,119711,119713,119716],{"className":119712},[83],[39,119714,11],{"className":119715},[83,84],[39,119717,119719],{"className":119718},[192],[39,119720,119722,119742],{"className":119721},[196,649],[39,119723,119725,119739],{"className":119724},[200],[39,119726,119728],{"className":119727,"style":1728},[204],[39,119729,119730,119733],{"style":966},[39,119731],{"className":119732,"style":213},[212],[39,119734,119736],{"className":119735},[217,218,219,220],[39,119737,118943],{"className":119738,"style":118985},[83,84,220],[39,119740,673],{"className":119741},[672],[39,119743,119745],{"className":119744},[200],[39,119746,119748],{"className":119747,"style":680},[204],[39,119749],{},[39,119751,136],{"className":119752},[182],[39,119754,118943],{"className":119755,"style":118985},[83,84],[39,119757,161],{"className":119758},[263],[39,119760],{"className":119761,"style":461},[230],[39,119763,5378],{"className":119764},[465],[39,119766],{"className":119767,"style":461},[230],[39,119769,119771,119774,119779,119816,119819,119822,119864,119867,119870,119873,119910],{"className":119770},[74],[39,119772],{"className":119773,"style":908},[78],[39,119775,92721,119777],{"className":119776},[1744],[39,119778,39209],{"style":35078},[39,119780,119782],{"className":119781},[182],[39,119783,119785],{"className":119784},[18520,85892],[39,119786,119788,119808],{"className":119787},[196,649],[39,119789,119791,119805],{"className":119790},[200],[39,119792,119794],{"className":119793,"style":928},[204],[39,119795,119796,119799],{"style":85904},[39,119797],{"className":119798,"style":85908},[212],[39,119800,119801],{"style":85911},[85913,119802,119803],{"xmlns":85915,"width":85916,"height":73875,"viewBox":85917},[85919,119804],{"d":85921},[39,119806,673],{"className":119807},[672],[39,119809,119811],{"className":119810},[200],[39,119812,119814],{"className":119813,"style":1217},[204],[39,119815],{},[39,119817,119143],{"className":119818},[1744],[39,119820],{"className":119821,"style":256},[230],[39,119823,119825,119829],{"className":119824},[83],[39,119826,119605],{"className":119827,"style":119828},[83,84],"margin-right:0.0962em;",[39,119830,119832],{"className":119831},[192],[39,119833,119835,119856],{"className":119834},[196,649],[39,119836,119838,119853],{"className":119837},[200],[39,119839,119841],{"className":119840,"style":3419},[204],[39,119842,119844,119847],{"style":119843},"top:-2.55em;margin-left:-0.0962em;margin-right:0.05em;",[39,119845],{"className":119846,"style":213},[212],[39,119848,119850],{"className":119849},[217,218,219,220],[39,119851,74335],{"className":119852,"style":107705},[83,84,220],[39,119854,673],{"className":119855},[672],[39,119857,119859],{"className":119858},[200],[39,119860,119862],{"className":119861,"style":35396},[204],[39,119863],{},[39,119865,136],{"className":119866},[182],[39,119868,118943],{"className":119869,"style":118985},[83,84],[39,119871,161],{"className":119872},[263],[39,119874,119876],{"className":119875},[263],[39,119877,119879],{"className":119878},[18520,85892],[39,119880,119882,119902],{"className":119881},[196,649],[39,119883,119885,119899],{"className":119884},[200],[39,119886,119888],{"className":119887,"style":928},[204],[39,119889,119890,119893],{"style":85904},[39,119891],{"className":119892,"style":85908},[212],[39,119894,119895],{"style":85911},[85913,119896,119897],{"xmlns":85915,"width":85916,"height":73875,"viewBox":85917},[85919,119898],{"d":85921},[39,119900,673],{"className":119901},[672],[39,119903,119905],{"className":119904},[200],[39,119906,119908],{"className":119907,"style":1217},[204],[39,119909],{},[39,119911,2903],{"className":119912},[83],[11,119914,100053,119915,119974],{},[39,119916,119918,119942],{"className":119917,"translate":43},[42],[39,119919,119921],{"className":119920},[47],[49,119922,119923],{"xmlns":51},[53,119924,119925,119939],{},[56,119926,119927,119929,119931,119933,119935,119937],{},[59,119928,92483],{},[133,119930,1690],{},[59,119932,150],{"mathvariant":145},[59,119934,119143],{},[133,119936,1690],{},[59,119938,150],{"mathvariant":145},[63,119940,119941],{"encoding":65},"\\log|\\det|",[39,119943,119945],{"className":119944,"ariaHidden":70},[69],[39,119946,119948,119951,119956,119959,119962,119965,119968,119971],{"className":119947},[74],[39,119949],{"className":119950,"style":245},[78],[39,119952,92721,119954],{"className":119953},[1744],[39,119955,39209],{"style":35078},[39,119957],{"className":119958,"style":256},[230],[39,119960,150],{"className":119961},[83],[39,119963],{"className":119964,"style":256},[230],[39,119966,119143],{"className":119967},[1744],[39,119969],{"className":119970,"style":256},[230],[39,119972,150],{"className":119973},[83]," 项就是“局部的伸缩率”：变换把一小块体积放大了多少，密度就要相应缩小多少。多层复合时 log-det 相加，所以加深网络不增加计算上的困难，只要每层的 Jacobian 行列式好算（比如三角矩阵）。",[11,119976,119977,119978,87840,120057,742,120148,120213,120214,120242,120243,742,120322,120436,120437,1607],{},"一维的情形可以推到极致：任意一维分布都能写成逆 CDF。取 ",[39,119979,119981,120009],{"className":119980,"translate":43},[42],[39,119982,119984],{"className":119983},[47],[49,119985,119986],{"xmlns":51},[53,119987,119988,120006],{},[56,119989,119990,119992,119994,119996,119998,120000,120002,120004],{},[59,119991,118943],{},[133,119993,119005],{},[59,119995,85394],{},[133,119997,136],{"stretchy":135},[539,119999,541],{},[133,120001,155],{"separator":70},[539,120003,546],{},[133,120005,161],{"stretchy":135},[63,120007,120008],{"encoding":65},"z\\sim U(0,1)",[39,120010,120012,120030],{"className":120011,"ariaHidden":70},[69],[39,120013,120015,120018,120021,120024,120027],{"className":120014},[74],[39,120016],{"className":120017,"style":290},[78],[39,120019,118943],{"className":120020,"style":118985},[83,84],[39,120022],{"className":120023,"style":231},[230],[39,120025,119005],{"className":120026},[235],[39,120028],{"className":120029,"style":231},[230],[39,120031,120033,120036,120039,120042,120045,120048,120051,120054],{"className":120032},[74],[39,120034],{"className":120035,"style":245},[78],[39,120037,85394],{"className":120038,"style":85420},[83,84],[39,120040,136],{"className":120041},[182],[39,120043,541],{"className":120044},[83],[39,120046,155],{"className":120047},[252],[39,120049],{"className":120050,"style":256},[230],[39,120052,546],{"className":120053},[83],[39,120055,161],{"className":120056},[263],[39,120058,120060,120086],{"className":120059,"translate":43},[42],[39,120061,120063],{"className":120062},[47],[49,120064,120065],{"xmlns":51},[53,120066,120067,120083],{},[56,120068,120069,120071,120073],{},[59,120070,74335],{},[133,120072,793],{},[138,120074,120075,120077],{},[59,120076,618],{},[56,120078,120079,120081],{},[133,120080,5378],{},[539,120082,546],{},[63,120084,120085],{"encoding":65},"f=F^{-1}",[39,120087,120089,120107],{"className":120088,"ariaHidden":70},[69],[39,120090,120092,120095,120098,120101,120104],{"className":120091},[74],[39,120093],{"className":120094,"style":51280},[78],[39,120096,74335],{"className":120097,"style":107705},[83,84],[39,120099],{"className":120100,"style":231},[230],[39,120102,793],{"className":120103},[235],[39,120105],{"className":120106,"style":231},[230],[39,120108,120110,120113],{"className":120109},[74],[39,120111],{"className":120112,"style":19710},[78],[39,120114,120116,120119],{"className":120115},[83],[39,120117,618],{"className":120118,"style":178},[83,84],[39,120120,120122],{"className":120121},[192],[39,120123,120125],{"className":120124},[196],[39,120126,120128],{"className":120127},[200],[39,120129,120131],{"className":120130,"style":19710},[204],[39,120132,120133,120136],{"style":208},[39,120134],{"className":120135,"style":213},[212],[39,120137,120139],{"className":120138},[217,218,219,220],[39,120140,120142,120145],{"className":120141},[83,220],[39,120143,5378],{"className":120144},[83,220],[39,120146,546],{"className":120147},[83,220],[39,120149,120151,120174],{"className":120150,"translate":43},[42],[39,120152,120154],{"className":120153},[47],[49,120155,120156],{"xmlns":51},[53,120157,120158,120172],{},[56,120159,120160,120162,120164,120166,120168,120170],{},[59,120161,1755],{},[133,120163,793],{},[59,120165,74335],{},[133,120167,136],{"stretchy":135},[59,120169,118943],{},[133,120171,161],{"stretchy":135},[63,120173,118948],{"encoding":65},[39,120175,120177,120195],{"className":120176,"ariaHidden":70},[69],[39,120178,120180,120183,120186,120189,120192],{"className":120179},[74],[39,120181],{"className":120182,"style":290},[78],[39,120184,1755],{"className":120185},[83,84],[39,120187],{"className":120188,"style":231},[230],[39,120190,793],{"className":120191},[235],[39,120193],{"className":120194,"style":231},[230],[39,120196,120198,120201,120204,120207,120210],{"className":120197},[74],[39,120199],{"className":120200,"style":245},[78],[39,120202,74335],{"className":120203,"style":107705},[83,84],[39,120205,136],{"className":120206},[182],[39,120208,118943],{"className":120209,"style":118985},[83,84],[39,120211,161],{"className":120212},[263]," 的分布就是 ",[39,120215,120217,120230],{"className":120216,"translate":43},[42],[39,120218,120220],{"className":120219},[47],[49,120221,120222],{"xmlns":51},[53,120223,120224,120228],{},[56,120225,120226],{},[59,120227,618],{},[63,120229,618],{"encoding":65},[39,120231,120233],{"className":120232,"ariaHidden":70},[69],[39,120234,120236,120239],{"className":120235},[74],[39,120237],{"className":120238,"style":79},[78],[39,120240,618],{"className":120241,"style":178},[83,84],"，这是精确的，不是近似。取 ",[39,120244,120246,120274],{"className":120245,"translate":43},[42],[39,120247,120249],{"className":120248},[47],[49,120250,120251],{"xmlns":51},[53,120252,120253,120271],{},[56,120254,120255,120257,120259,120261,120263,120265,120267,120269],{},[59,120256,118943],{},[133,120258,119005],{},[59,120260,77559],{"mathvariant":617},[133,120262,136],{"stretchy":135},[539,120264,541],{},[133,120266,155],{"separator":70},[539,120268,546],{},[133,120270,161],{"stretchy":135},[63,120272,120273],{"encoding":65},"z\\sim\\mathcal N(0,1)",[39,120275,120277,120295],{"className":120276,"ariaHidden":70},[69],[39,120278,120280,120283,120286,120289,120292],{"className":120279},[74],[39,120281],{"className":120282,"style":290},[78],[39,120284,118943],{"className":120285,"style":118985},[83,84],[39,120287],{"className":120288,"style":231},[230],[39,120290,119005],{"className":120291},[235],[39,120293],{"className":120294,"style":231},[230],[39,120296,120298,120301,120304,120307,120310,120313,120316,120319],{"className":120297},[74],[39,120299],{"className":120300,"style":245},[78],[39,120302,77559],{"className":120303,"style":77627},[83,641],[39,120305,136],{"className":120306},[182],[39,120308,541],{"className":120309},[83],[39,120311,155],{"className":120312},[252],[39,120314],{"className":120315,"style":256},[230],[39,120317,546],{"className":120318},[83],[39,120320,161],{"className":120321},[263],[39,120323,120325,120356],{"className":120324,"translate":43},[42],[39,120326,120328],{"className":120327},[47],[49,120329,120330],{"xmlns":51},[53,120331,120332,120353],{},[56,120333,120334,120336,120338,120348,120350],{},[59,120335,74335],{},[133,120337,793],{},[138,120339,120340,120342],{},[59,120341,618],{},[56,120343,120344,120346],{},[133,120345,5378],{},[539,120347,546],{},[133,120349,88763],{},[59,120351,120352],{"mathvariant":145},"Φ",[63,120354,120355],{"encoding":65},"f=F^{-1}\\circ\\Phi",[39,120357,120359,120377,120427],{"className":120358,"ariaHidden":70},[69],[39,120360,120362,120365,120368,120371,120374],{"className":120361},[74],[39,120363],{"className":120364,"style":51280},[78],[39,120366,74335],{"className":120367,"style":107705},[83,84],[39,120369],{"className":120370,"style":231},[230],[39,120372,793],{"className":120373},[235],[39,120375],{"className":120376,"style":231},[230],[39,120378,120380,120383,120418,120421,120424],{"className":120379},[74],[39,120381],{"className":120382,"style":19710},[78],[39,120384,120386,120389],{"className":120385},[83],[39,120387,618],{"className":120388,"style":178},[83,84],[39,120390,120392],{"className":120391},[192],[39,120393,120395],{"className":120394},[196],[39,120396,120398],{"className":120397},[200],[39,120399,120401],{"className":120400,"style":19710},[204],[39,120402,120403,120406],{"style":208},[39,120404],{"className":120405,"style":213},[212],[39,120407,120409],{"className":120408},[217,218,219,220],[39,120410,120412,120415],{"className":120411},[83,220],[39,120413,5378],{"className":120414},[83,220],[39,120416,546],{"className":120417},[83,220],[39,120419],{"className":120420,"style":461},[230],[39,120422,88763],{"className":120423},[465],[39,120425],{"className":120426,"style":461},[230],[39,120428,120430,120433],{"className":120429},[74],[39,120431],{"className":120432,"style":79},[78],[39,120434,120352],{"className":120435},[83]," 就得到任意一维分布。