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x_0",[1109,1368,1370,1389],{"className":1369,"ariaHidden":1139},[1138],[1109,1371,1373,1377,1380,1383,1386],{"className":1372},[1143],[1109,1374],{"className":1375,"style":1376},[1147],"height:0.7719em;vertical-align:-0.136em;",[1109,1378,1268],{"className":1379},[1152,1153],[1109,1381],{"className":1382,"style":1232},[1231],[1109,1384,1357],{"className":1385},[1236],[1109,1387],{"className":1388,"style":1232},[1231],[1109,1390,1392,1395],{"className":1391},[1143],[1109,1393],{"className":1394,"style":1283},[1147],[1109,1396,1398,1401],{"className":1397},[1152],[1109,1399,1268],{"className":1400},[1152,1153],[1109,1402,1404],{"className":1403},[1293],[1109,1405,1407,1427],{"className":1406},[1297,1298],[1109,1408,1410,1424],{"className":1409},[1302],[1109,1411,1413],{"className":1412,"style":1307},[1306],[1109,1414,1415,1418],{"style":1310},[1109,1416],{"className":1417,"style":1315},[1314],[1109,1419,1421],{"className":1420},[1319,1320,1321,1322],[1109,1422,1210],{"className":1423},[1152,1322],[1109,1425,1330],{"className":1426},[1329],[1109,1428,1430],{"className":1429},[1302],[1109,1431,1433],{"className":1432,"style":1337},[1306],[1109,1434],{}," 时 ",[1109,1437,1439,1489],{"className":1438},[1112],[1109,1440,1442],{"className":1441},[1116],[1118,1443,1444],{"xmlns":1120},[1122,1445,1446,1486],{},[1125,1447,1448,1452,1454,1458,1460,1463,1465,1468,1470,1474,1476,1478,1480,1482,1484],{},[1128,1449,1451],{"mathvariant":1450},"normal","∣",[1128,1453,1130],{},[1204,1455,1457],{"stretchy":1456},"false","(",[1128,1459,1268],{},[1204,1461,1462],{"stretchy":1456},")",[1128,1464,1451],{"mathvariant":1450},[1204,1466,1467],{},"≤",[1128,1469,1202],{},[1471,1472,1473],"mtext",{}," ",[1128,1475,1451],{"mathvariant":1450},[1128,1477,1170],{},[1204,1479,1457],{"stretchy":1456},[1128,1481,1268],{},[1204,1483,1462],{"stretchy":1456},[1128,1485,1451],{"mathvariant":1450},[1132,1487,1488],{"encoding":1134},"|f(x)| \\le 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，记 ",[1109,1561,1563,1601],{"className":1562},[1112],[1109,1564,1566],{"className":1565},[1116],[1118,1567,1568],{"xmlns":1120},[1122,1569,1570,1598],{},[1125,1571,1572,1574,1576,1578,1580,1583,1586,1588,1590,1592,1594,1596],{},[1128,1573,1130],{},[1204,1575,1457],{"stretchy":1456},[1128,1577,1268],{},[1204,1579,1462],{"stretchy":1456},[1204,1581,1582],{},"=",[1128,1584,1585],{},"O",[1204,1587,1457],{"stretchy":1456},[1128,1589,1170],{},[1204,1591,1457],{"stretchy":1456},[1128,1593,1268],{},[1204,1595,1462],{"stretchy":1456},[1204,1597,1462],{"stretchy":1456},[1132,1599,1600],{"encoding":1134},"f(x) = O(g(x))",[1109,1602,1604,1631],{"className":1603,"ariaHidden":1139},[1138],[1109,1605,1607,1610,1613,1616,1619,1622,1625,1628],{"className":1606},[1143],[1109,1608],{"className":1609,"style":1498},[1147],[1109,1611,1130],{"className":1612,"style":1154},[1152,1153],[1109,1614,1457],{"className":1615},[1508],[1109,1617,1268],{"className":1618},[1152,1153],[1109,1620,1462],{"className":1621},[1515],[1109,1623],{"className":1624,"style":1232},[1231],[1109,1626,1582],{"className":1627},[1236],[1109,1629],{"className":1630,"style":1232},[1231],[1109,1632,1634,1637,1641,1644,1647,1650,1653],{"className":1633},[1143],[1109,1635],{"className":1636,"style":1498},[1147],[1109,1638,1585],{"className":1639,"style":1640},[1152,1153],"margin-right:0.0278em;",[1109,1642,1457],{"className":1643},[1508],[1109,1645,1170],{"className":1646,"style":1186},[1152,1153],[1109,1648,1457],{"className":1649},[1508],[1109,1651,1268],{"className":1652},[1152,1153],[1109,1654,1656],{"className":1655},[1515],"))",[1101,1658,1659,1662,1663,1730,1731,1759,1760,1730,1826,1854,1855,1883],{},[1104,1660,1661],{},"性质",":",[1109,1664,1666,1691],{"className":1665},[1112],[1109,1667,1669],{"className":1668},[1116],[1118,1670,1671],{"xmlns":1120},[1122,1672,1673,1688],{},[1125,1674,1675,1677,1679,1681,1683,1686],{},[1128,1676,1130],{},[1204,1678,1582],{},[1128,1680,1585],{},[1204,1682,1457],{"stretchy":1456},[1208,1684,1685],{},"1",[1204,1687,1462],{"stretchy":1456},[1132,1689,1690],{"encoding":1134},"f = O(1)",[1109,1692,1694,1712],{"className":1693,"ariaHidden":1139},[1138],[1109,1695,1697,1700,1703,1706,1709],{"className":1696},[1143],[1109,1698],{"className":1699,"style":1148},[1147],[1109,1701,1130],{"className":1702,"style":1154},[1152,1153],[1109,1704],{"className":1705,"style":1232},[1231],[1109,1707,1582],{"className":1708},[1236],[1109,1710],{"className":1711,"style":1232},[1231],[1109,1713,1715,1718,1721,1724,1727],{"className":1714},[1143],[1109,1716],{"className":1717,"style":1498},[1147],[1109,1719,1585],{"className":1720,"style":1640},[1152,1153],[1109,1722,1457],{"className":1723},[1508],[1109,1725,1685],{"className":1726},[1152],[1109,1728,1462],{"className":1729},[1515]," 表示 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的增长不超过 ",[1109,1856,1858,1871],{"className":1857},[1112],[1109,1859,1861],{"className":1860},[1116],[1118,1862,1863],{"xmlns":1120},[1122,1864,1865,1869],{},[1125,1866,1867],{},[1128,1868,1170],{},[1132,1870,1170],{"encoding":1134},[1109,1872,1874],{"className":1873,"ariaHidden":1139},[1138],[1109,1875,1877,1880],{"className":1876},[1143],[1109,1878],{"className":1879,"style":1182},[1147],[1109,1881,1170],{"className":1882,"style":1186},[1152,1153]," 的量级。",[1101,1885,1886,1889,1890,2019,2020,2170,2171,2319,2320,2983,2984,3141,3142,3421],{},[1104,1887,1888],{},"证明思路",":假设存在 ",[1109,1891,1893,1921],{"className":1892},[1112],[1109,1894,1896],{"className":1895},[1116],[1118,1897,1898],{"xmlns":1120},[1122,1899,1900,1918],{},[1125,1901,1902,1908,1911],{},[1264,1903,1904,1906],{},[1128,1905,1202],{},[1208,1907,1685],{},[1204,1909,1910],{"separator":1139},",",[1264,1912,1913,1915],{},[1128,1914,1202],{},[1208,1916,1917],{},"2",[1132,1919,1920],{"encoding":1134},"C_1, C_2",[1109,1922,1924],{"className":1923,"ariaHidden":1139},[1138],[1109,1925,1927,1931,1972,1976,1979],{"className":1926},[1143],[1109,1928],{"className":1929,"style":1930},[1147],"height:0.8778em;vertical-align:-0.1944em;",[1109,1932,1934,1937],{"className":1933},[1152],[1109,1935,1202],{"className":1936,"style":1227},[1152,1153],[1109,1938,1940],{"className":1939},[1293],[1109,1941,1943,1964],{"className":1942},[1297,1298],[1109,1944,1946,1961],{"className":1945},[1302],[1109,1947,1949],{"className":1948,"style":1307},[1306],[1109,1950,1952,1955],{"style":1951},"top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;",[1109,1953],{"className":1954,"style":1315},[1314],[1109,1956,1958],{"className":1957},[1319,1320,1321,1322],[1109,1959,1685],{"className":1960},[1152,1322],[1109,1962,1330],{"className":1963},[1329],[1109,1965,1967],{"className":1966},[1302],[1109,1968,1970],{"className":1969,"style":1337},[1306],[1109,1971],{},[1109,1973,1910],{"className":1974},[1975],"mpunct",[1109,1977],{"className":1978,"style":1540},[1231],[1109,1980,1982,1985],{"className":1981},[1152],[1109,1983,1202],{"className":1984,"style":1227},[1152,1153],[1109,1986,1988],{"className":1987},[1293],[1109,1989,1991,2011],{"className":1990},[1297,1298],[1109,1992,1994,2008],{"className":1993},[1302],[1109,1995,1997],{"className":1996,"style":1307},[1306],[1109,1998,1999,2002],{"style":1951},[1109,2000],{"className":2001,"style":1315},[1314],[1109,2003,2005],{"className":2004},[1319,1320,1321,1322],[1109,2006,1917],{"className":2007},[1152,1322],[1109,2009,1330],{"className":2010},[1329],[1109,2012,2014],{"className":2013},[1302],[1109,2015,2017],{"className":2016,"style":1337},[1306],[1109,2018],{}," 使得 ",[1109,2021,2023,2055],{"className":2022},[1112],[1109,2024,2026],{"className":2025},[1116],[1118,2027,2028],{"xmlns":1120},[1122,2029,2030,2052],{},[1125,2031,2032,2038,2040,2042,2044,2050],{},[1264,2033,2034,2036],{},[1128,2035,1130],{},[1208,2037,1685],{},[1204,2039,1582],{},[1128,2041,1585],{},[1204,2043,1457],{"stretchy":1456},[1264,2045,2046,2048],{},[1128,2047,1170],{},[1208,2049,1685],{},[1204,2051,1462],{"stretchy":1456},[1132,2053,2054],{"encoding":1134},"f_1 = O(g_1)",[1109,2056,2058,2114],{"className":2057,"ariaHidden":1139},[1138],[1109,2059,2061,2064,2105,2108,2111],{"className":2060},[1143],[1109,2062],{"className":2063,"style":1148},[1147],[1109,2065,2067,2070],{"className":2066},[1152],[1109,2068,1130],{"className":2069,"style":1154},[1152,1153],[1109,2071,2073],{"className":2072},[1293],[1109,2074,2076,2097],{"className":2075},[1297,1298],[1109,2077,2079,2094],{"className":2078},[1302],[1109,2080,2082],{"className":2081,"style":1307},[1306],[1109,2083,2085,2088],{"style":2084},"top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;",[1109,2086],{"className":2087,"style":1315},[1314],[1109,2089,2091],{"className":2090},[1319,1320,1321,1322],[1109,2092,1685],{"className":2093},[1152,1322],[1109,2095,1330],{"className":2096},[1329],[1109,2098,2100],{"className":2099},[1302],[1109,2101,2103],{"className":2102,"style":1337},[1306],[1109,2104],{},[1109,2106],{"className":2107,"style":1232},[1231],[1109,2109,1582],{"className":2110},[1236],[1109,2112],{"className":2113,"style":1232},[1231],[1109,2115,2117,2120,2123,2126,2167],{"className":2116},[1143],[1109,2118],{"className":2119,"style":1498},[1147],[1109,2121,1585],{"className":2122,"style":1640},[1152,1153],[1109,2124,1457],{"className":2125},[1508],[1109,2127,2129,2132],{"className":2128},[1152],[1109,2130,1170],{"className":2131,"style":1186},[1152,1153],[1109,2133,2135],{"className":2134},[1293],[1109,2136,2138,2159],{"className":2137},[1297,1298],[1109,2139,2141,2156],{"className":2140},[1302],[1109,2142,2144],{"className":2143,"style":1307},[1306],[1109,2145,2147,2150],{"style":2146},"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;",[1109,2148],{"className":2149,"style":1315},[1314],[1109,2151,2153],{"className":2152},[1319,1320,1321,1322],[1109,2154,1685],{"className":2155},[1152,1322],[1109,2157,1330],{"className":2158},[1329],[1109,2160,2162],{"className":2161},[1302],[1109,2163,2165],{"className":2164,"style":1337},[1306],[1109,2166],{},[1109,2168,1462],{"className":2169},[1515],"、",[1109,2172,2174,2206],{"className":2173},[1112],[1109,2175,2177],{"className":2176},[1116],[1118,2178,2179],{"xmlns":1120},[1122,2180,2181,2203],{},[1125,2182,2183,2189,2191,2193,2195,2201],{},[1264,2184,2185,2187],{},[1128,2186,1130],{},[1208,2188,1917],{},[1204,2190,1582],{},[1128,2192,1585],{},[1204,2194,1457],{"stretchy":1456},[1264,2196,2197,2199],{},[1128,2198,1170],{},[1208,2200,1917],{},[1204,2202,1462],{"stretchy":1456},[1132,2204,2205],{"encoding":1134},"f_2 = O(g_2)",[1109,2207,2209,2264],{"className":2208,"ariaHidden":1139},[1138],[1109,2210,2212,2215,2255,2258,2261],{"className":2211},[1143],[1109,2213],{"className":2214,"style":1148},[1147],[1109,2216,2218,2221],{"className":2217},[1152],[1109,2219,1130],{"className":2220,"style":1154},[1152,1153],[1109,2222,2224],{"className":2223},[1293],[1109,2225,2227,2247],{"className":2226},[1297,1298],[1109,2228,2230,2244],{"className":2229},[1302],[1109,2231,2233],{"className":2232,"style":1307},[1306],[1109,2234,2235,2238],{"style":2084},[1109,2236],{"className":2237,"style":1315},[1314],[1109,2239,2241],{"className":2240},[1319,1320,1321,1322],[1109,2242,1917],{"className":2243},[1152,1322],[1109,2245,1330],{"className":2246},[1329],[1109,2248,2250],{"className":2249},[1302],[1109,2251,2253],{"className":2252,"style":1337},[1306],[1109,2254],{},[1109,2256],{"className":2257,"style":1232},[1231],[1109,2259,1582],{"className":2260},[1236],[1109,2262],{"className":2263,"style":1232},[1231],[1109,2265,2267,2270,2273,2276,2316],{"className":2266},[1143],[1109,2268],{"className":2269,"style":1498},[1147],[1109,2271,1585],{"className":2272,"style":1640},[1152,1153],[1109,2274,1457],{"className":2275},[1508],[1109,2277,2279,2282],{"className":2278},[1152],[1109,2280,1170],{"className":2281,"style":1186},[1152,1153],[1109,2283,2285],{"className":2284},[1293],[1109,2286,2288,2308],{"className":2287},[1297,1298],[1109,2289,2291,2305],{"className":2290},[1302],[1109,2292,2294],{"className":2293,"style":1307},[1306],[1109,2295,2296,2299],{"style":2146},[1109,2297],{"className":2298,"style":1315},[1314],[1109,2300,2302],{"className":2301},[1319,1320,1321,1322],[1109,2303,1917],{"className":2304},[1152,1322],[1109,2306,1330],{"className":2307},[1329],[1109,2309,2311],{"className":2310},[1302],[1109,2312,2314],{"className":2313,"style":1337},[1306],[1109,2315],{},[1109,2317,1462],{"className":2318},[1515],",则对和式有 ",[1109,2321,2323,2442],{"className":2322},[1112],[1109,2324,2326],{"className":2325},[1116],[1118,2327,2328],{"xmlns":1120},[1122,2329,2330,2439],{},[1125,2331,2332,2334,2340,2343,2349,2351,2353,2359,2361,2367,2369,2371,2377,2379,2385,2387,2389,2391,2397,2399,2405,2407,2410,2413,2415,2417,2423,2425,2427,2429,2435,2437],{},[1128,2333,1451],{"mathvariant":1450},[1264,2335,2336,2338],{},[1128,2337,1130],{},[1208,2339,1685],{},[1204,2341,2342],{},"+",[1264,2344,2345,2347],{},[1128,2346,1130],{},[1208,2348,1917],{},[1128,2350,1451],{"mathvariant":1450},[1204,2352,1467],{},[1264,2354,2355,2357],{},[1128,2356,1202],{},[1208,2358,1685],{},[1128,2360,1451],{"mathvariant":1450},[1264,2362,2363,2365],{},[1128,2364,1170],{},[1208,2366,1685],{},[1128,2368,1451],{"mathvariant":1450},[1204,2370,2342],{},[1264,2372,2373,2375],{},[1128,2374,1202],{},[1208,2376,1917],{},[1128,2378,1451],{"mathvariant":1450},[1264,2380,2381,2383],{},[1128,2382,1170],{},[1208,2384,1917],{},[1128,2386,1451],{"mathvariant":1450},[1204,2388,1467],{},[1204,2390,1457],{"stretchy":1456},[1264,2392,2393,2395],{},[1128,2394,1202],{},[1208,2396,1685],{},[1204,2398,2342],{},[1264,2400,2401,2403],{},[1128,2402,1202],{},[1208,2404,1917],{},[1204,2406,1462],{"stretchy":1456},[1128,2408,2409],{},"max",[1204,2411,2412],{},"⁡",[1204,2414,1457],{"stretchy":1456},[1128,2416,1451],{"mathvariant":1450},[1264,2418,2419,2421],{},[1128,2420,1170],{},[1208,2422,1685],{},[1128,2424,1451],{"mathvariant":1450},[1204,2426,1910],{"separator":1139},[1128,2428,1451],{"mathvariant":1450},[1264,2430,2431,2433],{},[1128,2432,1170],{},[1208,2434,1917],{},[1128,2436,1451],{"mathvariant":1450},[1204,2438,1462],{"stretchy":1456},[1132,2440,2441],{"encoding":1134},"|f_1+f_2| \\le C_1|g_1| + C_2|g_2| \\le (C_1+C_2)\\max(|g_1|,|g_2|)",[1109,2443,2445,2505,2563,2664,2765,2823],{"className":2444,"ariaHidden":1139},[1138],[1109,2446,2448,2451,2454,2494,2498,2502],{"className":2447},[1143],[1109,2449],{"className":2450,"style":1498},[1147],[1109,2452,1451],{"className":2453},[1152],[1109,2455,2457,2460],{"className":2456},[1152],[1109,2458,1130],{"className":2459,"style":1154},[1152,1153],[1109,2461,2463],{"className":2462},[1293],[1109,2464,2466,2486],{"className":2465},[1297,1298],[1109,2467,2469,2483],{"className":2468},[1302],[1109,2470,2472],{"className":2471,"style":1307},[1306],[1109,2473,2474,2477],{"style":2084},[1109,2475],{"className":2476,"style":1315},[1314],[1109,2478,2480],{"className":2479},[1319,1320,1321,1322],[1109,2481,1685],{"className":2482},[1152,1322],[1109,2484,1330],{"className":2485},[1329],[1109,2487,2489],{"className":2488},[1302],[1109,2490,2492],{"className":2491,"style":1337},[1306],[1109,2493],{},[1109,2495],{"className":2496,"style":2497},[1231],"margin-right:0.2222em;",[1109,2499,2342],{"className":2500},[2501],"mbin",[1109,2503],{"className":2504,"style":2497},[1231],[1109,2506,2508,2511,2551,2554,2557,2560],{"className":2507},[1143],[1109,2509],{"className":2510,"style":1498},[1147],[1109,2512,2514,2517],{"className":2513},[1152],[1109,2515,1130],{"className":2516,"style":1154},[1152,1153],[1109,2518,2520],{"className":2519},[1293],[1109,2521,2523,2543],{"className":2522},[1297,1298],[1109,2524,2526,2540],{"className":2525},[1302],[1109,2527,2529],{"className":2528,"style":1307},[1306],[1109,2530,2531,2534],{"style":2084},[1109,2532],{"className":2533,"style":1315},[1314],[1109,2535,2537],{"className":2536},[1319,1320,1321,1322],[1109,2538,1917],{"className":2539},[1152,1322],[1109,2541,1330],{"className":2542},[1329],[1109,2544,2546],{"className":2545},[1302],[1109,2547,2549],{"className":2548,"style":1337},[1306],[1109,2550],{},[1109,2552,1451],{"className":2553},[1152],[1109,2555],{"className":2556,"style":1232},[1231],[1109,2558,1467],{"className":2559},[1236],[1109,2561],{"className":2562,"style":1232},[1231],[1109,2564,2566,2569,2609,2612,2652,2655,2658,2661],{"className":2565},[1143],[1109,2567],{"className":2568,"style":1498},[1147],[1109,2570,2572,2575],{"className":2571},[1152],[1109,2573,1202],{"className":2574,"style":1227},[1152,1153],[1109,2576,2578],{"className":2577},[1293],[1109,2579,2581,2601],{"className":2580},[1297,1298],[1109,2582,2584,2598],{"className":2583},[1302],[1109,2585,2587],{"className":2586,"style":1307},[1306],[1109,2588,2589,2592],{"style":1951},[1109,2590],{"className":2591,"style":1315},[1314],[1109,2593,2595],{"className":2594},[1319,1320,1321,1322],[1109,2596,1685],{"className":2597},[1152,1322],[1109,2599,1330],{"className":2600},[1329],[1109,2602,2604],{"className":2603},[1302],[1109,2605,2607],{"className":2606,"style":1337},[1306],[1109,2608],{},[1109,2610,1451],{"className":2611},[1152],[1109,2613,2615,2618],{"className":2614},[1152],[1109,2616,1170],{"className":2617,"style":1186},[1152,1153],[1109,2619,2621],{"className":2620},[1293],[1109,2622,2624,2644],{"className":2623},[1297,1298],[1109,2625,2627,2641],{"className":2626},[1302],[1109,2628,2630],{"className":2629,"style":1307},[1306],[1109,2631,2632,2635],{"style":2146},[1109,2633],{"className":2634,"style":1315},[1314],[1109,2636,2638],{"className":2637},[1319,1320,1321,1322],[1109,2639,1685],{"className":2640},[1152,1322],[1109,2642,1330],{"className":2643},[1329],[1109,2645,2647],{"className":2646},[1302],[1109,2648,2650],{"className":2649,"style":1337},[1306],[1109,2651],{},[1109,2653,1451],{"className":2654},[1152],[1109,2656],{"className":2657,"style":2497},[1231],[1109,2659,2342],{"className":2660},[2501],[1109,2662],{"className":2663,"style":2497},[1231],[1109,2665,2667,2670,2710,2713,2753,2756,2759,2762],{"className":2666},[1143],[1109,2668],{"className":2669,"style":1498},[1147],[1109,2671,2673,2676],{"className":2672},[1152],[1109,2674,1202],{"className":2675,"style":1227},[1152,1153],[1109,2677,2679],{"className":2678},[1293],[1109,2680,2682,2702],{"className":2681},[1297,1298],[1109,2683,2685,2699],{"className":2684},[1302],[1109,2686,2688],{"className":2687,"style":1307},[1306],[1109,2689,2690,2693],{"style":1951},[1109,2691],{"className":2692,"style":1315},[1314],[1109,2694,2696],{"className":2695},[1319,1320,1321,1322],[1109,2697,1917],{"className":2698},[1152,1322],[1109,2700,1330],{"className":2701},[1329],[1109,2703,2705],{"className":2704},[1302],[1109,2706,2708],{"className":2707,"style":1337},[1306],[1109,2709],{},[1109,2711,1451],{"className":2712},[1152],[1109,2714,2716,2719],{"className":2715},[1152],[1109,2717,1170],{"className":2718,"style":1186},[1152,1153],[1109,2720,2722],{"className":2721},[1293],[1109,2723,2725,2745],{"className":2724},[1297,1298],[1109,2726,2728,2742],{"className":2727},[1302],[1109,2729,2731],{"className":2730,"style":1307},[1306],[1109,2732,2733,2736],{"style":2146},[1109,2734],{"className":2735,"style":1315},[1314],[1109,2737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",[1109,2985,2987,3017],{"className":2986},[1112],[1109,2988,2990],{"className":2989},[1116],[1118,2991,2992],{"xmlns":1120},[1122,2993,2994,3014],{},[1125,2995,2996,2998,3000,3006,3008],{},[1128,2997,1202],{},[1204,2999,1582],{},[1264,3001,3002,3004],{},[1128,3003,1202],{},[1208,3005,1685],{},[1204,3007,2342],{},[1264,3009,3010,3012],{},[1128,3011,1202],{},[1208,3013,1917],{},[1132,3015,3016],{"encoding":1134},"C = C_1+C_2",[1109,3018,3020,3039,3095],{"className":3019,"ariaHidden":1139},[1138],[1109,3021,3023,3027,3030,3033,3036],{"className":3022},[1143],[1109,3024],{"className":3025,"style":3026},[1147],"height:0.6833em;",[1109,3028,1202],{"className":3029,"style":1227},[1152,1153],[1109,3031],{"className":3032,"style":1232},[1231],[1109,3034,1582],{"className":3035},[1236],[1109,3037],{"className":3038,"style":1232},[1231],[1109,3040,3042,3046,3086,3089,3092],{"className":3041},[1143],[1109,3043],{"className":3044,"style":3045},[1147],"height:0.8333em;vertical-align:-0.15em;",[1109,3047,3049,3052],{"className":3048},[1152],[1109,3050,1202],{"className":3051,"style":1227},[1152,1153],[1109,3053,3055],{"className":3054},[1293],[1109,3056,3058,3078],{"className":3057},[1297,1298],[1109,3059,3061,3075],{"className":3060},[1302],[1109,3062,3064],{"className":3063,"style":1307},[1306],[1109,3065,3066,3069],{"style":1951},[1109,3067],{"className":3068,"style":1315},[1314],[1109,3070,3072],{"className":3071},[1319,1320,1321,1322],[1109,3073,1685],{"className":3074},[1152,1322],[1109,3076,1330],{"className":3077},[1329],[1109,3079,3081],{"className":3080},[1302],[1109,3082,3084],{"className":3083,"style":1337},[1306],[1109,3085],{},[1109,3087],{"className":3088,"style":2497},[1231],[1109,3090,2342],{"className":3091},[2501],[1109,3093],{"className":3094,"style":2497},[1231],[1109,3096,3098,3101],{"className":3097},[1143],[1109,3099],{"className":3100,"style":3045},[1147],[1109,3102,3104,3107],{"className":3103},[1152],[1109,3105,1202],{"className":3106,"style":1227},[1152,1153],[1109,3108,3110],{"className":3109},[1293],[1109,3111,3113,3133],{"className":3112},[1297,1298],[1109,3114,3116,3130],{"className":3115},[1302],[1109,3117,3119],{"className":3118,"style":1307},[1306],[1109,3120,3121,3124],{"style":1951},[1109,3122],{"className":3123,"style":1315},[1314],[1109,3125,3127],{"className":3126},[1319,1320,1321,1322],[1109,3128,1917],{"className":3129},[1152,1322],[1109,3131,1330],{"className":3132},[1329],[1109,3134,3136],{"className":3135},[1302],[1109,3137,3139],{"className":3138,"style":1337},[1306],[1109,3140],{}," 即得 ",[1109,3143,3145,3201],{"className":3144},[1112],[1109,3146,3148],{"className":3147},[1116],[1118,3149,3150],{"xmlns":1120},[1122,3151,3152,3198],{},[1125,3153,3154,3160,3162,3168,3170,3172,3174,3176,3178,3180,3186,3188,3194,3196],{},[1264,3155,3156,3158],{},[1128,3157,1130],{},[1208,3159,1685],{},[1204,3161,2342],{},[1264,3163,3164,3166],{},[1128,3165,1130],{},[1208,3167,1917],{},[1204,3169,1582],{},[1128,3171,1585],{},[1204,3173,1457],{"stretchy":1456},[1128,3175,2409],{},[1204,3177,2412],{},[1204,3179,1457],{"stretchy":1456},[1264,3181,3182,3184],{},[1128,3183,1170],{},[1208,3185,1685],{},[1204,3187,1910],{"separator":1139},[1264,3189,3190,3192],{},[1128,3191,1170],{},[1208,3193,1917],{},[1204,3195,1462],{"stretchy":1456},[1204,3197,1462],{"stretchy":1456},[1132,3199,3200],{"encoding":1134},"f_1+f_2 = O(\\max(g_1,g_2))",[1109,3202,3204,3259,3314],{"className":3203,"ariaHidden":1139},[1138],[1109,3205,3207,3210,3250,3253,3256],{"className":3206},[1143],[1109,3208],{"className":3209,"style":1148},[1147],[1109,3211,3213,3216],{"className":3212},[1152],[1109,3214,1130],{"className":3215,"style":1154},[1152,1153],[1109,3217,3219],{"className":3218},[1293],[1109,3220,3222,3242],{"className":3221},[1297,1298],[1109,3223,3225,3239],{"className":3224},[1302],[1109,3226,3228],{"className":3227,"style":1307},[1306],[1109,3229,3230,3233],{"style":2084},[1109,3231],{"className":3232,"style":1315},[1314],[1109,3234,3236],{"className":3235},[1319,1320,1321,1322],[1109,3237,1685],{"className":3238},[1152,1322],[1109,3240,1330],{"className":3241},[1329],[1109,3243,3245],{"className":3244},[1302],[1109,3246,3248],{"className":3247,"style":1337},[1306],[1109,3249],{},[1109,3251],{"className":3252,"style":2497},[1231],[1109,3254,2342],{"className":3255},[2501],[1109,3257],{"className":3258,"style":2497},[1231],[1109,3260,3262,3265,3305,3308,3311],{"className":3261},[1143],[1109,3263],{"className":3264,"style":1148},[1147],[1109,3266,3268,3271],{"className":3267},[1152],[1109,3269,1130],{"className":3270,"style":1154},[1152,1153],[1109,3272,3274],{"className":3273},[1293],[1109,3275,3277,3297],{"className":3276},[1297,1298],[1109,3278,3280,3294],{"className":3279},[1302],[1109,3281,3283],{"className":3282,"style":1307},[1306],[1109,3284,3285,3288],{"style":2084},[1109,3286],{"className":3287,"style":1315},[1314],[1109,3289,3291],{"className":3290},[1319,1320,1321,1322],[1109,3292,1917],{"className":3293},[1152,1322],[1109,3295,1330],{"className":3296},[1329],[1109,3298,3300],{"className":3299},[1302],[1109,3301,3303],{"className":3302,"style":1337},[1306],[1109,3304],{},[1109,3306],{"className":3307,"style":1232},[1231],[1109,3309,1582],{"className":3310},[1236],[1109,3312],{"className":3313,"style":1232},[1231],[1109,3315,3317,3320,3323,3326,3329,3332,3372,3375,3378,3418],{"className":3316},[1143],[1109,3318],{"className":3319,"style":1498},[1147],[1109,3321,1585],{"className":3322,"style":1640},[1152,1153],[1109,3324,1457],{"className":3325},[1508],[1109,3327,2409],{"className":3328},[2878],[1109,3330,1457],{"className":3331},[1508],[1109,3333,3335,3338],{"className":3334},[1152],[1109,3336,1170],{"className":3337,"style":1186},[1152,1153],[1109,3339,3341],{"className":3340},[1293],[1109,3342,3344,3364],{"className":3343},[1297,1298],[1109,3345,3347,3361],{"className":3346},[1302],[1109,3348,3350],{"className":3349,"style":1307},[1306],[1109,3351,3352,3355],{"style":2146},[1109,3353],{"className":3354,"style":1315},[1314],[1109,3356,3358],{"className":3357},[1319,1320,1321,1322],[1109,3359,1685],{"className":3360},[1152,1322],[1109,3362,1330],{"className":3363},[1329],[1109,3365,3367],{"className":3366},[1302],[1109,3368,3370],{"className":3369,"style":1337},[1306],[1109,3371],{},[1109,3373,1910],{"className":3374},[1975],[1109,3376],{"className":3377,"style":1540},[1231],[1109,3379,3381,3384],{"className":3380},[1152],[1109,3382,1170],{"className":3383,"style":1186},[1152,1153],[1109,3385,3387],{"className":3386},[1293],[1109,3388,3390,3410],{"className":3389},[1297,1298],[1109,3391,3393,3407],{"className":3392},[1302],[1109,3394,3396],{"className":3395,"style":1307},[1306],[1109,3397,3398,3401],{"style":2146},[1109,3399],{"className":3400,"style":1315},[1314],[1109,3402,3404],{"className":3403},[1319,1320,1321,1322],[1109,3405,1917],{"className":3406},[1152,1322],[1109,3408,1330],{"className":3409},[1329],[1109,3411,3413],{"className":3412},[1302],[1109,3414,3416],{"className":3415,"style":1337},[1306],[1109,3417],{},[1109,3419,1656],{"className":3420},[1515],",得证。",[1101,3423,3424,1662,3427,1910,3736,4024],{},[1104,3425,3426],{},"一些简单关系",[1109,3428,3430,3498],{"className":3429},[1112],[1109,3431,3433],{"className":3432},[1116],[1118,3434,3435],{"xmlns":1120},[1122,3436,3437,3495],{},[1125,3438,3439,3441,3443,3449,3451,3453,3455,3457,3463,3465,3467,3469,3471,3473,3475,3477,3483,3485,3491,3493],{},[1128,3440,1585],{},[1204,3442,1457],{"stretchy":1456},[1264,3444,3445,3447],{},[1128,3446,1170],{},[1208,3448,1685],{},[1204,3450,1462],{"stretchy":1456},[1204,3452,2342],{},[1128,3454,1585],{},[1204,3456,1457],{"stretchy":1456},[1264,3458,3459,3461],{},[1128,3460,1170],{},[1208,3462,1917],{},[1204,3464,1462],{"stretchy":1456},[1204,3466,1582],{},[1128,3468,1585],{},[1204,3470,1457],{"stretchy":1456},[1128,3472,2409],{},[1204,3474,2412],{},[1204,3476,1457],{"stretchy":1456},[1264,3478,3479,3481],{},[1128,3480,1170],{},[1208,3482,1685],{},[1204,3484,1910],{"separator":1139},[1264,3486,3487,3489],{},[1128,3488,1170],{},[1208,3490,1917],{},[1204,3492,1462],{"stretchy":1456},[1204,3494,1462],{"stretchy":1456},[1132,3496,3497],{"encoding":1134},"O(g_1) + O(g_2) = O(\\max(g_1, g_2))",[1109,3499,3501,3565,3629],{"className":3500,"ariaHidden":1139},[1138],[1109,3502,3504,3507,3510,3513,3553,3556,3559,3562],{"className":3503},[1143],[1109,3505],{"className":3506,"style":1498},[1147],[1109,3508,1585],{"className":3509,"style":1640},[1152,1153],[1109,3511,1457],{"className":3512},[1508],[1109,3514,3516,3519],{"className":3515},[1152],[1109,3517,1170],{"className":3518,"style":1186},[1152,1153],[1109,3520,3522],{"className":3521},[1293],[1109,3523,3525,3545],{"className":3524},[1297,1298],[1109,3526,3528,3542],{"className":3527},[1302],[1109,3529,3531],{"className":3530,"style":1307},[1306],[1109,3532,3533,3536],{"style":2146},[1109,3534],{"className":3535,"style":1315},[1314],[1109,3537,3539],{"className":3538},[1319,1320,1321,1322],[1109,3540,1685],{"className":3541},[1152,1322],[1109,3543,1330],{"className":3544},[1329],[1109,3546,3548],{"className":3547},[1302],[1109,3549,3551],{"className":3550,"style":1337},[1306],[1109,3552],{},[1109,3554,1462],{"className":3555},[1515],[1109,3557],{"className":3558,"style":2497},[1231],[1109,3560,2342],{"className":3561},[2501],[1109,3563],{"className":3564,"style":2497},[1231],[1109,3566,3568,3571,3574,3577,3617,3620,3623,3626],{"className":3567},[1143],[1109,3569],{"className":3570,"style":1498},[1147],[1109,3572,1585],{"className":3573,"style":1640},[1152,1153],[1109,3575,1457],{"className":3576},[1508],[1109,3578,3580,3583],{"className":3579},[1152],[1109,3581,1170],{"className":3582,"style":1186},[1152,1153],[1109,3584,3586],{"className":3585},[1293],[1109,3587,3589,3609],{"className":3588},[1297,1298],[1109,3590,3592,3606],{"className":3591},[1302],[1109,3593,3595],{"className":3594,"style":1307},[1306],[1109,3596,3597,3600],{"style":2146},[1109,3598],{"className":3599,"style":1315},[1314],[1109,3601,3603],{"className":3602},[1319,1320,1321,1322],[1109,3604,1917],{"className":3605},[1152,1322],[1109,3607,1330],{"className":3608},[1329],[1109,3610,3612],{"className":3611},[1302],[1109,3613,3615],{"className":3614,"style":1337},[1306],[1109,3616],{},[1109,3618,1462],{"className":3619},[1515],[1109,3621],{"className":3622,"style":1232},[1231],[1109,3624,1582],{"className":3625},[1236],[1109,3627],{"className":3628,"style":1232},[1231],[1109,3630,3632,3635,3638,3641,3644,3647,3687,3690,3693,3733],{"className":3631},[1143],[1109,3633],{"className":3634,"style":1498},[1147],[1109,3636,1585],{"className":3637,"style":1640},[1152,1153],[1109,3639,1457],{"className":3640},[1508],[1109,3642,2409],{"className":3643},[2878],[1109,3645,1457],{"className":3646},[1508],[1109,3648,3650,3653],{"className":3649},[1152],[1109,3651,1170],{"className":3652,"style":1186},[1152,1153],[1109,3654,3656],{"className":3655},[1293],[1109,3657,3659,3679],{"className":3658},[1297,1298],[1109,3660,3662,3676],{"className":3661},[1302],[1109,3663,3665],{"className":3664,"style":1307},[1306],[1109,3666,3667,3670],{"style":2146},[1109,3668],{"className":3669,"style":1315},[1314],[1109,3671,3673],{"className":3672},[1319,1320,1321,1322],[1109,3674,1685],{"className":3675},[1152,1322],[1109,3677,1330],{"className":3678},[1329],[1109,3680,3682],{"className":3681},[1302],[1109,3683,3685],{"className":3684,"style":1337},[1306],[1109,3686],{},[1109,3688,1910],{"className":3689},[1975],[1109,3691],{"className":3692,"style":1540},[1231],[1109,3694,3696,3699],{"className":3695},[1152],[1109,3697,1170],{"className":3698,"style":1186},[1152,1153],[1109,3700,3702],{"className":3701},[1293],[1109,3703,3705,3725],{"className":3704},[1297,1298],[1109,3706,3708,3722],{"className":3707},[1302],[1109,3709,3711],{"className":3710,"style":1307},[1306],[1109,3712,3713,3716],{"style":2146},[1109,3714],{"className":3715,"style":1315},[1314],[1109,3717,3719],{"className":3718},[1319,1320,1321,1322],[1109,3720,1917],{"className":3721},[1152,1322],[1109,3723,1330],{"className":3724},[1329],[1109,3726,3728],{"className":3727},[1302],[1109,3729,3731],{"className":3730,"style":1337},[1306],[1109,3732],{},[1109,3734,1656],{"className":3735},[1515],[1109,3737,3739,3798],{"className":3738},[1112],[1109,3740,3742],{"className":3741},[1116],[1118,3743,3744],{"xmlns":1120},[1122,3745,3746,3795],{},[1125,3747,3748,3750,3752,3758,3760,3763,3765,3767,3773,3775,3777,3779,3781,3787,3793],{},[1128,3749,1585],{},[1204,3751,1457],{"stretchy":1456},[1264,3753,3754,3756],{},[1128,3755,1170],{},[1208,3757,1685],{},[1204,3759,1462],{"stretchy":1456},[1204,3761,3762],{},"⋅",[1128,3764,1585],{},[1204,3766,1457],{"stretchy":1456},[1264,3768,3769,3771],{},[1128,3770,1170],{},[1208,3772,1917],{},[1204,3774,1462],{"stretchy":1456},[1204,3776,1582],{},[1128,3778,1585],{},[1204,3780,1457],{"stretchy":1456},[1264,3782,3783,3785],{},[1128,3784,1170],{},[1208,3786,1685],{},[1264,3788,3789,3791],{},[1128,3790,1170],{},[1208,3792,1917],{},[1204,3794,1462],{"stretchy":1456},[1132,3796,3797],{"encoding":1134},"O(g_1)\\cdot