上面那个指数分布的例子就是 ",[39,120438,120440,120490],{"className":120439,"translate":43},[42],[39,120441,120443],{"className":120442},[47],[49,120444,120445],{"xmlns":51},[53,120446,120447,120487],{},[56,120448,120449,120459,120461,120463,120465,120467,120469,120471,120473,120475,120477,120479,120481,120483,120485],{},[138,120450,120451,120453],{},[59,120452,618],{},[56,120454,120455,120457],{},[133,120456,5378],{},[539,120458,546],{},[133,120460,136],{"stretchy":135},[59,120462,35321],{},[133,120464,161],{"stretchy":135},[133,120466,793],{},[133,120468,5378],{},[59,120470,92483],{},[133,120472,1690],{},[133,120474,136],{"stretchy":135},[539,120476,546],{},[133,120478,5378],{},[59,120480,35321],{},[133,120482,161],{"stretchy":135},[59,120484,45691],{"mathvariant":145},[59,120486,118199],{},[63,120488,120489],{"encoding":65},"F^{-1}(u)=-\\log(1-u)\u002F\\lambda",[39,120491,120493,120552,120584],{"className":120492,"ariaHidden":70},[69],[39,120494,120496,120499,120534,120537,120540,120543,120546,120549],{"className":120495},[74],[39,120497],{"className":120498,"style":84615},[78],[39,120500,120502,120505],{"className":120501},[83],[39,120503,618],{"className":120504,"style":178},[83,84],[39,120506,120508],{"className":120507},[192],[39,120509,120511],{"className":120510},[196],[39,120512,120514],{"className":120513},[200],[39,120515,120517],{"className":120516,"style":19710},[204],[39,120518,120519,120522],{"style":208},[39,120520],{"className":120521,"style":213},[212],[39,120523,120525],{"className":120524},[217,218,219,220],[39,120526,120528,120531],{"className":120527},[83,220],[39,120529,5378],{"className":120530},[83,220],[39,120532,546],{"className":120533},[83,220],[39,120535,136],{"className":120536},[182],[39,120538,35321],{"className":120539},[83,84],[39,120541,161],{"className":120542},[263],[39,120544],{"className":120545,"style":231},[230],[39,120547,793],{"className":120548},[235],[39,120550],{"className":120551,"style":231},[230],[39,120553,120555,120558,120561,120564,120569,120572,120575,120578,120581],{"className":120554},[74],[39,120556],{"className":120557,"style":245},[78],[39,120559,5378],{"className":120560},[83],[39,120562],{"className":120563,"style":256},[230],[39,120565,92721,120567],{"className":120566},[1744],[39,120568,39209],{"style":35078},[39,120570,136],{"className":120571},[182],[39,120573,546],{"className":120574},[83],[39,120576],{"className":120577,"style":461},[230],[39,120579,5378],{"className":120580},[465],[39,120582],{"className":120583,"style":461},[230],[39,120585,120587,120590,120593,120596,120599],{"className":120586},[74],[39,120588],{"className":120589,"style":245},[78],[39,120591,35321],{"className":120592},[83,84],[39,120594,161],{"className":120595},[263],[39,120597,45691],{"className":120598},[83],[39,120600,118199],{"className":120601},[83,84],[11,120603,120604,120605,120897,120898,120926,120927,121219,121220,121332,121333,121385,121386,121453,121454,121482,121483,121550],{},"举个能算的例子。",[39,120606,120608,120668],{"className":120607,"translate":43},[42],[39,120609,120611],{"className":120610},[47],[49,120612,120613],{"xmlns":51},[53,120614,120615,120665],{},[56,120616,120617,120619,120621,120623,120625,120627,120635,120637,120639,120641,120643,120645,120653,120655,120657,120659,120661,120663],{},[59,120618,11],{},[133,120620,136],{"stretchy":135},[59,120622,1755],{},[133,120624,161],{"stretchy":135},[133,120626,793],{},[770,120628,120629],{"scriptlevel":541,"displaystyle":135},[5362,120630,120631,120633],{},[539,120632,546],{},[539,120634,17359],{},[59,120636,80648],{},[133,120638,136],{"stretchy":135},[59,120640,1755],{},[133,120642,161],{"stretchy":135},[133,120644,788],{},[770,120646,120647],{"scriptlevel":541,"displaystyle":135},[5362,120648,120649,120651],{},[539,120650,546],{},[539,120652,17359],{},[59,120654,80648],{},[133,120656,136],{"stretchy":135},[59,120658,1755],{},[133,120660,5378],{},[539,120662,59187],{},[133,120664,161],{"stretchy":135},[63,120666,120667],{"encoding":65},"p(x)=\\tfrac12\\varphi(x)+\\tfrac12\\varphi(x-3)",[39,120669,120671,120698,120793,120885],{"className":120670,"ariaHidden":70},[69],[39,120672,120674,120677,120680,120683,120686,120689,120692,120695],{"className":120673},[74],[39,120675],{"className":120676,"style":245},[78],[39,120678,11],{"className":120679},[83,84],[39,120681,136],{"className":120682},[182],[39,120684,1755],{"className":120685},[83,84],[39,120687,161],{"className":120688},[263],[39,120690],{"className":120691,"style":231},[230],[39,120693,793],{"className":120694},[235],[39,120696],{"className":120697,"style":231},[230],[39,120699,120701,120704,120772,120775,120778,120781,120784,120787,120790],{"className":120700},[74],[39,120702],{"className":120703,"style":74794},[78],[39,120705,120707,120710,120769],{"className":120706},[83],[39,120708],{"className":120709},[182,5463],[39,120711,120713],{"className":120712},[5362],[39,120714,120716,120761],{"className":120715},[196,649],[39,120717,120719,120758],{"className":120718},[200],[39,120720,120722,120736,120744],{"className":120721,"style":74813},[204],[39,120723,120724,120727],{"style":74816},[39,120725],{"className":120726,"style":935},[212],[39,120728,120730],{"className":120729},[217,218,219,220],[39,120731,120733],{"className":120732},[83,220],[39,120734,17359],{"className":120735},[83,220],[39,120737,120738,120741],{"style":5503},[39,120739],{"className":120740,"style":935},[212],[39,120742],{"className":120743,"style":5511},[5510],[39,120745,120746,120749],{"style":74839},[39,120747],{"className":120748,"style":935},[212],[39,120750,120752],{"className":120751},[217,218,219,220],[39,120753,120755],{"className":120754},[83,220],[39,120756,546],{"className":120757},[83,220],[39,120759,673],{"className":120760},[672],[39,120762,120764],{"className":120763},[200],[39,120765,120767],{"className":120766,"style":74861},[204],[39,120768],{},[39,120770],{"className":120771},[263,5463],[39,120773,80648],{"className":120774},[83,84],[39,120776,136],{"className":120777},[182],[39,120779,1755],{"className":120780},[83,84],[39,120782,161],{"className":120783},[263],[39,120785],{"className":120786,"style":461},[230],[39,120788,788],{"className":120789},[465],[39,120791],{"className":120792,"style":461},[230],[39,120794,120796,120799,120867,120870,120873,120876,120879,120882],{"className":120795},[74],[39,120797],{"className":120798,"style":74794},[78],[39,120800,120802,120805,120864],{"className":120801},[83],[39,120803],{"className":120804},[182,5463],[39,120806,120808],{"className":120807},[5362],[39,120809,120811,120856],{"className":120810},[196,649],[39,120812,120814,120853],{"className":120813},[200],[39,120815,120817,120831,120839],{"className":120816,"style":74813},[204],[39,120818,120819,120822],{"style":74816},[39,120820],{"className":120821,"style":935},[212],[39,120823,120825],{"className":120824},[217,218,219,220],[39,120826,120828],{"className":120827},[83,220],[39,120829,17359],{"className":120830},[83,220],[39,120832,120833,120836],{"style":5503},[39,120834],{"className":120835,"style":935},[212],[39,120837],{"className":120838,"style":5511},[5510],[39,120840,120841,120844],{"style":74839},[39,120842],{"className":120843,"style":935},[212],[39,120845,120847],{"className":120846},[217,218,219,220],[39,120848,120850],{"className":120849},[83,220],[39,120851,546],{"className":120852},[83,220],[39,120854,673],{"className":120855},[672],[39,120857,120859],{"className":120858},[200],[39,120860,120862],{"className":120861,"style":74861},[204],[39,120863],{},[39,120865],{"className":120866},[263,5463],[39,120868,80648],{"className":120869},[83,84],[39,120871,136],{"className":120872},[182],[39,120874,1755],{"className":120875},[83,84],[39,120877],{"className":120878,"style":461},[230],[39,120880,5378],{"className":120881},[465],[39,120883],{"className":120884,"style":461},[230],[39,120886,120888,120891,120894],{"className":120887},[74],[39,120889],{"className":120890,"style":245},[78],[39,120892,59187],{"className":120893},[83],[39,120895,161],{"className":120896},[263],"，两个单位方差高斯，中心相距 ",[39,120899,120901,120914],{"className":120900,"translate":43},[42],[39,120902,120904],{"className":120903},[47],[49,120905,120906],{"xmlns":51},[53,120907,120908,120912],{},[56,120909,120910],{},[539,120911,59187],{},[63,120913,59187],{"encoding":65},[39,120915,120917],{"className":120916,"ariaHidden":70},[69],[39,120918,120920,120923],{"className":120919},[74],[39,120921],{"className":120922,"style":2473},[78],[39,120924,59187],{"className":120925},[83],"。它的 CDF 是 ",[39,120928,120930,120990],{"className":120929,"translate":43},[42],[39,120931,120933],{"className":120932},[47],[49,120934,120935],{"xmlns":51},[53,120936,120937,120987],{},[56,120938,120939,120941,120943,120945,120947,120949,120957,120959,120961,120963,120965,120967,120975,120977,120979,120981,120983,120985],{},[59,120940,618],{},[133,120942,136],{"stretchy":135},[59,120944,1755],{},[133,120946,161],{"stretchy":135},[133,120948,793],{},[770,120950,120951],{"scriptlevel":541,"displaystyle":135},[5362,120952,120953,120955],{},[539,120954,546],{},[539,120956,17359],{},[59,120958,120352],{"mathvariant":145},[133,120960,136],{"stretchy":135},[59,120962,1755],{},[133,120964,161],{"stretchy":135},[133,120966,788],{},[770,120968,120969],{"scriptlevel":541,"displaystyle":135},[5362,120970,120971,120973],{},[539,120972,546],{},[539,120974,17359],{},[59,120976,120352],{"mathvariant":145},[133,120978,136],{"stretchy":135},[59,120980,1755],{},[133,120982,5378],{},[539,120984,59187],{},[133,120986,161],{"stretchy":135},[63,120988,120989],{"encoding":65},"F(x)=\\tfrac12\\Phi(x)+\\tfrac12\\Phi(x-3)",[39,120991,120993,121020,121115,121207],{"className":120992,"ariaHidden":70},[69],[39,120994,120996,120999,121002,121005,121008,121011,121014,121017],{"className":120995},[74],[39,120997],{"className":120998,"style":245},[78],[39,121000,618],{"className":121001,"style":178},[83,84],[39,121003,136],{"className":121004},[182],[39,121006,1755],{"className":121007},[83,84],[39,121009,161],{"className":121010},[263],[39,121012],{"className":121013,"style":231},[230],[39,121015,793],{"className":121016},[235],[39,121018],{"className":121019,"style":231},[230],[39,121021,121023,121026,121094,121097,121100,121103,121106,121109,121112],{"className":121022},[74],[39,121024],{"className":121025,"style":74794},[78],[39,121027,121029,121032,121091],{"className":121028},[83],[39,121030],{"className":121031},[182,5463],[39,121033,121035],{"className":121034},[5362],[39,121036,121038,121083],{"className":121037},[196,649],[39,121039,121041,121080],{"className":121040},[200],[39,121042,121044,121058,121066],{"className":121043,"style":74813},[204],[39,121045,121046,121049],{"style":74816},[39,121047],{"className":121048,"style":935},[212],[39,121050,121052],{"className":121051},[217,218,219,220],[39,121053,121055],{"className":121054},[83,220],[39,121056,17359],{"className":121057},[83,220],[39,121059,121060,121063],{"style":5503},[39,121061],{"className":121062,"style":935},[212],[39,121064],{"className":121065,"style":5511},[5510],[39,121067,121068,121071],{"style":74839},[39,121069],{"className":121070,"style":935},[212],[39,121072,121074],{"className":121073},[217,218,219,220],[39,121075,121077],{"className":121076},[83,220],[39,121078,546],{"className":121079},[83,220],[39,121081,673],{"className":121082},[672],[39,121084,121086],{"className":121085},[200],[39,121087,121089],{"className":121088,"style":74861},[204],[39,121090],{},[39,121092],{"className":121093},[263,5463],[39,121095,120352],{"className":121096},[83],[39,121098,136],{"className":121099},[182],[39,121101,1755],{"className":121102},[83,84],[39,121104,161],{"className":121105},[263],[39,121107],{"className":121108,"style":461},[230],[39,121110,788],{"className":121111},[465],[39,121113],{"className":121114,"style":461},[230],[39,121116,121118,121121,121189,121192,121195,121198,121201,121204],{"className":121117},[74],[39,121119],{"className":121120,"style":74794},[78],[39,121122,121124,121127,121186],{"className":121123},[83],[39,121125],{"className":121126},[182,5463],[39,121128,121130],{"className":121129},[5362],[39,121131,121133,121178],{"className":121132},[196,649],[39,121134,121136,121175],{"className":121135},[200],[39,121137,121139,121153,121161],{"className":121138,"style":74813},[204],[39,121140,121141,121144],{"style":74816},[39,121142],{"className":121143,"style":935},[212],[39,121145,121147],{"className":121146},[217,218,219,220],[39,121148,121150],{"className":121149},[83,220],[39,121151,17359],{"className":121152},[83,220],[39,121154,121155,121158],{"style":5503},[39,121156],{"className":121157,"style":935},[212],[39,121159],{"className":121160,"style":5511},[5510],[39,121162,121163,121166],{"style":74839},[39,121164],{"className":121165,"style":935},[212],[39,121167,121169],{"className":121168},[217,218,219,220],[39,121170,121172],{"className":121171},[83,220],[39,121173,546],{"className":121174},[83,220],[39,121176,673],{"className":121177},[672],[39,121179,121181],{"className":121180},[200],[39,121182,121184],{"className":121183,"style":74861},[204],[39,121185],{},[39,121187],{"className":121188},[263,5463],[39,121190,120352],{"className":121191},[83],[39,121193,136],{"className":121194},[182],[39,121196,1755],{"className":121197},[83,84],[39,121199],{"className":121200,"style":461},[230],[39,121202,5378],{"className":121203},[465],[39,121205],{"className":121206,"style":461},[230],[39,121208,121210,121213,121216],{"className":121209},[74],[39,121211],{"className":121212,"style":245},[78],[39,121214,59187],{"className":121215},[83],[39,121217,161],{"className":121218},[263],"，流就是 ",[39,121221,121223,121252],{"className":121222,"translate":43},[42],[39,121224,121226],{"className":121225},[47],[49,121227,121228],{"xmlns":51},[53,121229,121230,121250],{},[56,121231,121232,121234,121236,121246,121248],{},[59,121233,74335],{},[133,121235,793],{},[138,121237,121238,121240],{},[59,121239,618],{},[56,121241,121242,121244],{},[133,121243,5378],{},[539,121245,546],{},[133,121247,88763],{},[59,121249,120352],{"mathvariant":145},[63,121251,120355],{"encoding":65},[39,121253,121255,121273,121323],{"className":121254,"ariaHidden":70},[69],[39,121256,121258,121261,121264,121267,121270],{"className":121257},[74],[39,121259],{"className":121260,"style":51280},[78],[39,121262,74335],{"className":121263,"style":107705},[83,84],[39,121265],{"className":121266,"style":231},[230],[39,121268,793],{"className":121269},[235],[39,121271],{"className":121272,"style":231},[230],[39,121274,121276,121279,121314,121317,121320],{"className":121275},[74],[39,121277],{"className":121278,"style":19710},[78],[39,121280,121282,121285],{"className":121281},[83],[39,121283,618],{"className":121284,"style":178},[83,84],[39,121286,121288],{"className":121287},[192],[39,121289,121291],{"className":121290},[196],[39,121292,121294],{"className":121293},[200],[39,121295,121297],{"className":121296,"style":19710},[204],[39,121298,121299,121302],{"style":208},[39,121300],{"className":121301,"style":213},[212],[39,121303,121305],{"className":121304},[217,218,219,220],[39,121306,121308,121311],{"className":121307},[83,220],[39,121309,5378],{"className":121310},[83,220],[39,121312,546],{"className":121313},[83,220],[39,121315],{"className":121316,"style":461},[230],[39,121318,88763],{"className":121319},[465],[39,121321],{"className":121322,"style":461},[230],[39,121324,121326,121329],{"className":121325},[74],[39,121327],{"className":121328,"style":79},[78],[39,121330,120352],{"className":121331},[83],"。两峰之间 ",[39,121334,121336,121355],{"className":121335,"translate":43},[42],[39,121337,121339],{"className":121338},[47],[49,121340,121341],{"xmlns":51},[53,121342,121343,121352],{},[56,121344,121345,121347,121349],{},[59,121346,1755],{},[133,121348,793],{},[539,121350,121351],{},"1.5",[63,121353,121354],{"encoding":65},"x=1.5",[39,121356,121358,121376],{"className":121357,"ariaHidden":70},[69],[39,121359,121361,121364,121367,121370,121373],{"className":121360},[74],[39,121362],{"className":121363,"style":290},[78],[39,121365,1755],{"className":121366},[83,84],[39,121368],{"className":121369,"style":231},[230],[39,121371,793],{"className":121372},[235],[39,121374],{"className":121375,"style":231},[230],[39,121377,121379,121382],{"className":121378},[74],[39,121380],{"className":121381,"style":2473},[78],[39,121383,121351],{"className":121384},[83]," 处的真实密度是 ",[39,121387,121389,121414],{"className":121388,"translate":43},[42],[39,121390,121392],{"className":121391},[47],[49,121393,121394],{"xmlns":51},[53,121395,121396,121411],{},[56,121397,121398,121400,121402,121404,121406,121408],{},[59,121399,80648],{},[133,121401,136],{"stretchy":135},[539,121403,121351],{},[133,121405,161],{"stretchy":135},[133,121407,99744],{},[539,121409,121410],{},"0.1295",[63,121412,121413],{"encoding":65},"\\varphi(1.5)\\approx0.1295",[39,121415,121417,121444],{"className":121416,"ariaHidden":70},[69],[39,121418,121420,121423,121426,121429,121432,121435,121438,121441],{"className":121419},[74],[39,121421],{"className":121422,"style":245},[78],[39,121424,80648],{"className":121425},[83,84],[39,121427,136],{"className":121428},[182],[39,121430,121351],{"className":121431},[83],[39,121433,161],{"className":121434},[263],[39,121436],{"className":121437,"style":231},[230],[39,121439,99744],{"className":121440},[235],[39,121442],{"className":121443,"style":231},[230],[39,121445,121447,121450],{"className":121446},[74],[39,121448],{"className":121449,"style":2473},[78],[39,121451,121410],{"className":121452},[83],"；如果错当成单个高斯（中心在 ",[39,121455,121457,121470],{"className":121456,"translate":43},[42],[39,121458,121460],{"className":121459},[47],[49,121461,121462],{"xmlns":51},[53,121463,121464,121468],{},[56,121465,121466],{},[539,121467,121351],{},[63,121469,121351],{"encoding":65},[39,121471,121473],{"className":121472,"ariaHidden":70},[69],[39,121474,121476,121479],{"className":121475},[74],[39,121477],{"className":121478,"style":2473},[78],[39,121480,121351],{"className":121481},[83],"、单峰），会给出 ",[39,121484,121486,121511],{"className":121485,"translate":43},[42],[39,121487,121489],{"className":121488},[47],[49,121490,121491],{"xmlns":51},[53,121492,121493,121508],{},[56,121494,121495,121497,121499,121501,121503,121505],{},[59,121496,80648],{},[133,121498,136],{"stretchy":135},[539,121500,541],{},[133,121502,161],{"stretchy":135},[133,121504,99744],{},[539,121506,121507],{},"0.3989",[63,121509,121510],{"encoding":65},"\\varphi(0)\\approx0.3989",[39,121512,121514,121541],{"className":121513,"ariaHidden":70},[69],[39,121515,121517,121520,121523,121526,121529,121532,121535,121538],{"className":121516},[74],[39,121518],{"className":121519,"style":245},[78],[39,121521,80648],{"className":121522},[83,84],[39,121524,136],{"className":121525},[182],[39,121527,541],{"className":121528},[83],[39,121530,161],{"className":121531},[263],[39,121533],{"className":121534,"style":231},[230],[39,121536,99744],{"className":121537},[235],[39,121539],{"className":121540,"style":231},[230],[39,121542,121544,121547],{"className":121543},[74],[39,121545],{"className":121546,"style":2473},[78],[39,121548,121507],{"className":121549},[83],"，高估三倍。流模型能把这个“鞍部”学出来，靠的正是分段可逆映射，而不是更大的方差。",[11,121552,121553,121554,121653],{},"连续情形就是 Neural ODE：",[39,121555,121557,121593],{"className":121556,"translate":43},[42],[39,121558,121560],{"className":121559},[47],[49,121561,121562],{"xmlns":51},[53,121563,121564,121590],{},[56,121565,121566,121568,121570,121572,121574,121576,121578,121580,121582,121584,121586,121588],{},[59,121567,29425],{},[59,121569,118943],{},[59,121571,45691],{"mathvariant":145},[59,121573,29425],{},[59,121575,621],{},[133,121577,793],{},[59,121579,77971],{},[133,121581,136],{"stretchy":135},[59,121583,118943],{},[133,121585,155],{"separator":70},[59,121587,621],{},[133,121589,161],{"stretchy":135},[63,121591,121592],{"encoding":65},"dz\u002Fdt=v(z,t)",[39,121594,121596,121626],{"className":121595,"ariaHidden":70},[69],[39,121597,121599,121602,121605,121608,121611,121614,121617,121620,121623],{"className":121598},[74],[39,121600],{"className":121601,"style":245},[78],[39,121603,29425],{"className":121604},[83,84],[39,121606,118943],{"className":121607,"style":118985},[83,84],[39,121609,45691],{"className":121610},[83],[39,121612,29425],{"className":121613},[83,84],[39,121615,621],{"className":121616},[83,84],[39,121618],{"className":121619,"style":231},[230],[39,121621,793],{"className":121622},[235],[39,121624],{"className":121625,"style":231},[230],[39,121627,121629,121632,121635,121638,121641,121644,121647,121650],{"className":121628},[74],[39,121630],{"className":121631,"style":245},[78],[39,121633,77971],{"className":121634,"style":2114},[83,84],[39,121636,136],{"className":121637},[182],[39,121639,118943],{"className":121640,"style":118985},[83,84],[39,121642,155],{"className":121643},[252],[39,121645],{"className":121646,"style":256},[230],[39,121648,621],{"className":121649},[83,84],[39,121651,161],{"className":121652},[263],"，密度沿轨迹演化，",[39,121655,121657],{"className":121656,"translate":43},[746],[39,121658,121660,121732],{"className":121659,"translate":43},[42],[39,121661,121663],{"className":121662},[47],[49,121664,121665],{"xmlns":51,"display":755},[53,121666,121667,121729],{},[56,121668,121669,121679,121681,121683,121689,121691,121697,121699,121701,121703,121709,121711,121725,121727],{},[5362,121670,121671,121673],{},[59,121672,29425],{},[56,121674,121675,121677],{},[59,121676,29425],{},[59,121678,621],{},[59,121680,92483],{},[133,121682,1690],{},[613,121684,121685,121687],{},[59,121686,11],{},[59,121688,621],{},[133,121690,136],{"stretchy":135},[613,121692,121693,121695],{},[59,121694,118943],{},[59,121696,621],{},[133,121698,161],{"stretchy":135},[133,121700,793],{},[133,121702,5378],{},[56,121704,121705,121707],{},[59,121706,621],{"mathvariant":145},[59,121708,401],{"mathvariant":145},[133,121710,136],{"fence":70,"stretchy":70,"minsize":65017,"maxsize":65017},[5362,121712,121713,121719],{},[56,121714,121715,121717],{},[59,121716,96436],{"mathvariant":145},[59,121718,77971],{},[56,121720,121721,121723],{},[59,121722,96436],{"mathvariant":145},[59,121724,118943],{},[133,121726,161],{"fence":70,"stretchy":70,"minsize":65017,"maxsize":65017},[133,121728,155],{"separator":70},[63,121730,121731],{"encoding":65},"\\frac{d}{dt}\\log