O(g_2) = O(g_1 g_2)",[1109,3799,3801,3865,3929],{"className":3800,"ariaHidden":1139},[1138],[1109,3802,3804,3807,3810,3813,3853,3856,3859,3862],{"className":3803},[1143],[1109,3805],{"className":3806,"style":1498},[1147],[1109,3808,1585],{"className":3809,"style":1640},[1152,1153],[1109,3811,1457],{"className":3812},[1508],[1109,3814,3816,3819],{"className":3815},[1152],[1109,3817,1170],{"className":3818,"style":1186},[1152,1153],[1109,3820,3822],{"className":3821},[1293],[1109,3823,3825,3845],{"className":3824},[1297,1298],[1109,3826,3828,3842],{"className":3827},[1302],[1109,3829,3831],{"className":3830,"style":1307},[1306],[1109,3832,3833,3836],{"style":2146},[1109,3834],{"className":3835,"style":1315},[1314],[1109,3837,3839],{"className":3838},[1319,1320,1321,1322],[1109,3840,1685],{"className":3841},[1152,1322],[1109,3843,1330],{"className":3844},[1329],[1109,3846,3848],{"className":3847},[1302],[1109,3849,3851],{"className":3850,"style":1337},[1306],[1109,3852],{},[1109,3854,1462],{"className":3855},[1515],[1109,3857],{"className":3858,"style":2497},[1231],[1109,3860,3762],{"className":3861},[2501],[1109,3863],{"className":3864,"style":2497},[1231],[1109,3866,3868,3871,3874,3877,3917,3920,3923,3926],{"className":3867},[1143],[1109,3869],{"className":3870,"style":1498},[1147],[1109,3872,1585],{"className":3873,"style":1640},[1152,1153],[1109,3875,1457],{"className":3876},[1508],[1109,3878,3880,3883],{"className":3879},[1152],[1109,3881,1170],{"className":3882,"style":1186},[1152,1153],[1109,3884,3886],{"className":3885},[1293],[1109,3887,3889,3909],{"className":3888},[1297,1298],[1109,3890,3892,3906],{"className":3891},[1302],[1109,3893,3895],{"className":3894,"style":1307},[1306],[1109,3896,3897,3900],{"style":2146},[1109,3898],{"className":3899,"style":1315},[1314],[1109,3901,3903],{"className":3902},[1319,1320,1321,1322],[1109,3904,1917],{"className":3905},[1152,1322],[1109,3907,1330],{"className":3908},[1329],[1109,3910,3912],{"className":3911},[1302],[1109,3913,3915],{"className":3914,"style":1337},[1306],[1109,3916],{},[1109,3918,1462],{"className":3919},[1515],[1109,3921],{"className":3922,"style":1232},[1231],[1109,3924,1582],{"className":3925},[1236],[1109,3927],{"className":3928,"style":1232},[1231],[1109,3930,3932,3935,3938,3941,3981,4021],{"className":3931},[1143],[1109,3933],{"className":3934,"style":1498},[1147],[1109,3936,1585],{"className":3937,"style":1640},[1152,1153],[1109,3939,1457],{"className":3940},[1508],[1109,3942,3944,3947],{"className":3943},[1152],[1109,3945,1170],{"className":3946,"style":1186},[1152,1153],[1109,3948,3950],{"className":3949},[1293],[1109,3951,3953,3973],{"className":3952},[1297,1298],[1109,3954,3956,3970],{"className":3955},[1302],[1109,3957,3959],{"className":3958,"style":1307},[1306],[1109,3960,3961,3964],{"style":2146},[1109,3962],{"className":3963,"style":1315},[1314],[1109,3965,3967],{"className":3966},[1319,1320,1321,1322],[1109,3968,1685],{"className":3969},[1152,1322],[1109,3971,1330],{"className":3972},[1329],[1109,3974,3976],{"className":3975},[1302],[1109,3977,3979],{"className":3978,"style":1337},[1306],[1109,3980],{},[1109,3982,3984,3987],{"className":3983},[1152],[1109,3985,1170],{"className":3986,"style":1186},[1152,1153],[1109,3988,3990],{"className":3989},[1293],[1109,3991,3993,4013],{"className":3992},[1297,1298],[1109,3994,3996,4010],{"className":3995},[1302],[1109,3997,3999],{"className":3998,"style":1307},[1306],[1109,4000,4001,4004],{"style":2146},[1109,4002],{"className":4003,"style":1315},[1314],[1109,4005,4007],{"className":4006},[1319,1320,1321,1322],[1109,4008,1917],{"className":4009},[1152,1322],[1109,4011,1330],{"className":4012},[1329],[1109,4014,4016],{"className":4015},[1302],[1109,4017,4019],{"className":4018,"style":1337},[1306],[1109,4020],{},[1109,4022,1462],{"className":4023},[1515],"。",[1096,4026,4028],{"id":4027},"小-o-符号","小 o 符号",[1101,4030,4031,4033,4034,4087,4088,1340,4157,4248,4249,4360,4361],{},[1104,4032,1106],{},":如果对每一个常数 ",[1109,4035,4037,4056],{"className":4036},[1112],[1109,4038,4040],{"className":4039},[1116],[1118,4041,4042],{"xmlns":1120},[1122,4043,4044,4053],{},[1125,4045,4046,4049,4051],{},[1128,4047,4048],{},"ϵ",[1204,4050,1206],{},[1208,4052,1210],{},[1132,4054,4055],{"encoding":1134},"\\epsilon > 0",[1109,4057,4059,4078],{"className":4058,"ariaHidden":1139},[1138],[1109,4060,4062,4066,4069,4072,4075],{"className":4061},[1143],[1109,4063],{"className":4064,"style":4065},[1147],"height:0.5782em;vertical-align:-0.0391em;",[1109,4067,4048],{"className":4068},[1152,1153],[1109,4070],{"className":4071,"style":1232},[1231],[1109,4073,1206],{"className":4074},[1236],[1109,4076],{"className":4077,"style":1232},[1231],[1109,4079,4081,4084],{"className":4080},[1143],[1109,4082],{"className":4083,"style":1246},[1147],[1109,4085,1210],{"className":4086},[1152]," 都存在 ",[1109,4089,4091,4108],{"className":4090},[1112],[1109,4092,4094],{"className":4093},[1116],[1118,4095,4096],{"xmlns":1120},[1122,4097,4098,4106],{},[1125,4099,4100],{},[1264,4101,4102,4104],{},[1128,4103,1268],{},[1208,4105,1210],{},[1132,4107,1273],{"encoding":1134},[1109,4109,4111],{"className":4110,"ariaHidden":1139},[1138],[1109,4112,4114,4117],{"className":4113},[1143],[1109,4115],{"className":4116,"style":1283},[1147],[1109,4118,4120,4123],{"className":4119},[1152],[1109,4121,1268],{"className":4122},[1152,1153],[1109,4124,4126],{"className":4125},[1293],[1109,4127,4129,4149],{"className":4128},[1297,1298],[1109,4130,4132,4146],{"className":4131},[1302],[1109,4133,4135],{"className":4134,"style":1307},[1306],[1109,4136,4137,4140],{"style":1310},[1109,4138],{"className":4139,"style":1315},[1314],[1109,4141,4143],{"className":4142},[1319,1320,1321,1322],[1109,4144,1210],{"className":4145},[1152,1322],[1109,4147,1330],{"className":4148},[1329],[1109,4150,4152],{"className":4151},[1302],[1109,4153,4155],{"className":4154,"style":1337},[1306],[1109,4156],{},[1109,4158,4160,4181],{"className":4159},[1112],[1109,4161,4163],{"className":4162},[1116],[1118,4164,4165],{"xmlns":1120},[1122,4166,4167,4179],{},[1125,4168,4169,4171,4173],{},[1128,4170,1268],{},[1204,4172,1357],{},[1264,4174,4175,4177],{},[1128,4176,1268],{},[1208,4178,1210],{},[1132,4180,1366],{"encoding":1134},[1109,4182,4184,4202],{"className":4183,"ariaHidden":1139},[1138],[1109,4185,4187,4190,4193,4196,4199],{"className":4186},[1143],[1109,4188],{"className":4189,"style":1376},[1147],[1109,4191,1268],{"className":4192},[1152,1153],[1109,4194],{"className":4195,"style":1232},[1231],[1109,4197,1357],{"className":4198},[1236],[1109,4200],{"className":4201,"style":1232},[1231],[1109,4203,4205,4208],{"className":4204},[1143],[1109,4206],{"className":4207,"style":1283},[1147],[1109,4209,4211,4214],{"className":4210},[1152],[1109,4212,1268],{"className":4213},[1152,1153],[1109,4215,4217],{"className":4216},[1293],[1109,4218,4220,4240],{"className":4219},[1297,1298],[1109,4221,4223,4237],{"className":4222},[1302],[1109,4224,4226],{"className":4225,"style":1307},[1306],[1109,4227,4228,4231],{"style":1310},[1109,4229],{"className":4230,"style":1315},[1314],[1109,4232,4234],{"className":4233},[1319,1320,1321,1322],[1109,4235,1210],{"className":4236},[1152,1322],[1109,4238,1330],{"className":4239},[1329],[1109,4241,4243],{"className":4242},[1302],[1109,4244,4246],{"className":4245,"style":1337},[1306],[1109,4247],{}," 