p_t(z_t)=-\\mathrm{tr}\\Bigl(\\frac{\\partial v}{\\partial z}\\Bigr),",[39,121733,121735,121914],{"className":121734,"ariaHidden":70},[69],[39,121736,121738,121742,121807,121810,121815,121818,121858,121861,121902,121905,121908,121911],{"className":121737},[74],[39,121739],{"className":121740,"style":121741},[78],"height:2.0574em;vertical-align:-0.686em;",[39,121743,121745,121748,121804],{"className":121744},[83],[39,121746],{"className":121747},[182,5463],[39,121749,121751],{"className":121750},[5362],[39,121752,121754,121796],{"className":121753},[196,649],[39,121755,121757,121793],{"className":121756},[200],[39,121758,121760,121774,121782],{"className":121759,"style":118749},[204],[39,121761,121762,121765],{"style":5479},[39,121763],{"className":121764,"style":935},[212],[39,121766,121768,121771],{"className":121767},[83],[39,121769,29425],{"className":121770},[83,84],[39,121772,621],{"className":121773},[83,84],[39,121775,121776,121779],{"style":5503},[39,121777],{"className":121778,"style":935},[212],[39,121780],{"className":121781,"style":5511},[5510],[39,121783,121784,121787],{"style":5514},[39,121785],{"className":121786,"style":935},[212],[39,121788,121790],{"className":121789},[83],[39,121791,29425],{"className":121792},[83,84],[39,121794,673],{"className":121795},[672],[39,121797,121799],{"className":121798},[200],[39,121800,121802],{"className":121801,"style":19487},[204],[39,121803],{},[39,121805],{"className":121806},[263,5463],[39,121808],{"className":121809,"style":256},[230],[39,121811,92721,121813],{"className":121812},[1744],[39,121814,39209],{"style":35078},[39,121816],{"className":121817,"style":256},[230],[39,121819,121821,121824],{"className":121820},[83],[39,121822,11],{"className":121823},[83,84],[39,121825,121827],{"className":121826},[192],[39,121828,121830,121850],{"className":121829},[196,649],[39,121831,121833,121847],{"className":121832},[200],[39,121834,121836],{"className":121835,"style":656},[204],[39,121837,121838,121841],{"style":966},[39,121839],{"className":121840,"style":213},[212],[39,121842,121844],{"className":121843},[217,218,219,220],[39,121845,621],{"className":121846},[83,84,220],[39,121848,673],{"className":121849},[672],[39,121851,121853],{"className":121852},[200],[39,121854,121856],{"className":121855,"style":680},[204],[39,121857],{},[39,121859,136],{"className":121860},[182],[39,121862,121864,121867],{"className":121863},[83],[39,121865,118943],{"className":121866,"style":118985},[83,84],[39,121868,121870],{"className":121869},[192],[39,121871,121873,121894],{"className":121872},[196,649],[39,121874,121876,121891],{"className":121875},[200],[39,121877,121879],{"className":121878,"style":656},[204],[39,121880,121882,121885],{"style":121881},"top:-2.55em;margin-left:-0.044em;margin-right:0.05em;",[39,121883],{"className":121884,"style":213},[212],[39,121886,121888],{"className":121887},[217,218,219,220],[39,121889,621],{"className":121890},[83,84,220],[39,121892,673],{"className":121893},[672],[39,121895,121897],{"className":121896},[200],[39,121898,121900],{"className":121899,"style":680},[204],[39,121901],{},[39,121903,161],{"className":121904},[263],[39,121906],{"className":121907,"style":231},[230],[39,121909,793],{"className":121910},[235],[39,121912],{"className":121913,"style":231},[230],[39,121915,121917,121920,121923,121929,121935,122003,122009],{"className":121916},[74],[39,121918],{"className":121919,"style":121741},[78],[39,121921,5378],{"className":121922},[83],[39,121924,121926],{"className":121925},[83],[39,121927,89525],{"className":121928},[83,35077],[39,121930,121932],{"className":121931},[182],[39,121933,136],{"className":121934},[18520,33204],[39,121936,121938,121941,122000],{"className":121937},[83],[39,121939],{"className":121940},[182,5463],[39,121942,121944],{"className":121943},[5362],[39,121945,121947,121992],{"className":121946},[196,649],[39,121948,121950,121989],{"className":121949},[200],[39,121951,121953,121967,121975],{"className":121952,"style":118749},[204],[39,121954,121955,121958],{"style":5479},[39,121956],{"className":121957,"style":935},[212],[39,121959,121961,121964],{"className":121960},[83],[39,121962,96436],{"className":121963,"style":565},[83],[39,121965,118943],{"className":121966,"style":118985},[83,84],[39,121968,121969,121972],{"style":5503},[39,121970],{"className":121971,"style":935},[212],[39,121973],{"className":121974,"style":5511},[5510],[39,121976,121977,121980],{"style":5514},[39,121978],{"className":121979,"style":935},[212],[39,121981,121983,121986],{"className":121982},[83],[39,121984,96436],{"className":121985,"style":565},[83],[39,121987,77971],{"className":121988,"style":2114},[83,84],[39,121990,673],{"className":121991},[672],[39,121993,121995],{"className":121994},[200],[39,121996,121998],{"className":121997,"style":19487},[204],[39,121999],{},[39,122001],{"className":122002},[263,5463],[39,122004,122006],{"className":122005},[263],[39,122007,161],{"className":122008},[18520,33204],[39,122010,155],{"className":122011},[252],[11,122013,122014,122015,742,122184,122212],{},"这其实是变量替换公式的无穷小版本，行列式取对数后变成迹。迹在高维不好算，实践中用 Hutchinson 估计 ",[39,122016,122018,122064],{"className":122017,"translate":43},[42],[39,122019,122021],{"className":122020},[47],[49,122022,122023],{"xmlns":51},[53,122024,122025,122061],{},[56,122026,122027,122033,122035,122037,122039,122041,122047,122049,122055,122057,122059],{},[56,122028,122029,122031],{},[59,122030,621],{"mathvariant":145},[59,122032,401],{"mathvariant":145},[133,122034,136],{"stretchy":135},[59,122036,101],{},[133,122038,161],{"stretchy":135},[133,122040,99744],{},[613,122042,122043,122045],{},[59,122044,4253],{"mathvariant":421},[59,122046,58009],{},[133,122048,537],{"stretchy":135},[138,122050,122051,122053],{},[59,122052,58009],{},[59,122054,93341],{"mathvariant":145},[59,122056,101],{},[59,122058,58009],{},[133,122060,4280],{"stretchy":135},[63,122062,122063],{"encoding":65},"\\mathrm{tr}(A)\\approx\\mathbb E_\\varepsilon[\\varepsilon^\\top A\\varepsilon]",[39,122065,122067,122097],{"className":122066,"ariaHidden":70},[69],[39,122068,122070,122073,122079,122082,122085,122088,122091,122094],{"className":122069},[74],[39,122071],{"className":122072,"style":245},[78],[39,122074,122076],{"className":122075},[83],[39,122077,89525],{"className":122078},[83,35077],[39,122080,136],{"className":122081},[182],[39,122083,101],{"className":122084},[83,84],[39,122086,161],{"className":122087},[263],[39,122089],{"className":122090,"style":231},[230],[39,122092,99744],{"className":122093},[235],[39,122095],{"className":122096,"style":231},[230],[39,122098,122100,122103,122143,122146,122175,122178,122181],{"className":122099},[74],[39,122101],{"className":122102,"style":15555},[78],[39,122104,122106,122109],{"className":122105},[83],[39,122107,4253],{"className":122108},[83,515],[39,122110,122112],{"className":122111},[192],[39,122113,122115,122135],{"className":122114},[196,649],[39,122116,122118,122132],{"className":122117},[200],[39,122119,122121],{"className":122120,"style":1728},[204],[39,122122,122123,122126],{"style":966},[39,122124],{"className":122125,"style":213},[212],[39,122127,122129],{"className":122128},[217,218,219,220],[39,122130,58009],{"className":122131},[83,84,220],[39,122133,673],{"className":122134},[672],[39,122136,122138],{"className":122137},[200],[39,122139,122141],{"className":122140,"style":680},[204],[39,122142],{},[39,122144,537],{"className":122145},[182],[39,122147,122149,122152],{"className":122148},[83],[39,122150,58009],{"className":122151},[83,84],[39,122153,122155],{"className":122154},[192],[39,122156,122158],{"className":122157},[196],[39,122159,122161],{"className":122160},[200],[39,122162,122164],{"className":122163,"style":15577},[204],[39,122165,122166,122169],{"style":208},[39,122167],{"className":122168,"style":213},[212],[39,122170,122172],{"className":122171},[217,218,219,220],[39,122173,93341],{"className":122174},[83,220],[39,122176,101],{"className":122177},[83,84],[39,122179,58009],{"className":122180},[83,84],[39,122182,4280],{"className":122183},[263],[39,122185,122187,122200],{"className":122186,"translate":43},[42],[39,122188,122190],{"className":122189},[47],[49,122191,122192],{"xmlns":51},[53,122193,122194,122198],{},[56,122195,122196],{},[59,122197,58009],{},[63,122199,77422],{"encoding":65},[39,122201,122203],{"className":122202,"ariaHidden":70},[69],[39,122204,122206,122209],{"className":122205},[74],[39,122207],{"className":122208,"style":290},[78],[39,122210,58009],{"className":122211},[83,84]," 取 Rademacher 向量，代价是方差。",[11,122214,122215,122216,122249,122250,122283],{},"流与最优传输有一个漂亮的接口。Brenier 说最优映射是 ",[39,122217,122219,122234],{"className":122218,"translate":43},[42],[39,122220,122222],{"className":122221},[47],[49,122223,122224],{"xmlns":51},[53,122225,122226,122232],{},[56,122227,122228,122230],{},[59,122229,84725],{"mathvariant":145},[59,122231,35321],{},[63,122233,84812],{"encoding":65},[39,122235,122237],{"className":122236,"ariaHidden":70},[69],[39,122238,122240,122243,122246],{"className":122239},[74],[39,122241],{"className":122242,"style":79},[78],[39,122244,84725],{"className":122245},[83],[39,122247,35321],{"className":122248},[83,84],"，流说数据由可逆映射得到；当映射取 ",[39,122251,122253,122268],{"className":122252,"translate":43},[42],[39,122254,122256],{"className":122255},[47],[49,122257,122258],{"xmlns":51},[53,122259,122260,122266],{},[56,122261,122262,122264],{},[59,122263,84725],{"mathvariant":145},[59,122265,35321],{},[63,122267,84812],{"encoding":65},[39,122269,122271],{"className":122270,"ariaHidden":70},[69],[39,122272,122274,122277,122280],{"className":122273},[74],[39,122275],{"className":122276,"style":79},[78],[39,122278,84725],{"className":122279},[83],[39,122281,35321],{"className":122282},[83,84]," 时两者重合，于是可以问：能不能直接学那个“最优”的流？Flow matching 就是这么做的。