时，",[1109,4250,4252,4294],{"className":4251},[1112],[1109,4253,4255],{"className":4254},[1116],[1118,4256,4257],{"xmlns":1120},[1122,4258,4259,4291],{},[1125,4260,4261,4263,4265,4267,4269,4271,4273,4275,4277,4279,4281,4283,4285,4287,4289],{},[1128,4262,1451],{"mathvariant":1450},[1128,4264,1130],{},[1204,4266,1457],{"stretchy":1456},[1128,4268,1268],{},[1204,4270,1462],{"stretchy":1456},[1128,4272,1451],{"mathvariant":1450},[1204,4274,1467],{},[1128,4276,4048],{},[1471,4278,1473],{},[1128,4280,1451],{"mathvariant":1450},[1128,4282,1170],{},[1204,4284,1457],{"stretchy":1456},[1128,4286,1268],{},[1204,4288,1462],{"stretchy":1456},[1128,4290,1451],{"mathvariant":1450},[1132,4292,4293],{"encoding":1134},"|f(x)| \\le \\epsilon\\,|g(x)|",[1109,4295,4297,4330],{"className":4296,"ariaHidden":1139},[1138],[1109,4298,4300,4303,4306,4309,4312,4315,4318,4321,4324,4327],{"className":4299},[1143],[1109,4301],{"className":4302,"style":1498},[1147],[1109,4304,1451],{"className":4305},[1152],[1109,4307,1130],{"className":4308,"style":1154},[1152,1153],[1109,4310,1457],{"className":4311},[1508],[1109,4313,1268],{"className":4314},[1152,1153],[1109,4316,1462],{"className":4317},[1515],[1109,4319,1451],{"className":4320},[1152],[1109,4322],{"className":4323,"style":1232},[1231],[1109,4325,1467],{"className":4326},[1236],[1109,4328],{"className":4329,"style":1232},[1231],[1109,4331,4333,4336,4339,4342,4345,4348,4351,4354,4357],{"className":4332},[1143],[1109,4334],{"className":4335,"style":1498},[1147],[1109,4337,4048],{"className":4338},[1152,1153],[1109,4340],{"className":4341,"style":1540},[1231],[1109,4343,1451],{"className":4344},[1152],[1109,4346,1170],{"className":4347,"style":1186},[1152,1153],[1109,4349,1457],{"className":4350},[1508],[1109,4352,1268],{"className":4353},[1152,1153],[1109,4355,1462],{"className":4356},[1515],[1109,4358,1451],{"className":4359},[1152]," ， 记 ",[1109,4362,4364,4401],{"className":4363},[1112],[1109,4365,4367],{"className":4366},[1116],[1118,4368,4369],{"xmlns":1120},[1122,4370,4371,4398],{},[1125,4372,4373,4375,4377,4379,4381,4383,4386,4388,4390,4392,4394,4396],{},[1128,4374,1130],{},[1204,4376,1457],{"stretchy":1456},[1128,4378,1268],{},[1204,4380,1462],{"stretchy":1456},[1204,4382,1582],{},[1128,4384,4385],{},"o",[1204,4387,1457],{"stretchy":1456},[1128,4389,1170],{},[1204,4391,1457],{"stretchy":1456},[1128,4393,1268],{},[1204,4395,1462],{"stretchy":1456},[1204,4397,1462],{"stretchy":1456},[1132,4399,4400],{"encoding":1134},"f(x) = o(g(x))",[1109,4402,4404,4431],{"className":4403,"ariaHidden":1139},[1138],[1109,4405,4407,4410,4413,4416,4419,4422,4425,4428],{"className":4406},[1143],[1109,4408],{"className":4409,"style":1498},[1147],[1109,4411,1130],{"className":4412,"style":1154},[1152,1153],[1109,4414,1457],{"className":4415},[1508],[1109,4417,1268],{"className":4418},[1152,1153],[1109,4420,1462],{"className":4421},[1515],[1109,4423],{"className":4424,"style":1232},[1231],[1109,4426,1582],{"className":4427},[1236],[1109,4429],{"className":4430,"style":1232},[1231],[1109,4432,4434,4437,4440,4443,4446,4449,4452],{"className":4433},[1143],[1109,4435],{"className":4436,"style":1498},[1147],[1109,4438,4385],{"className":4439},[1152,1153],[1109,4441,1457],{"className":4442},[1508],[1109,4444,1170],{"className":4445,"style":1186},[1152,1153],[1109,4447,1457],{"className":4448},[1508],[1109,4450,1268],{"className":4451},[1152,1153],[1109,4453,1656],{"className":4454},[1515],[1101,4456,4457,4458,4696,4697,4725,4726,4729,4730],{},"等价于 ",[1109,4459,4461,4519],{"className":4460},[1112],[1109,4462,4464],{"className":4463},[1116],[1118,4465,4466],{"xmlns":1120},[1122,4467,4468,4516],{},[1125,4469,4470,4489,4512,4514],{},[1264,4471,4472,4479],{},[1125,4473,4474,4477],{},[1128,4475,4476],{},"lim",[1204,4478,2412],{},[1125,4480,4481,4483,4486],{},[1128,4482,1268],{},[1204,4484,4485],{},"→",[1128,4487,4488],{"mathvariant":1450},"∞",[4490,4491,4492,4502],"mfrac",{},[1125,4493,4494,4496,4498,4500],{},[1128,4495,1130],{},[1204,4497,1457],{"stretchy":1456},[1128,4499,1268],{},[1204,4501,1462],{"stretchy":1456},[1125,4503,4504,4506,4508,4510],{},[1128,4505,1170],{},[1204,4507,1457],{"stretchy":1456},[1128,4509,1268],{},[1204,4511,1462],{"stretchy":1456},[1204,4513,1582],{},[1208,4515,1210],{},[1132,4517,4518],{"encoding":1134},"\\lim_{x\\to\\infty}\\frac{f(x)}{g(x)} = 0",[1109,4520,4522,4687],{"className":4521,"ariaHidden":1139},[1138],[1109,4523,4525,4529,4580,4583,4678,4681,4684],{"className":4524},[1143],[1109,4526],{"className":4527,"style":4528},[1147],"height:1.53em;vertical-align:-0.52em;",[1109,4530,4532,4535],{"className":4531},[2878],[1109,4533,4476],{"className":4534},[2878],[1109,4536,4538],{"className":4537},[1293],[1109,4539,4541,4572],{"className":4540},[1297,1298],[1109,4542,4544,4569],{"className":4543},[1302],[1109,4545,4548],{"className":4546,"style":4547},[1306],"height:0.1514em;",[1109,4549,4551,4554],{"style":4550},"top:-2.55em;margin-right:0.05em;",[1109,4552],{"className":4553,"style":1315},[1314],[1109,4555,4557],{"className":4556},[1319,1320,1321,1322],[1109,4558,4560,4563,4566],{"className":4559},[1152,1322],[1109,4561,1268],{"className":4562},[1152,1153,1322],[1109,4564,4485],{"className":4565},[1236,1322],[1109,4567,4488],{"className":4568},[1152,1322],[1109,4570,1330],{"className":4571},[1329],[1109,4573,4575],{"className":4574},[1302],[1109,4576,4578],{"className":4577,"style":1337},[1306],[1109,4579],{},[1109,4581],{"className":4582,"style":1540},[1231],[1109,4584,4586,4590,4675],{"className":4585},[1152],[1109,4587],{"className":4588},[1508,4589],"nulldelimiter",[1109,4591,4593],{"className":4592},[4490],[1109,4594,4596,4666],{"className":4595},[1297,1298],[1109,4597,4599,4663],{"className":4598},[1302],[1109,4600,4603,4628,4639],{"className":4601,"style":4602},[1306],"height:1.01em;",[1109,4604,4606,4610],{"style":4605},"top:-2.655em;",[1109,4607],{"className":4608,"style":4609},[1314],"height:3em;",[1109,4611,4613],{"className":4612},[1319,1320,1321,1322],[1109,4614,4616,4619,4622,4625],{"className":4615},[1152,1322],[1109,4617,1170],{"className":4618,"style":1186},[1152,1153,1322],[1109,4620,1457],{"className":4621},[1508,1322],[1109,4623,1268],{"className":4624},[1152,1153,1322],[1109,4626,1462],{"className":4627},[1515,1322],[1109,4629,4631,4634],{"style":4630},"top:-3.23em;",[1109,4632],{"className":4633,"style":4609},[1314],[1109,4635],{"className":4636,"style":4638},[4637],"frac-line","border-bottom-width:0.04em;",[1109,4640,4642,4645],{"style":4641},"top:-3.485em;",[1109,4643],{"className":4644,"style":4609},[1314],[1109,4646,4648],{"className":4647},[1319,1320,1321,1322],[1109,4649,4651,4654,4657,4660],{"className":4650},[1152,1322],[1109,4652,1130],{"className":4653,"style":1154},[1152,1153,1322],[1109,4655,1457],{"className":4656},[1508,1322],[1109,4658,1268],{"className":4659},[1152,1153,1322],[1109,4661,1462],{"className":4662},[1515,1322],[1109,4664,1330],{"className":4665},[1329],[1109,4667,4669],{"className":4668},[1302],[1109,4670,4673],{"className":4671,"style":4672},[1306],"height:0.52em;",[1109,4674],{},[1109,4676],{"className":4677},[1515,4589],[1109,4679],{"className":4680,"style":1232},[1231],[1109,4682,1582],{"className":4683},[1236],[1109,4685],{"className":4686,"style":1232},[1231],[1109,4688,4690,4693],{"className":4689},[1143],[1109,4691],{"className":4692,"style":1246},[1147],[1109,4694,1210],{"className":4695},[1152],",即 ",[1109,4698,4700,4713],{"className":4699},[1112],[1109,4701,4703],{"className":4702},[1116],[1118,4704,4705],{"xmlns":1120},[1122,4706,4707,4711],{},[1125,4708,4709],{},[1128,4710,1130],{},[1132,4712,1130],{"encoding":1134},[1109,4714,4716],{"className":4715,"ariaHidden":1139},[1138],[1109,4717,4719,4722],{"className":4718},[1143],[1109,4720],{"className":4721,"style":1148},[1147],[1109,4723,1130],{"className":4724,"style":1154},[1152,1153]," 的量级",[1104,4727,4728],{},"严格低于"," ",[1109,4731,4733,4746],{"className":4732},[1112],[1109,4734,4736],{"className":4735},[1116],[1118,4737,4738],{"xmlns":1120},[1122,4739,4740,4744],{},[1125,4741,4742],{},[1128,4743,1170],{},[1132,4745,1170],{"encoding":1134},[1109,4747,4749],{"className":4748,"ariaHidden":1139},[1138],[1109,4750,4752,4755],{"className":4751},[1143],[1109,4753],{"className":4754,"style":1182},[1147],[1109,4756,1170],{"className":4757,"style":1186},[1152,1153],[1101,4759,4760,4762,4763,4829,4830,4895],{},[1104,4761,1661],{},":如果 ",[1109,4764,4766,4790],{"className":4765},[1112],[1109,4767,4769],{"className":4768},[1116],[1118,4770,4771],{"xmlns":1120},[1122,4772,4773,4787],{},[1125,4774,4775,4777,4779,4781,4783,4785],{},[1128,4776,1130],{},[1204,4778,1582],{},[1128,4780,4385],{},[1204,4782,1457],{"stretchy":1456},[1128,4784,1170],{},[1204,4786,1462],{"stretchy":1456},[1132,4788,4789],{"encoding":1134},"f = o(g)",[1109,4791,4793,4811],{"className":4792,"ariaHidden":1139},[1138],[1109,4794,4796,4799,4802,4805,4808],{"className":4795},[1143],[1109,4797],{"className":4798,"style":1148},[1147],[1109,4800,1130],{"className":4801,"style":1154},[1152,1153],[1109,4803],{"className":4804,"style":1232},[1231],[1109,4806,1582],{"className":4807},[1236],[1109,4809],{"className":4810,"style":1232},[1231],[1109,4812,4814,4817,4820,4823,4826],{"className":4813},[1143],[1109,4815],{"className":4816,"style":1498},[1147],[1109,4818,4385],{"className":4819},[1152,1153],[1109,4821,1457],{"className":4822},[1508],[1109,4824,1170],{"className":4825,"style":1186},[1152,1153],[1109,4827,1462],{"className":4828},[1515],",则也有 ",[1109,4831,4833,4856],{"className":4832},[1112],[1109,4834,4836],{"className":4835},[1116],[1118,4837,4838],{"xmlns":1120},[1122,4839,4840,4854],{},[1125,4841,4842,4844,4846,4848,4850,4852],{},[1128,4843,1130],{},[1204,4845,1582],{},[1128,4847,1585],{},[1204,4849,1457],{"stretchy":1456},[1128,4851,1170],{},[1204,4853,1462],{"stretchy":1456},[1132,4855,1786],{"encoding":1134},[1109,4857,4859,4877],{"className":4858,"ariaHidden":1139},[1138],[1109,4860,4862,4865,4868,4871,4874],{"className":4861},[1143],[1109,4863],{"className":4864,"style":1148},[1147],[1109,4866,1130],{"className":4867,"style":1154},[1152,1153],[1109,4869],{"className":4870,"style":1232},[1231],[1109,4872,1582],{"className":4873},[1236],[1109,4875],{"className":4876,"style":1232},[1231],[1109,4878,4880,4883,4886,4889,4892],{"className":4879},[1143],[1109,4881],{"className":4882,"style":1498},[1147],[1109,4884,1585],{"className":4885,"style":1640},[1152,1153],[1109,4887,1457],{"className":4888},[1508],[1109,4890,1170],{"className":4891,"style":1186},[1152,1153],[1109,4893,1462],{"className":4894},[1515],"(小 o 比大 O 更强)。",[1101,4897,4898,1662,4900,1910,5209,4024],{},[1104,4899,3426],{},[1109,4901,4903,4971],{"className":4902},[1112],[1109,4904,4906],{"className":4905},[1116],[1118,4907,4908],{"xmlns":1120},[1122,4909,4910,4968],{},[1125,4911,4912,4914,4916,4922,4924,4926,4928,4930,4936,4938,4940,4942,4944,4946,4948,4950,4956,4958,4964,4966],{},[1128,4913,4385],{},[1204,4915,1457],{"stretchy":1456},[1264,4917,4918,4920],{},[1128,4919,1170],{},[1208,4921,1685],{},[1204,4923,1462],{"stretchy":1456},[1204,4925,2342],{},[1128,4927,4385],{},[1204,4929,1457],{"stretchy":1456},[1264,4931,4932,4934],{},[1128,4933,1170],{},[1208,4935,1917],{},[1204,4937,1462],{"stretchy":1456},[1204,4939,1582],{},[1128,4941,4385],{},[1204,4943,1457],{"stretchy":1456},[1128,4945,2409],{},[1204,4947,2412],{},[1204,4949,1457],{"stretchy":1456},[1264,4951,4952,4954],{},[1128,4953,1170],{},[1208,4955,1685],{},[1204,4957,1910],{"separator":1139},[1264,4959,4960,4962],{},[1128,4961,1170],{},[1208,4963,1917],{},[1204,4965,1462],{"stretchy":1456},[1204,4967,1462],{"stretchy":1456},[1132,4969,4970],{"encoding":1134},"o(g_1) + o(g_2) = o(\\max(g_1,g_2))",[1109,4972,4974,5038,5102],{"className":4973,"ariaHidden":1139},[1138],[1109,4975,4977,4980,4983,4986,5026,5029,5032,5035],{"className":4976},[1143],[1109,4978],{"className":4979,"style":1498},[1147],[1109,4981,4385],{"className":4982},[1152,1153],[1109,4984,1457],{"className":4985},[1508],[1109,4987,4989,4992],{"className":4988},[1152],[1109,4990,1170],{"className":4991,"style":1186},[1152,1153],[1109,4993,4995],{"className":4994},[1293],[1109,4996,4998,5018],{"className":4997},[1297,1298],[1109,4999,5001,5015],{"className":5000},[1302],[1109,5002,5004],{"className":5003,"style":1307},[1306],[1109,5005,5006,5009],{"style":2146},[1109,5007],{"className":5008,"style":1315},[1314],[1109,5010,5012],{"className":5011},[1319,1320,1321,1322],[1109,5013,1685],{"className":5014},[1152,1322],[1109,5016,1330],{"className":5017},[1329],[1109,5019,5021],{"className":5020},[1302],[1109,5022,5024],{"className":5023,"style":1337},[1306],[1109,5025],{},[1109,5027,1462],{"className":5028},[1515],[1109,5030],{"className":5031,"style":2497},[1231],[1109,5033,2342],{"className":5034},[2501],[1109,5036],{"className":5037,"style":2497},[1231],[1109,5039,5041,5044,5047,5050,5090,5093,5096,5099],{"className":5040},[1143],[1109,5042],{"className":5043,"style":1498},[1147],[1109,5045,4385],{"className":5046},[1152,1153],[1109,5048,1457],{"className":5049},[1508],[1109,5051,5053,5056],{"className":5052},[1152],[1109,5054,1170],{"className":5055,"style":1186},[1152,1153],[1109,5057,5059],{"className":5058},[1293],[1109,5060,5062,5082],{"className":5061},[1297,1298],[1109,5063,5065,5079],{"className":5064},[1302],[1109,5066,5068],{"className":5067,"style":1307},[1306],[1109,5069,5070,5073],{"style":2146},[1109,5071],{"className":5072,"style":1315},[1314],[1109,5074,5076],{"className":5075},[1319,1320,1321,1322],[1109,5077,1917],{"className":5078},[1152,1322],[1109,5080,1330],{"className":5081},[1329],[1109,5083,5085],{"className":5084},[1302],[1109,5086,5088],{"className":5087,"style":1337},[1306],[1109,5089],{},[1109,5091,1462],{"className":5092},[1515],[1109,5094],{"className":5095,"style":1232},[1231],[1109,5097,1582],{"className":5098},[1236],[1109,5100],{"className":5101,"style":1232},[1231],[1109,5103,5105,5108,5111,5114,5117,5120,5160,5163,5166,5206],{"className":5104},[1143],[1109,5106],{"className":5107,"style":1498},[1147],[1109,5109,4385],{"className":5110},[1152,1153],[1109,5112,1457],{"className":5113},[1508],[1109,5115,2409],{"className":5116},[2878],[1109,5118,1457],{"className":5119},[1508],[1109,5121,5123,5126],{"className":5122},[1152],[1109,5124,1170],{"className":5125,"style":1186},[1152,1153],[1109,5127,5129],{"className":5128},[1293],[1109,5130,5132,5152],{"className":5131},[1297,1298],[1109,5133,5135,5149],{"className":5134},[1302],[1109,5136,5138],{"className":5137,"style":1307},[1306],[1109,5139,5140,5143],{"style":2146},[1109,5141],{"className":5142,"style":1315},[1314],[1109,5144,5146],{"className":5145},[1319,1320,1321,1322],[1109,5147,1685],{"className":5148},[1152,1322],[1109,5150,1330],{"className":5151},[1329],[1109,5153,5155],{"className":5154},[1302],[1109,5156,5158],{"className":5157,"style":1337},[1306],[1109,5159],{},[1109,5161,1910],{"className":5162},[1975],[1109,5164],{"className":5165,"style":1540},[1231],[1109,5167,5169,5172],{"className":5168},[1152],[1109,5170,1170],{"className":5171,"style":1186},[1152,1153],[1109,5173,5175],{"className":5174},[1293],[1109,5176,5178,5198],{"className":5177},[1297,1298],[1109,5179,5181,5195],{"className":5180},[1302],[1109,5182,5184],{"className":5183,"style":1307},[1306],[1109,5185,5186,5189],{"style":2146},[1109,5187],{"className":5188,"style":1315},[1314],[1109,5190,5192],{"className":5191},[1319,1320,1321,1322],[1109,5193,1917],{"className":5194},[1152,1322],[1109,5196,1330],{"className":5197},[1329],[1109,5199,5201],{"className":5200},[1302],[1109,5202,5204],{"className":5203,"style":1337},[1306],[1109,5205],{},[1109,5207,1656],{"className":5208},[1515],[1109,5210,5212,5270],{"className":5211},[1112],[1109,5213,5215],{"className":5214},[1116],[1118,5216,5217],{"xmlns":1120},[1122,5218,5219,5267],{},[1125,5220,5221,5223,5225,5231,5233,5235,5237,5239,5245,5247,5249,5251,5253,5259,5265],{},[1128,5222,4385],{},[1204,5224,1457],{"stretchy":1456},[1264,5226,5227,5229],{},[1128,5228,1170],{},[1208,5230,1685],{},[1204,5232,1462],{"stretchy":1456},[1204,5234,3762],{},[1128,5236,4385],{},[1204,5238,1457],{"stretchy":1456},[1264,5240,5241,5243],{},[1128,5242,1170],{},[1208,5244,1917],{},[1204,5246,1462],{"stretchy":1456},[1204,5248,1582],{},[1128,5250,4385],{},[1204,5252,1457],{"stretchy":1456},[1264,5254,5255,5257],{},[1128,5256,1170],{},[1208,5258,1685],{},[1264,5260,5261,5263],{},[1128,5262,1170],{},[1208,5264,1917],{},[1204,5266,1462],{"stretchy":1456},[1132,5268,5269],{"encoding":1134},"o(g_1)\\cdot