它的条件目标是",[39,122285,122287],{"className":122286,"translate":43},[746],[39,122288,122290,122418],{"className":122289,"translate":43},[42],[39,122291,122293],{"className":122292},[47],[49,122294,122295],{"xmlns":51,"display":755},[53,122296,122297,122415],{},[56,122298,122299,122301,122303,122327,122329,122335,122337,122343,122345,122347,122349,122351,122353,122359,122361,122367,122369,122375,122377,122379,122385,122387,122389,122391,122393,122395,122397,122403,122405,122407,122413],{},[59,122300,77190],{"mathvariant":617},[133,122302,793],{},[613,122304,122305,122307],{},[59,122306,4253],{"mathvariant":421},[56,122308,122309,122311,122313,122319,122321],{},[59,122310,621],{},[133,122312,155],{"separator":70},[613,122314,122315,122317],{},[59,122316,1755],{},[539,122318,541],{},[133,122320,155],{"separator":70},[613,122322,122323,122325],{},[59,122324,1755],{},[539,122326,546],{},[133,122328,15389],{"fence":70,"stretchy":70,"minsize":73875,"maxsize":73875},[613,122330,122331,122333],{},[59,122332,77971],{},[59,122334,96325],{},[133,122336,136],{"stretchy":135},[613,122338,122339,122341],{},[59,122340,1755],{},[59,122342,621],{},[133,122344,155],{"separator":70},[59,122346,621],{},[133,122348,161],{"stretchy":135},[133,122350,5378],{},[133,122352,136],{"stretchy":135},[613,122354,122355,122357],{},[59,122356,1755],{},[539,122358,546],{},[133,122360,5378],{},[613,122362,122363,122365],{},[59,122364,1755],{},[539,122366,541],{},[133,122368,161],{"stretchy":135},[138,122370,122371,122373],{},[133,122372,15389],{"fence":70,"stretchy":70,"minsize":73875,"maxsize":73875},[539,122374,17359],{},[133,122376,155],{"separator":70},[230,122378],{"width":10251},[613,122380,122381,122383],{},[59,122382,1755],{},[59,122384,621],{},[133,122386,793],{},[133,122388,136],{"stretchy":135},[539,122390,546],{},[133,122392,5378],{},[59,122394,621],{},[133,122396,161],{"stretchy":135},[613,122398,122399,122401],{},[59,122400,1755],{},[539,122402,541],{},[133,122404,788],{},[59,122406,621],{},[613,122408,122409,122411],{},[59,122410,1755],{},[539,122412,546],{},[133,122414,155],{"separator":70},[63,122416,122417],{"encoding":65},"\\mathcal L=\\mathbb E_{t,x_0,x_1}\\bigl\\|v_\\theta(x_t,t)-(x_1-x_0)\\bigr\\|^2,\n\\qquad x_t=(1-t)x_0+tx_1,",[39,122419,122421,122439,122717,122775,122945,122966,123027],{"className":122420,"ariaHidden":70},[69],[39,122422,122424,122427,122430,122433,122436],{"className":122423},[74],[39,122425],{"className":122426,"style":79},[78],[39,122428,77190],{"className":122429},[83,641],[39,122431],{"className":122432,"style":231},[230],[39,122434,793],{"className":122435},[235],[39,122437],{"className":122438,"style":231},[230],[39,122440,122442,122445,122576,122613,122653,122656,122696,122699,122702,122705,122708,122711,122714],{"className":122441},[74],[39,122443],{"className":122444,"style":908},[78],[39,122446,122448,122451],{"className":122447},[83],[39,122449,4253],{"className":122450},[83,515],[39,122452,122454],{"className":122453},[192],[39,122455,122457,122568],{"className":122456},[196,649],[39,122458,122460,122565],{"className":122459},[200],[39,122461,122463],{"className":122462,"style":656},[204],[39,122464,122465,122468],{"style":966},[39,122466],{"className":122467,"style":213},[212],[39,122469,122471],{"className":122470},[217,218,219,220],[39,122472,122474,122477,122480,122522,122525],{"className":122473},[83,220],[39,122475,621],{"className":122476},[83,84,220],[39,122478,155],{"className":122479},[252,220],[39,122481,122483,122486],{"className":122482},[83,220],[39,122484,1755],{"className":122485},[83,84,220],[39,122487,122489],{"className":122488},[192],[39,122490,122492,122514],{"className":122491},[196,649],[39,122493,122495,122511],{"className":122494},[200],[39,122496,122499],{"className":122497,"style":122498},[204],"height:0.3173em;",[39,122500,122502,122505],{"style":122501},"top:-2.357em;margin-left:0em;margin-right:0.0714em;",[39,122503],{"className":122504,"style":4004},[212],[39,122506,122508],{"className":122507},[217,4008,4009,220],[39,122509,541],{"className":122510},[83,220],[39,122512,673],{"className":122513},[672],[39,122515,122517],{"className":122516},[200],[39,122518,122520],{"className":122519,"style":64389},[204],[39,122521],{},[39,122523,155],{"className":122524},[252,220],[39,122526,122528,122531],{"className":122527},[83,220],[39,122529,1755],{"className":122530},[83,84,220],[39,122532,122534],{"className":122533},[192],[39,122535,122537,122557],{"className":122536},[196,649],[39,122538,122540,122554],{"className":122539},[200],[39,122541,122543],{"className":122542,"style":122498},[204],[39,122544,122545,122548],{"style":122501},[39,122546],{"className":122547,"style":4004},[212],[39,122549,122551],{"className":122550},[217,4008,4009,220],[39,122552,546],{"className":122553},[83,220],[39,122555,673],{"className":122556},[672],[39,122558,122560],{"className":122559},[200],[39,122561,122563],{"className":122562,"style":64389},[204],[39,122564],{},[39,122566,673],{"className":122567},[672],[39,122569,122571],{"className":122570},[200],[39,122572,122574],{"className":122573,"style":35396},[204],[39,122575],{},[39,122577,122579],{"className":122578},[182],[39,122580,122582],{"className":122581},[18520,85892],[39,122583,122585,122605],{"className":122584},[196,649],[39,122586,122588,122602],{"className":122587},[200],[39,122589,122591],{"className":122590,"style":928},[204],[39,122592,122593,122596],{"style":85904},[39,122594],{"className":122595,"style":85908},[212],[39,122597,122598],{"style":105368},[85913,122599,122600],{"xmlns":85915,"width":105371,"height":73875,"viewBox":105372},[85919,122601],{"d":105375},[39,122603,673],{"className":122604},[672],[39,122606,122608],{"className":122607},[200],[39,122609,122611],{"className":122610,"style":1217},[204],[39,122612],{},[39,122614,122616,122619],{"className":122615},[83],[39,122617,77971],{"className":122618,"style":2114},[83,84],[39,122620,122622],{"className":122621},[192],[39,122623,122625,122645],{"className":122624},[196,649],[39,122626,122628,122642],{"className":122627},[200],[39,122629,122631],{"className":122630,"style":3419},[204],[39,122632,122633,122636],{"style":41316},[39,122634],{"className":122635,"style":213},[212],[39,122637,122639],{"className":122638},[217,218,219,220],[39,122640,96325],{"className":122641,"style":438},[83,84,220],[39,122643,673],{"className":122644},[672],[39,122646,122648],{"className":122647},[200],[39,122649,122651],{"className":122650,"style":680},[204],[39,122652],{},[39,122654,136],{"className":122655},[182],[39,122657,122659,122662],{"className":122658},[83],[39,122660,1755],{"className":122661},[83,84],[39,122663,122665],{"className":122664},[192],[39,122666,122668,122688],{"className":122667},[196,649],[39,122669,122671,122685],{"className":122670},[200],[39,122672,122674],{"className":122673,"style":656},[204],[39,122675,122676,122679],{"style":966},[39,122677],{"className":122678,"style":213},[212],[39,122680,122682],{"className":122681},[217,218,219,220],[39,122683,621],{"className":122684},[83,84,220],[39,122686,673],{"className":122687},[672],[39,122689,122691],{"className":122690},[200],[39,122692,122694],{"className":122693,"style":680},[204],[39,122695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是噪声与数据的配对。如果配对来自 OT 计划，这条直线就是 ",[39,123218,123220,123237],{"className":123219,"translate":43},[42],[39,123221,123223],{"className":123222},[47],[49,123224,123225],{"xmlns":51},[53,123226,123227,123235],{},[56,123228,123229],{},[613,123230,123231,123233],{},[59,123232,23703],{},[539,123234,17359],{},[63,123236,91634],{"encoding":65},[39,123238,123240],{"className":123239,"ariaHidden":70},[69],[39,123241,123243,123246],{"className":123242},[74],[39,123244],{"className":123245,"style":634},[78],[39,123247,123249,123252],{"className":123248},[83],[39,123250,23703],{"className":123251,"style":178},[83,84],[39,123253,123255],{"className":123254},[192],[39,123256,123258,123278],{"className":123257},[196,649],[39,123259,123261,123275],{"className":123260},[200],[39,123262,123264],{"className":123263,"style":963},[204],[39,123265,123266,123269],{"style":73765},[39,123267],{"className":123268,"style":213},[212],[39,123270,123272],{"className":123271},[217,218,219,220],[39,123273,17359],{"className":123274},[83,220],[39,123276,673],{"className":123277},[672],[39,123279,123281],{"className":123280},[200],[39,123282,123284],{"className":123283,"style":680},[204],[39,123285],{}," 的测地线，轨迹最直，采样需要的步数最少。这是我看到的“最优传输”与“生成模型”结合得最干净的一处：OT 在这里不是损失函数，而是给流指定了走哪条路。",[11,123288,123289],{},"不过我要泼一盆冷水。流模型能给出很好的密度，但似然不等于“属于这个类”。Nalisnick 等人在 2019 年观察到，在图像上训练得很好的流模型给 OOD 样本的似然反而更高；原因还是典型集——模型把质量摊在一个高维薄壳上，OOD 样本落在壳外，却可能恰好落在密度不低的地方。所以流更适合做生成、做密度比、做精确的变量替换，而不是拿来当 OOD 分数直接用。",[21,123291,123292],{"id":123292},"落地",[11,123294,123295],{},"现在把这些工具放回机器学习里，按我实际见过、用过的顺序说。",[11,123297,123298,123299,123340,123341,123410],{},"生成模型是最直接的一站。GAN 的原始目标用 JS 散度，而 JS 在两个分布支撑不交时是个常数 ",[39,123300,123302,123320],{"className":123301,"translate":43},[42],[39,123303,123305],{"className":123304},[47],[49,123306,123307],{"xmlns":51},[53,123308,123309,123317],{},[56,123310,123311,123313,123315],{},[59,123312,92483],{},[133,123314,1690],{},[539,123316,17359],{},[63,123318,123319],{"encoding":65},"\\log2",[39,123321,123323],{"className":123322,"ariaHidden":70},[69],[39,123324,123326,123329,123334,123337],{"className":123325},[74],[39,123327],{"className":123328,"style":51280},[78],[39,123330,92721,123332],{"className":123331},[1744],[39,123333,39209],{"style":35078},[39,123335],{"className":123336,"style":256},[230],[39,123338,17359],{"className":123339},[83],"，梯度为零——这是早期 GAN 训练不稳定的理论解释之一。WGAN 换成 ",[39,123342,123344,123361],{"className":123343,"translate":43},[42],[39,123345,123347],{"className":123346},[47],[49,123348,123349],{"xmlns":51},[53,123350,123351,123359],{},[56,123352,123353],{},[613,123354,123355,123357],{},[59,123356,23703],{},[539,123358,546],{},[63,123360,77839],{"encoding":65},[39,123362,123364],{"className":123363,"ariaHidden":70},[69],[39,123365,123367,123370],{"className":123366},[74],[39,123368],{"className":123369,"style":634},[78],[39,123371,123373,123376],{"className":123372},[83],[39,123374,23703],{"className":123375,"style":178},[83,84],[39,123377,123379],{"className":123378},[192],[39,123380,123382,123402],{"className":123381},[196,649],[39,123383,123385,123399],{"className":123384},[200],[39,123386,123388],{"className":123387,"style":963},[204],[39,123389,123390,123393],{"style":73765},[39,123391],{"className":123392,"style":213},[212],[39,123394,123396],{"className":123395},[217,218,219,220],[39,123397,546],{"className":123398},[83,220],[39,123400,673],{"className":123401},[672],[39,123403,123405],{"className":123404},[200],[39,123406,123408],{"className":123407,"style":680},[204],[39,123409],{},"，用对偶式",[39,123412,123414],{"className":123413,"translate":43},[746],[39,123415,123417,123519],{"className":123416,"translate":43},[42],[39,123418,123420],{"className":123419},[47],[49,123421,123422],{"xmlns":51,"display":755},[53,123423,123424,123516],{},[56,123425,123426,123436,123466,123468,123478,123480,123482,123484,123486,123488,123490,123492,123502,123504,123506,123508,123510,123512,123514],{},[6556,123427,123428,123434],{},[56,123429,123430,123432],{},[59,123431,92422],{},[133,123433,1690],{},[59,123435,3181],{},[6556,123437,123438,123444],{},[56,123439,123440,123442],{},[59,123441,1687],{},[133,123443,1690],{},[56,123445,123446,123448,123450,123462,123464],{},[59,123447,15389],{"mathvariant":145},[59,123449,74335],{},[613,123451,123452,123454],{},[59,123453,15389],{"mathvariant":145},[56,123455,123456,123458,123460],{},[59,123457,77190],{"mathvariant":145},[59,123459,13377],{"mathvariant":145},[59,123461,11],{"mathvariant":145},[133,123463,1824],{},[539,123465,546],{},[35841,123467,74223],{},[613,123469,123470,123472],{},[59,123471,4253],{"mathvariant":421},[613,123473,123474,123476],{},[59,123475,131],{},[59,123477,401],{},[133,123479,537],{"stretchy":135},[59,123481,74335],{},[133,123483,136],{"stretchy":135},[59,123485,1755],{},[133,123487,161],{"stretchy":135},[133,123489,4280],{"stretchy":135},[133,123491,5378],{},[613,123493,123494,123496],{},[59,123495,4253],{"mathvariant":421},[613,123497,123498,123500],{},[59,123499,131],{},[59,123501,39209],{},[133,123503,537],{"stretchy":135},[59,123505,74335],{},[133,123507,136],{"stretchy":135},[59,123509,1755],{},[133,123511,161],{"stretchy":135},[133,123513,4280],{"stretchy":135},[133,123515,155],{"separator":70},[63,123517,123518],{"encoding":65},"\\min_G\\max_{\\|f\\|_{\\mathrm{Lip}}\\le1}\\ \\mathbb E_{P_r}[f(x)]-\\mathbb E_{P_g}[f(x)],",[39,123520,123522,123791],{"className":123521,"ariaHidden":70},[69],[39,123523,123525,123529,123573,123576,123679,123682,123685,123767,123770,123773,123776,123779,123782,123785,123788],{"className":123524},[74],[39,123526],{"className":123527,"style":123528},[78],"height:1.7385em;vertical-align:-0.9885em;",[39,123530,123532],{"className":123531},[1744,3294],[39,123533,123535,123565],{"className":123534},[196,649],[39,123536,123538,123562],{"className":123537},[200],[39,123539,123541,123552],{"className":123540,"style":92523},[204],[39,123542,123543,123546],{"style":74308},[39,123544],{"className":123545,"style":935},[212],[39,123547,123549],{"className":123548},[217,218,219,220],[39,123550,3181],{"className":123551},[83,84,220],[39,123553,123554,123557],{"style":21599},[39,123555],{"className":123556,"style":935},[212],[39,123558,123559],{},[39,123560,92422],{"className":123561},[1744],[39,123563,673],{"className":123564},[672],[39,123566,123568],{"className":123567},[200],[39,123569,123571],{"className":123570,"style":74345},[204],[39,123572],{},[39,123574],{"className":123575,"style":256},[230],[39,123577,123579],{"className":123578},[1744,3294],[39,123580,123582,123670],{"className":123581},[196,649],[39,123583,123585,123667],{"className":123584},[200],[39,123586,123588,123657],{"className":123587,"style":290},[204],[39,123589,123590,123593],{"style":75939},[39,123591],{"className":123592,"style":935},[212],[39,123594,123596],{"className":123595},[217,218,219,220],[39,123597,123599,123602,123605,123651,123654],{"className":123598},[83,220],[39,123600,15389],{"className":123601},[83,220],[39,123603,74335],{"className":123604,"style":107705},[83,84,220],[39,123606,123608,123611],{"className":123607},[83,220],[39,123609,15389],{"className":123610},[83,220],[39,123612,123614],{"className":123613},[192],[39,123615,123617,123643],{"className":123616},[196,649],[39,123618,123620,123640],{"className":123619},[200],[39,123621,123623],{"className":123622,"style":84344},[204],[39,123624,123625,123628],{"style":84347},[39,123626],{"className":123627,"style":4004},[212],[39,123629,123631],{"className":123630},[217,4008,4009,220],[39,123632,123634],{"className":123633},[83,220],[39,123635,123637],{"className":123636},[83,220],[39,123638,84363],{"className":123639},[83,35077,220],[39,123641,673],{"className":123642},[672],[39,123644,123646],{"className":123645},[200],[39,123647,123649],{"className":123648,"style":84373},[204],[39,123650],{},[39,123652,1824],{"className":123653},[235,220],[39,123655,546],{"className":123656},[83,220],[39,123658,123659,123662],{"style":21599},[39,123660],{"className":123661,"style":935},[212],[39,123663,123664],{},[39,123665,1687],{"className":123666},[1744],[39,123668,673],{"className":123669},[672],[39,123671,123673],{"className":123672},[200],[39,123674,123677],{"className":123675,"style":123676},[204],"height:0.9885em;",[39,123678],{},[39,123680,74223],{"className":123681},[230],[39,123683],{"className":123684,"style":256},[230],[39,123686,123688,123691],{"className":123687},[83],[39,123689,4253],{"className":123690},[83,515],[39,123692,123694],{"className":123693},[192],[39,123695,123697,123758],{"className":123696},[196,649],[39,123698,123700,123755],{"className":123699},[200],[39,123701,123703],{"className":123702,"style":62921},[204],[39,123704,123705,123708],{"style":966},[39,123706],{"className":123707,"style":213},[212],[39,123709,123711],{"className":123710},[217,218,219,220],[39,123712,123714],{"className":123713},[83,220],[39,123715,123717,123720],{"className":123716},[83,220],[39,123718,131],{"className":123719,"style":178},[83,84,220],[39,123721,123723],{"className":123722},[192],[39,123724,123726,123747],{"className":123725},[196,649],[39,123727,123729,123744],{"className":123728},[200],[39,123730,123732],{"className":123731,"style":42105},[204],[39,123733,123735,123738],{"style":123734},"top:-2.357em;margin-left:-0.1389em;margin-right:0.0714em;",[39,123736],{"className":123737,"style":4004},[212],[39,123739,123741],{"className":123740},[217,4008,4009,220],[39,123742,401],{"className":123743,"style":438},[83,84,220],[39,123745,673],{"className":123746},[672],[39,123748,123750],{"className":123749},[200],[39,123751,123753],{"className":123752,"style":64389},[204],[39,123754],{},[39,123756,673],{"className":123757},[672],[39,123759,123761],{"className":123760},[200],[39,123762,123765],{"className":123763,"style":123764},[204],"height:0.2501em;",[39,123766],{},[39,123768,537],{"className":123769},[182],[39,123771,74335],{"className":123772,"style":107705},[83,84],[39,123774,136],{"className":123775},[182],[39,123777,1755],{"className":123778},[83,84],[39,123780,102832],{"className":123781},[263],[39,123783],{"className":123784,"style":461},[230],[39,123786,5378],{"className":123787},[465],[39,123789],{"className":123790,"style":461},[230],[39,123792,123794,123798,123879,123882,123885,123888,123891,123894],{"className":123793},[74],[39,123795],{"className":123796,"style":123797},[78],"height:1.0973em;vertical-align:-0.3473em;",[39,123799,123801,123804],{"className":123800},[83],[39,123802,4253],{"className":123803},[83,515],[39,123805,123807],{"className":123806},[192],[39,123808,123810,123870],{"className":123809},[196,649],[39,123811,123813,123867],{"className":123812},[200],[39,123814,123816],{"className":123815,"style":62921},[204],[39,123817,123818,123821],{"style":966},[39,123819],{"className":123820,"style":213},[212],[39,123822,123824],{"className":123823},[217,218,219,220],[39,123825,123827],{"className":123826},[83,220],[39,123828,123830,123833],{"className":123829},[83,220],[39,123831,131],{"className":123832,"style":178},[83,84,220],[39,123834,123836],{"className":123835},[192],[39,123837,123839,123859],{"className":123838},[196,649],[39,123840,123842,123856],{"className":123841},[200],[39,123843,123845],{"className":123844,"style":42105},[204],[39,123846,123847,123850],{"style":123734},[39,123848],{"className":123849,"style":4004},[212],[39,123851,123853],{"className":123852},[217,4008,4009,220],[39,123854,39209],{"className":123855,"style":2114},[83,84,220],[39,123857,673],{"className":123858},[672],[39,123860,123862],{"className":123861},[200],[39,123863,123865],{"className":123864,"style":42127},[204],[39,123866],{},[39,123868,673],{"className":123869},[672],[39,123871,123873],{"className":123872},[200],[39,123874,123877],{"className":123875,"style":123876},[204],"height:0.3473em;",[39,123878],{},[39,123880,537],{"className":123881},[182],[39,123883,74335],{"className":123884,"style":107705},[83,84],[39,123886,136],{"className":123887},[182],[39,123889,1755],{"className":123890},[83,84],[39,123892,102832],{"className":123893},[263],[39,123895,155],{"className":123896},[252],[11,123898,123899,123900,123928],{},"critic ",[39,123901,123903,123916],{"className":123902,"translate":43},[42],[39,123904,123906],{"className":123905},[47],[49,123907,123908],{"xmlns":51},[53,123909,123910,123914],{},[56,123911,123912],{},[59,123913,74335],{},[63,123915,74335],{"encoding":65},[39,123917,123919],{"className":123918,"ariaHidden":70},[69],[39,123920,123922,123925],{"className":123921},[74],[39,123923],{"className":123924,"style":51280},[78],[39,123926,74335],{"className":123927,"style":107705},[83,84]," 就是 Kantorovich 势函数，Lipschitz 约束用 weight clipping 或 gradient penalty 近似。这个改动的实质是把一个饱和的散度换成一个不饱和的度量。",[11,123930,123931],{},"扩散模型和流匹配是同一件事的两种写法，都是把噪声分布运输到数据分布。区别在路径：扩散用固定的前向过程，流匹配可以自己选，选了 OT 路径就得到更直的轨迹和更少的采样步数。",[11,123933,123934,123935,123963,123964,124088,124089,124228,124229,124257,124258,124437,124438,124507],{},"域适应与分布偏移是 ",[39,123936,123938,123951],{"className":123937,"translate":43},[42],[39,123939,123941],{"className":123940},[47],[49,123942,123943],{"xmlns":51},[53,123944,123945,123949],{},[56,123946,123947],{},[59,123948,23703],{},[63,123950,23703],{"encoding":65},[39,123952,123954],{"className":123953,"ariaHidden":70},[69],[39,123955,123957,123960],{"className":123956},[74],[39,123958],{"className":123959,"style":79},[78],[39,123961,23703],{"className":123962,"style":178},[83,84]," 最能发挥的地方。给定源域和目标域的特征分布 ",[39,123965,123967,123993],{"className":123966,"translate":43},[42],[39,123968,123970],{"className":123969},[47],[49,123971,123972],{"xmlns":51},[53,123973,123974,123990],{},[56,123975,123976,123982,123984],{},[613,123977,123978,123980],{},[59,123979,73578],{},[59,123981,61],{},[133,123983,155],{"separator":70},[613,123985,123986,123988],{},[59,123987,73578],{},[59,123989,11772],{},[63,123991,123992],{"encoding":65},"\\mu_S,\\mu_T",[39,123994,123996],{"className":123995,"ariaHidden":70},[69],[39,123997,123999,124002,124042,124045,124048],{"className":123998},[74],[39,124000],{"className":124001,"style":2654},[78],[39,124003,124005,124008],{"className":124004},[83],[39,124006,73578],{"className":124007},[83,84],[39,124009,124011],{"className":124010},[192],[39,124012,124014,124034],{"className":124013},[196,649],[39,124015,124017,124031],{"className":124016},[200],[39,124018,124020],{"className":124019,"style":62921},[204],[39,124021,124022,124025],{"style":966},[39,124023],{"className":124024,"style":213},[212],[39,124026,124028],{"className":124027},[217,218,219,220],[39,124029,61],{"className":124030,"style":85},[83,84,220],[39,124032,673],{"className":124033},[672],[39,124035,124037],{"className":124036},[200],[39,124038,124040],{"className":124039,"style":680},[204],[39,124041],{},[39,124043,155],{"className":124044},[252],[39,124046],{"className":124047,"style":256},[230],[39,124049,124051,124054],{"className":124050},[83],[39,124052,73578],{"className":124053},[83,84],[39,124055,124057],{"className":124056},[192],[39,124058,124060,124080],{"className":124059},[196,649],[39,124061,124063,124077],{"className":124062},[200],[39,124064,124066],{"className":124065,"style":62921},[204],[39,124067,124068,124071],{"style":966},[39,124069],{"className":124070,"style":213},[212],[39,124072,124074],{"className":124073},[217,218,219,220],[39,124075,11772],{"className":124076,"style":178},[83,84,220],[39,124078,673],{"className":124079},[672],[39,124081,124083],{"className":124082},[200],[39,124084,124086],{"className":124085,"style":680},[204],[39,124087],{},"，直接最小化 ",[39,124090,124092,124124],{"className":124091,"translate":43},[42],[39,124093,124095],{"className":124094},[47],[49,124096,124097],{"xmlns":51},[53,124098,124099,124121],{},[56,124100,124101,124103,124105,124111,124113,124119],{},[59,124102,23703],{},[133,124104,136],{"stretchy":135},[613,124106,124107,124109],{},[59,124108,73578],{},[59,124110,61],{},[133,124112,155],{"separator":70},[613,124114,124115,124117],{},[59,124116,73578],{},[59,124118,11772],{},[133,124120,161],{"stretchy":135},[63,124122,124123],{"encoding":65},"W(\\mu_S,\\mu_T)",[39,124125,124127],{"className":124126,"ariaHidden":70},[69],[39,124128,124130,124133,124136,124139,124179,124182,124185,124225],{"className":124129},[74],[39,124131],{"className":124132,"style":245},[78],[39,124134,23703],{"className":124135,"style":178},[83,84],[39,124137,136],{"className":124138},[182],[39,124140,124142,124145],{"className":124141},[83],[39,124143,73578],{"className":124144},[83,84],[39,124146,124148],{"className":124147},[192],[39,124149,124151,124171],{"className":124150},[196,649],[39,124152,124154,124168],{"className":124153},[200],[39,124155,124157],{"className":124156,"style":62921},[204],[39,124158,124159,124162],{"style":966},[39,124160],{"className":124161,"style":213},[212],[39,124163,124165],{"className":124164},[217,218,219,220],[39,124166,61],{"className":124167,"style":85},[83,84,220],[39,124169,673],{"className":124170},[672],[39,124172,124174],{"className":124173},[200],[39,124175,124177],{"className":124176,"style":680},[204],[39,124178],{},[39,124180,155],{"className":124181},[252],[39,124183],{"className":124184,"style":256},[230],[39,124186,124188,124191],{"className":124187},[83],[39,124189,73578],{"className":124190},[83,84],[39,124192,124194],{"className":124193},[192],[39,124195,124197,124217],{"className":124196},[196,649],[39,124198,124200,124214],{"className":124199},[200],[39,124201,124203],{"className":124202,"style":62921},[204],[39,124204,124205,124208],{"style":966},[39,124206],{"className":124207,"style":213},[212],[39,124209,124211],{"className":124210},[217,218,219,220],[39,124212,11772],{"className":124213,"style":178},[83,84,220],[39,124215,673],{"className":124216},[672],[39,124218,124220],{"className":124219},[200],[39,124221,124223],{"className":124222,"style":680},[204],[39,124224],{},[39,124226,161],{"className":124227},[263]," 就能对齐两个域，且不需要标签。对偶形式又让它变成对抗训练：一个网络学 ",[39,124230,124232,124245],{"className":124231,"translate":43},[42],[39,124233,124235],{"className":124234},[47],[49,124236,124237],{"xmlns":51},[53,124238,124239,124243],{},[56,124240,124241],{},[59,124242,80648],{},[63,124244,84085],{"encoding":65},[39,124246,124248],{"className":124247,"ariaHidden":70},[69],[39,124249,124251,124254],{"className":124250},[74],[39,124252],{"className":124253,"style":2654},[78],[39,124255,80648],{"className":124256},[83,84]," 去最大化 ",[39,124259,124261,124303],{"className":124260,"translate":43},[42],[39,124262,124264],{"className":124263},[47],[49,124265,124266],{"xmlns":51},[53,124267,124268,124300],{},[56,124269,124270,124272,124274,124276,124278,124284,124286,124288,124290,124292,124294],{},[133,124271,69087],{},[59,124273,80648],{},[35841,124275,35843],{},[59,124277,29425],{},[613,124279,124280,124282],{},[59,124281,73578],{},[59,124283,61],{},[133,124285,5378],{},[133,124287,69087],{},[59,124289,80648],{},[35841,124291,35843],{},[59,124293,29425],{},[613,124295,124296,124298],{},[59,124297,73578],{},[59,124299,11772],{},[63,124301,124302],{"encoding":65},"\\int\\varphi\\,d\\mu_S-\\int\\varphi\\,d\\mu_T",[39,124304,124306,124376],{"className":124305,"ariaHidden":70},[69],[39,124307,124309,124312,124315,124318,124321,124324,124327,124367,124370,124373],{"className":124308},[74],[39,124310],{"className":124311,"style":74140},[78],[39,124313,69087],{"className":124314,"style":74144},[1744,3339,3952],[39,124316],{"className":124317,"style":256},[230],[39,124319,80648],{"className":124320},[83,84],[39,124322],{"className":124323,"style":256},[230],[39,124325,29425],{"className":124326},[83,84],[39,124328,124330,124333],{"className":124329},[83],[39,124331,73578],{"className":124332},[83,84],[39,124334,124336],{"className":124335},[192],[39,124337,124339,124359],{"className":124338},[196,649],[39,124340,124342,124356],{"className":124341},[200],[39,124343,124345],{"className":124344,"style":62921},[204],[39,124346,124347,124350],{"style":966},[39,124348],{"className":124349,"style":213},[212],[39,124351,124353],{"className":124352},[217,218,219,220],[39,124354,61],{"className":124355,"style":85},[83,84,220],[39,124357,673],{"className":124358},[672],[39,124360,124362],{"className":124361},[200],[39,124363,124365],{"className":124364,"style":680},[204],[39,124366],{},[39,124368],{"className":124369,"style":461},[230],[39,124371,5378],{"className":124372},[465],[39,124374],{"className":124375,"style":461},[230],[39,124377,124379,124382,124385,124388,124391,124394,124397],{"className":124378},[74],[39,124380],{"className":124381,"style":74140},[78],[39,124383,69087],{"className":124384,"style":74144},[1744,3339,3952],[39,124386],{"className":124387,"style":256},[230],[39,124389,80648],{"className":124390},[83,84],[39,124392],{"className":124393,"style":256},[230],[39,124395,29425],{"className":124396},[83,84],[39,124398,124400,124403],{"className":124399},[83],[39,124401,73578],{"className":124402},[83,84],[39,124404,124406],{"className":124405},[192],[39,124407,124409,124429],{"className":124408},[196,649],[39,124410,124412,124426],{"className":124411},[200],[39,124413,124415],{"className":124414,"style":62921},[204],[39,124416,124417,124420],{"style":966},[39,124418],{"className":124419,"style":213},[212],[39,124421,124423],{"className":124422},[217,218,219,220],[39,124424,11772],{"className":124425,"style":178},[83,84,220],[39,124427,673],{"className":124428},[672],[39,124430,124432],{"className":124431},[200],[39,124433,124435],{"className":124434,"style":680},[204],[39,124436],{},"，另一个网络调整特征表示去最小化这个差。多个源域时用 Wasserstein 重心，在 ",[39,124439,124441,124458],{"className":124440,"translate":43},[42],[39,124442,124444],{"className":124443},[47],[49,124445,124446],{"xmlns":51},[53,124447,124448,124456],{},[56,124449,124450],{},[613,124451,124452,124454],{},[59,124453,23703],{},[539,124455,17359],{},[63,124457,91634],{"encoding":65},[39,124459,124461],{"className":124460,"ariaHidden":70},[69],[39,124462,124464,124467],{"className":124463},[74],[39,124465],{"className":124466,"style":634},[78],[39,124468,124470,124473],{"className":124469},[83],[39,124471,23703],{"className":124472,"style":178},[83,84],[39,124474,124476],{"className":124475},[192],[39,124477,124479,124499],{"className":124478},[196,649],[39,124480,124482,124496],{"className":124481},[200],[39,124483,124485],{"className":124484,"style":963},[204],[39,124486,124487,124490],{"style":73765},[39,124488],{"className":124489,"style":213},[212],[39,124491,124493],{"className":124492},[217,218,219,220],[39,124494,17359],{"className":124495},[83,220],[39,124497,673],{"className":124498},[672],[39,124500,124502],{"className":124501},[200],[39,124503,124505],{"className":124504,"style":680},[204],[39,124506],{}," 空间里求“平均分布”，这正是 Fréchet 均值在分布空间里的实例。",[11,124509,124510],{},"OOD 检测，也就是我开头那个 0.99 的问题，我会分三层来做。",[11,124512,124513],{},"设想一条质检流水线，训练时只有“正常”“划痕”“凹陷”三类。现在来了一个从没见过的新缺陷。",[11,124515,124516,124517,124665],{},"点态一层比较样本密度：对每个类拟合 KDE 或流，取 ",[39,124518,124520,124554],{"className":124519,"translate":43},[42],[39,124521,124523],{"className":124522},[47],[49,124524,124525],{"xmlns":51},[53,124526,124527,124551],{},[56,124528,124529,124531,124533,124535,124545,124547,124549],{},[133,124530,5378],{},[59,124532,92483],{},[133,124534,1690],{},[613,124536,124537,124543],{},[39689,124538,124539,124541],{"accent":70},[59,124540,11],{},[133,124542,114909],{},[59,124544,3196],{},[133,124546,136],{"stretchy":135},[59,124548,1755],{},[133,124550,161],{"stretchy":135},[63,124552,124553],{"encoding":65},"-\\log\\hat p_k(x)",[39,124555,124557],{"className":124556,"ariaHidden":70},[69],[39,124558,124560,124563,124566,124569,124574,124577,124656,124659,124662],{"className":124559},[74],[39,124561],{"className":124562,"style":245},[78],[39,124564,5378],{"className":124565},[83],[39,124567],{"className":124568,"style":256},[230],[39,124570,92721,124572],{"className":124571},[1744],[39,124573,39209],{"style":35078},[39,124575],{"className":124576,"style":256},[230],[39,124578,124580,124622],{"className":124579},[83],[39,124581,124583],{"className":124582},[83,39712],[39,124584,124586,124614],{"className":124585},[196,649],[39,124587,124589,124611],{"className":124588},[200],[39,124590,124592,124600],{"className":124591,"style":28824},[204],[39,124593,124594,124597],{"style":21599},[39,124595],{"className":124596,"style":935},[212],[39,124598,11],{"className":124599},[83,84],[39,124601,124602,124605],{"style":21599},[39,124603],{"className":124604,"style":935},[212],[39,124606,124608],{"className":124607,"style":115045},[39739],[39,124609,114909],{"className":124610},[83],[39,124612,673],{"className":124613},[672],[39,124615,124617],{"className":124616},[200],[39,124618,124620],{"className":124619,"style":115058},[204],[39,124621],{},[39,124623,124625],{"className":124624},[192],[39,124626,124628,124648],{"className":124627},[196,649],[39,124629,124631,124645],{"className":124630},[200],[39,124632,124634],{"className":124633,"style":3419},[204],[39,124635,124636,124639],{"style":966},[39,124637],{"className":124638,"style":213},[212],[39,124640,124642],{"className":124641},[217,218,219,220],[39,124643,3196],{"className":124644,"style":3321},[83,84,220],[39,124646,673],{"className":124647},[672],[39,124649,124651],{"className":124650},[200],[39,124652,124654],{"className":124653,"style":680},[204],[39,124655],{},[39,124657,136],{"className":124658},[182],[39,124660,1755],{"className":124661},[83,84],[39,124663,161],{"className":124664},[263]," 作为分数。它便宜、可解释，但受典型集悖论和带宽影响，高维不可靠——上一节那个“空洞比簇心更密”的例子就是反例。",[11,124667,124668,124669,124738,124739,124815,124816,124844,124845,742,124873,124917],{},"分布一层算测试批（或测试样本的邻域）到每个类条件分布的 ",[39,124670,124672,124689],{"className":124671,"translate":43},[42],[39,124673,124675],{"className":124674},[47],[49,124676,124677],{"xmlns":51},[53,124678,124679,124687],{},[56,124680,124681],{},[613,124682,124683,124685],{},[59,124684,23703],{},[539,124686,17359],{},[63,124688,91634],{"encoding":65},[39,124690,124692],{"className":124691,"ariaHidden":70},[69],[39,124693,124695,124698],{"className":124694},[74],[39,124696],{"className":124697,"style":634},[78],[39,124699,124701,124704],{"className":124700},[83],[39,124702,23703],{"className":124703,"style":178},[83,84],[39,124705,124707],{"className":124706},[192],[39,124708,124710,124730],{"className":124709},[196,649],[39,124711,124713,124727],{"className":124712},[200],[39,124714,124716],{"className":124715,"style":963},[204],[39,124717,124718,124721],{"style":73765},[39,124719],{"className":124720,"style":213},[212],[39,124722,124724],{"className":124723},[217,218,219,220],[39,124725,17359],{"className":124726},[83,220],[39,124728,673],{"className":124729},[672],[39,124731,124733],{"className":124732},[200],[39,124734,124736],{"className":124735,"style":680},[204],[39,124737],{},"，取最小者作为“离最近已知类的距离”。优点是它看整片质量而不是单个点；缺点是它需要一批样本，还有 ",[39,124740,124742,124768],{"className":124741,"translate":43},[42],[39,124743,124745],{"className":124744},[47],[49,124746,124747],{"xmlns":51},[53,124748,124749,124765],{},[56,124750,124751],{},[138,124752,124753,124755],{},[59,124754,12962],{},[56,124756,124757,124759,124761,124763],{},[133,124758,5378],{},[539,124760,546],{},[59,124762,45691],{"mathvariant":145},[59,124764,29425],{},[63,124766,124767],{"encoding":65},"n^{-1\u002Fd}",[39,124769,124771],{"className":124770,"ariaHidden":70},[69],[39,124772,124774,124777],{"className":124773},[74],[39,124775],{"className":124776,"style":95385},[78],[39,124778,124780,124783],{"className":124779},[83],[39,124781,12962],{"className":124782},[83,84],[39,124784,124786],{"className":124785},[192],[39,124787,124789],{"className":124788},[196],[39,124790,124792],{"className":124791},[200],[39,124793,124795],{"className":124794,"style":95385},[204],[39,124796,124797,124800],{"style":208},[39,124798],{"className":124799,"style":213},[212],[39,124801,124803],{"className":124802},[217,218,219,220],[39,124804,124806,124809,124812],{"className":124805},[83,220],[39,124807,5378],{"className":124808},[83,220],[39,124810,76672],{"className":124811},[83,220],[39,124813,29425],{"className":124814},[83,84,220]," 的样本复杂度。实践中的折中是：在低维隐空间里算，或者用对偶势 ",[39,124817,124819,124832],{"className":124818,"translate":43},[42],[39,124820,124822],{"className":124821},[47],[49,124823,124824],{"xmlns":51},[53,124825,124826,124830],{},[56,124827,124828],{},[59,124829,80648],{},[63,124831,84085],{"encoding":65},[39,124833,124835],{"className":124834,"ariaHidden":70},[69],[39,124836,124838,124841],{"className":124837},[74],[39,124839],{"className":124840,"style":2654},[78],[39,124842,80648],{"className":124843},[83,84],"——一旦在一批数据上解出 ",[39,124846,124848,124861],{"className":124847,"translate":43},[42],[39,124849,124851],{"className":124850},[47],[49,124852,124853],{"xmlns":51},[53,124854,124855,124859],{},[56,124856,124857],{},[59,124858,80648],{},[63,124860,84085],{"encoding":65},[39,124862,124864],{"className":124863,"ariaHidden":70},[69],[39,124865,124867,124870],{"className":124866},[74],[39,124868],{"className":124869,"style":2654},[78],[39,124871,80648],{"className":124872},[83,84],[39,124874,124876,124896],{"className":124875,"translate":43},[42],[39,124877,124879],{"className":124878},[47],[49,124880,124881],{"xmlns":51},[53,124882,124883,124893],{},[56,124884,124885,124887,124889,124891],{},[59,124886,80648],{},[133,124888,136],{"stretchy":135},[59,124890,1755],{},[133,124892,161],{"stretchy":135},[63,124894,124895],{"encoding":65},"\\varphi(x)",[39,124897,124899],{"className":124898,"ariaHidden":70},[69],[39,124900,124902,124905,124908,124911,124914],{"className":124901},[74],[39,124903],{"className":124904,"style":245},[78],[39,124906,80648],{"className":124907},[83,84],[39,124909,136],{"className":124910},[182],[39,124912,1755],{"className":124913},[83,84],[39,124915,161],{"className":124916},[263]," 本身就是逐样本分数，可以拿去做单点判定。我很喜欢这个技巧，它把“批级别的距离”翻译成了“点级别的分数”，两个需求一次满足。",[11,124919,124920],{},"输出一层比较类别概率分布的差异：训练时记录每个类在验证集上的平均预测分布（单纯形上的一个点），测试时算测试批的平均预测分布与它的 Fisher–Rao 距离。距离大，说明模型在这批数据上的“说话方式”变了，即使每个样本的置信度都不低。这一层不需要标签也不需要密度估计，我觉得是性价比最高的一个。",[11,124922,124923],{},"分类与损失那一侧前面已经说了：交叉熵是 Bregman，自然梯度是 Fisher 预条件，镜像下降是 Bregman 投影。这三个东西实现上都很朴素，但把它们在理论上认成同一族，能省掉很多“为什么这个 trick 有效”的困惑。",[21,124925,124926],{"id":124926},"回到开头",[11,124928,124929],{},"回到那个 0.99。",[11,124931,124932,124933,124961],{},"置信度回答的是“在这几个类里最像哪个”，它是一次相对排序，分母被 softmax 强行归一化成 ",[39,124934,124936,124949],{"className":124935,"translate":43},[42],[39,124937,124939],{"className":124938},[47],[49,124940,124941],{"xmlns":51},[53,124942,124943,124947],{},[56,124944,124945],{},[539,124946,546],{},[63,124948,546],{"encoding":65},[39,124950,124952],{"className":124951,"ariaHidden":70},[69],[39,124953,124955,124958],{"className":124954},[74],[39,124956],{"className":124957,"style":2473},[78],[39,124959,546],{"className":124960},[83],"。分布几何回答的是“这个点离已知类别的概率质量有多远”，它是一个绝对量，不依赖“一定属于某个类”这个假设。这两件事不该互相替代，而应该互补：分类器给候选，分布几何判越界。",[11,124963,124964],{},"如果只留一个式子，我会留 Kantorovich 对偶：",[39,124966,124968],{"className":124967,"translate":43},[746],[39,124969,124971,125051],{"className":124970,"translate":43},[42],[39,124972,124974],{"className":124973},[47],[49,124975,124976],{"xmlns":51,"display":755},[53,124977,124978,125048],{},[56,124979,124980,124986,124988,124990,124992,124994,124996,124998,125028,125030,125032,125034,125036,125038,125040,125042,125044,125046],{},[613,124981,124982,124984],{},[59,124983,23703],{},[539,124985,546],{},[133,124987,136],{"stretchy":135},[59,124989,73578],{},[133,124991,155],{"separator":70},[59,124993,73609],{},[133,124995,161],{"stretchy":135},[133,124997,793],{},[6556,124999,125000,125006],{},[56,125001,125002,125004],{},[59,125003,21465],{},[133,125005,1690],{},[56,125007,125008,125010,125012,125024,125026],{},[59,125009,15389],{"mathvariant":145},[59,125011,80648],{},[613,125013,125014,125016],{},[59,125015,15389],{"mathvariant":145},[56,125017,125018,125020,125022],{},[59,125019,77190],{"mathvariant":145},[59,125021,13377],{"mathvariant":145},[59,125023,11],{"mathvariant":145},[133,125025,1824],{},[539,125027,546],{},[133,125029,69087],{},[59,125031,80648],{},[35841,125033,35843],{},[59,125035,29425],{},[133,125037,136],{"stretchy":135},[59,125039,73578],{},[133,125041,5378],{},[59,125043,73609],{},[133,125045,161],{"stretchy":135},[59,125047,2903],{"mathvariant":145},[63,125049,125050],{"encoding":65},"W_1(\\mu,\\nu)=\\sup_{\\|\\varphi\\|_{\\mathrm{Lip}}\\le1}\\int\\varphi\\,d(\\mu-\\nu).",[39,125052,125054,125127,125268],{"className":125053,"ariaHidden":70},[69],[39,125055,125057,125060,125100,125103,125106,125109,125112,125115,125118,125121,125124],{"className":125056},[74],[39,125058],{"className":125059,"style":245},[78],[39,125061,125063,125066],{"className":125062},[83],[39,125064,23703],{"className":125065,"style":178},[83,84],[39,125067,125069],{"className":125068},[192],[39,125070,125072,125092],{"className":125071},[196,649],[39,125073,125075,125089],{"className":125074},[200],[39,125076,125078],{"className":125077,"style":963},[204],[39,125079,125080,125083],{"style":73765},[39,125081],{"className":125082,"style":213},[212],[39,125084,125086],{"className":125085},[217,218,219,220],[39,125087,546],{"className":125088},[83,220],[39,125090,673],{"className":125091},[672],[39,125093,125095],{"className":125094},[200],[39,125096,125098],{"className":125097,"style":680},[204],[39,125099],{},[39,125101,136],{"className":125102},[182],[39,125104,73578],{"className":125105},[83,84],[39,125107,155],{"className":125108},[252],[39,125110],{"className":125111,"style":256},[230],[39,125113,73609],{"className":125114,"style":73625},[83,84],[39,125116,161],{"className":125117},[263],[39,125119],{"className":125120,"style":231},[230],[39,125122,793],{"className":125123},[235],[39,125125],{"className":125126,"style":231},[230],[39,125128,125130,125133,125235,125238,125241,125244,125247,125250,125253,125256,125259,125262,125265],{"className":125129},[74],[39,125131],{"className":125132,"style":84295},[78],[39,125134,125136],{"className":125135},[1744,3294],[39,125137,125139,125227],{"className":125138},[196,649],[39,125140,125142,125224],{"className":125141},[200],[39,125143,125145,125214],{"className":125144,"style":290},[204],[39,125146,125147,125150],{"style":84310},[39,125148],{"className":125149,"style":935},[212],[39,125151,125153],{"className":125152},[217,218,219,220],[39,125154,125156,125159,125162,125208,125211],{"className":125155},[83,220],[39,125157,15389],{"className":125158},[83,220],[39,125160,80648],{"className":125161},[83,84,220],[39,125163,125165,125168],{"className":125164},[83,220],[39,125166,15389],{"className":125167},[83,220],[39,125169,125171],{"className":125170},[192],[39,125172,125174,125200],{"className":125173},[196,649],[39,125175,125177,125197],{"className":125176},[200],[39,125178,125180],{"className":125179,"style":84344},[204],[39,125181,125182,125185],{"style":84347},[39,125183],{"className":125184,"style":4004},[212],[39,125186,125188],{"className":125187},[217,4008,4009,220],[39,125189,125191],{"className":125190},[83,220],[39,125192,125194],{"className":125193},[83,220],[39,125195,84363],{"className":125196},[83,35077,220],[39,125198,673],{"className":125199},[672],[39,125201,125203],{"className":125202},[200],[39,125204,125206],{"className":125205,"style":84373},[204],[39,125207],{},[39,125209,1824],{"className":125210},[235,220],[39,125212,546],{"className":125213},[83,220],[39,125215,125216,125219],{"style":21599},[39,125217],{"className":125218,"style":935},[212],[39,125220,125221],{},[39,125222,21465],{"className":125223},[1744],[39,125225,673],{"className":125226},[672],[39,125228,125230],{"className":125229},[200],[39,125231,125233],{"className":125232,"style":84401},[204],[39,125234],{},[39,125236],{"className":125237,"style":256},[230],[39,125239,69087],{"className":125240,"style":69380},[1744,3339,3340],[39,125242],{"className":125243,"style":256},[230],[39,125245,80648],{"className":125246},[83,84],[39,125248],{"className":125249,"style":256},[230],[39,125251,29425],{"className":125252},[83,84],[39,125254,136],{"className":125255},[182],[39,125257,73578],{"className":125258},[83,84],[39,125260],{"className":125261,"style":461},[230],[39,125263,5378],{"className":125264},[465],[39,125266],{"className":125267,"style":461},[230],[39,125269,125271,125274,125277,125280],{"className":125270},[74],[39,125272],{"className":125273,"style":245},[78],[39,125275,73609],{"className":125276,"style":73625},[83,84],[39,125278,161],{"className":125279},[263],[39,125281,2903],{"className":125282},[83],[11,125284,125285],{},"它把一个关于分布的优化问题，变成了一个关于函数的优化问题。分布是无穷维的、难参数化的；函数是可以拿神经网络拟合的。这一步转换，才是最优传输能从 1781 年的搬土问题走进今天每一篇 WGAN、每一次域适应、每一个流匹配模型的真正原因。",[11,125287,125288],{},"至于 Fisher–Rao 和 Bregman，它们给我的东西不太一样：不是算法，而是一种校准过的直觉——当我说“两个分布很近”的时候，我最好知道自己是拿哪把尺子在量，以及那把尺子在局部是不是真的像一个度量。",{"title":72720,"searchDepth":72721,"depth":72721,"links":125290},[125291,125292,125293,125294,125295,125296,125297,125298,125299,125300,125301],{"id":72794,"depth":72721,"text":72794},{"id":77921,"depth":72721,"text":77921},{"id":84830,"depth":72721,"text":84830},{"id":91986,"depth":72721,"text":91986},{"id":95424,"depth":72721,"text":95424},{"id":101925,"depth":72721,"text":101925},{"id":106422,"depth":72721,"text":106423},{"id":114424,"depth":72721,"text":114424},{"id":118092,"depth":72721,"text":118092},{"id":123292,"depth":72721,"text":123292},{"id":124926,"depth":72721,"text":124926},{},"\u002Fblog\u002F2026\u002F0014-zui-you-chuan-shu-ji-he-jian-lun",{"title":72783,"description":72788},"blog\u002F2026\u002F0014-zui-you-chuan-shu-ji-he-jian-lun","从 Kantorovich 对偶到 Fisher–Rao 度量、Bregman 散度、核密度估计与标准化流，用同一套分布几何的语言解释为什么分布层面的判据比分类器置信度更稳健。",[72778,72777,125308],"statistics","2026-10-31T20:17:23+08:00","IAT5rYeOtscnEe8Mhh1eCeIF4PJR1Ewyc86Z_G_0ncI",1794239909349]