o(g_2) = o(g_1 g_2)",[1109,5271,5273,5337,5401],{"className":5272,"ariaHidden":1139},[1138],[1109,5274,5276,5279,5282,5285,5325,5328,5331,5334],{"className":5275},[1143],[1109,5277],{"className":5278,"style":1498},[1147],[1109,5280,4385],{"className":5281},[1152,1153],[1109,5283,1457],{"className":5284},[1508],[1109,5286,5288,5291],{"className":5287},[1152],[1109,5289,1170],{"className":5290,"style":1186},[1152,1153],[1109,5292,5294],{"className":5293},[1293],[1109,5295,5297,5317],{"className":5296},[1297,1298],[1109,5298,5300,5314],{"className":5299},[1302],[1109,5301,5303],{"className":5302,"style":1307},[1306],[1109,5304,5305,5308],{"style":2146},[1109,5306],{"className":5307,"style":1315},[1314],[1109,5309,5311],{"className":5310},[1319,1320,1321,1322],[1109,5312,1685],{"className":5313},[1152,1322],[1109,5315,1330],{"className":5316},[1329],[1109,5318,5320],{"className":5319},[1302],[1109,5321,5323],{"className":5322,"style":1337},[1306],[1109,5324],{},[1109,5326,1462],{"className":5327},[1515],[1109,5329],{"className":5330,"style":2497},[1231],[1109,5332,3762],{"className":5333},[2501],[1109,5335],{"className":5336,"style":2497},[1231],[1109,5338,5340,5343,5346,5349,5389,5392,5395,5398],{"className":5339},[1143],[1109,5341],{"className":5342,"style":1498},[1147],[1109,5344,4385],{"className":5345},[1152,1153],[1109,5347,1457],{"className":5348},[1508],[1109,5350,5352,5355],{"className":5351},[1152],[1109,5353,1170],{"className":5354,"style":1186},[1152,1153],[1109,5356,5358],{"className":5357},[1293],[1109,5359,5361,5381],{"className":5360},[1297,1298],[1109,5362,5364,5378],{"className":5363},[1302],[1109,5365,5367],{"className":5366,"style":1307},[1306],[1109,5368,5369,5372],{"style":2146},[1109,5370],{"className":5371,"style":1315},[1314],[1109,5373,5375],{"className":5374},[1319,1320,1321,1322],[1109,5376,1917],{"className":5377},[1152,1322],[1109,5379,1330],{"className":5380},[1329],[1109,5382,5384],{"className":5383},[1302],[1109,5385,5387],{"className":5386,"style":1337},[1306],[1109,5388],{},[1109,5390,1462],{"className":5391},[1515],[1109,5393],{"className":5394,"style":1232},[1231],[1109,5396,1582],{"className":5397},[1236],[1109,5399],{"className":5400,"style":1232},[1231],[1109,5402,5404,5407,5410,5413,5453,5493],{"className":5403},[1143],[1109,5405],{"className":5406,"style":1498},[1147],[1109,5408,4385],{"className":5409},[1152,1153],[1109,5411,1457],{"className":5412},[1508],[1109,5414,5416,5419],{"className":5415},[1152],[1109,5417,1170],{"className":5418,"style":1186},[1152,1153],[1109,5420,5422],{"className":5421},[1293],[1109,5423,5425,5445],{"className":5424},[1297,1298],[1109,5426,5428,5442],{"className":5427},[1302],[1109,5429,5431],{"className":5430,"style":1307},[1306],[1109,5432,5433,5436],{"style":2146},[1109,5434],{"className":5435,"style":1315},[1314],[1109,5437,5439],{"className":5438},[1319,1320,1321,1322],[1109,5440,1685],{"className":5441},[1152,1322],[1109,5443,1330],{"className":5444},[1329],[1109,5446,5448],{"className":5447},[1302],[1109,5449,5451],{"className":5450,"style":1337},[1306],[1109,5452],{},[1109,5454,5456,5459],{"className":5455},[1152],[1109,5457,1170],{"className":5458,"style":1186},[1152,1153],[1109,5460,5462],{"className":5461},[1293],[1109,5463,5465,5485],{"className":5464},[1297,1298],[1109,5466,5468,5482],{"className":5467},[1302],[1109,5469,5471],{"className":5470,"style":1307},[1306],[1109,5472,5473,5476],{"style":2146},[1109,5474],{"className":5475,"style":1315},[1314],[1109,5477,5479],{"className":5478},[1319,1320,1321,1322],[1109,5480,1917],{"className":5481},[1152,1322],[1109,5483,1330],{"className":5484},[1329],[1109,5486,5488],{"className":5487},[1302],[1109,5489,5491],{"className":5490,"style":1337},[1306],[1109,5492],{},[1109,5494,1462],{"className":5495},[1515],[1096,5497,5499],{"id":5498},"weierstrass-定理","Weierstrass 定理",[1101,5501,5502,5504,5505,5561,5562,5590],{},[1104,5503,5499],{},":取值于闭区间 ",[1109,5506,5508,5534],{"className":5507},[1112],[1109,5509,5511],{"className":5510},[1116],[1118,5512,5513],{"xmlns":1120},[1122,5514,5515,5531],{},[1125,5516,5517,5520,5523,5525,5528],{},[1204,5518,5519],{"stretchy":1456},"[",[1128,5521,5522],{},"a",[1204,5524,1910],{"separator":1139},[1128,5526,5527],{},"b",[1204,5529,5530],{"stretchy":1456},"]",[1132,5532,5533],{"encoding":1134},"[a,b]",[1109,5535,5537],{"className":5536,"ariaHidden":1139},[1138],[1109,5538,5540,5543,5546,5549,5552,5555,5558],{"className":5539},[1143],[1109,5541],{"className":5542,"style":1498},[1147],[1109,5544,5519],{"className":5545},[1508],[1109,5547,5522],{"className":5548},[1152,1153],[1109,5550,1910],{"className":5551},[1975],[1109,5553],{"className":5554,"style":1540},[1231],[1109,5556,5527],{"className":5557},[1152,1153],[1109,5559,5530],{"className":5560},[1515]," 上的连续函数 ",[1109,5563,5565,5578],{"className":5564},[1112],[1109,5566,5568],{"className":5567},[1116],[1118,5569,5570],{"xmlns":1120},[1122,5571,5572,5576],{},[1125,5573,5574],{},[1128,5575,1130],{},[1132,5577,1130],{"encoding":1134},[1109,5579,5581],{"className":5580,"ariaHidden":1139},[1138],[1109,5582,5584,5587],{"className":5583},[1143],[1109,5585],{"className":5586,"style":1148},[1147],[1109,5588,1130],{"className":5589,"style":1154},[1152,1153],",其最大值、最小值一定存在,并且在区间中取得。",[1101,5592,5593,5596,5597,5669,5670,4729,5741,5744],{},[1104,5594,5595],{},"平衡点(驻点)",":使得梯度 ",[1109,5598,5600,5627],{"className":5599},[1112],[1109,5601,5603],{"className":5602},[1116],[1118,5604,5605],{"xmlns":1120},[1122,5606,5607,5624],{},[1125,5608,5609,5612,5614,5616,5618,5620,5622],{},[1128,5610,5611],{"mathvariant":1450},"∇",[1128,5613,1130],{},[1204,5615,1457],{"stretchy":1456},[1128,5617,1268],{},[1204,5619,1462],{"stretchy":1456},[1204,5621,1582],{},[1208,5623,1210],{},[1132,5625,5626],{"encoding":1134},"\\nabla f(x) = 0",[1109,5628,5630,5660],{"className":5629,"ariaHidden":1139},[1138],[1109,5631,5633,5636,5639,5642,5645,5648,5651,5654,5657],{"className":5632},[1143],[1109,5634],{"className":5635,"style":1498},[1147],[1109,5637,5611],{"className":5638},[1152],[1109,5640,1130],{"className":5641,"style":1154},[1152,1153],[1109,5643,1457],{"className":5644},[1508],[1109,5646,1268],{"className":5647},[1152,1153],[1109,5649,1462],{"className":5650},[1515],[1109,5652],{"className":5653,"style":1232},[1231],[1109,5655,1582],{"className":5656},[1236],[1109,5658],{"className":5659,"style":1232},[1231],[1109,5661,5663,5666],{"className":5662},[1143],[1109,5664],{"className":5665,"style":1246},[1147],[1109,5667,1210],{"className":5668},[1152]," 的点称为平衡点。但 ",[1109,5671,5673,5699],{"className":5672},[1112],[1109,5674,5676],{"className":5675},[1116],[1118,5677,5678],{"xmlns":1120},[1122,5679,5680,5696],{},[1125,5681,5682,5684,5686,5688,5690,5692,5694],{},[1128,5683,5611],{"mathvariant":1450},[1128,5685,1130],{},[1204,5687,1457],{"stretchy":1456},[1128,5689,1268],{},[1204,5691,1462],{"stretchy":1456},[1204,5693,1582],{},[1208,5695,1210],{},[1132,5697,5698],{"encoding":1134},"\\nabla f(x)=0",[1109,5700,5702,5732],{"className":5701,"ariaHidden":1139},[1138],[1109,5703,5705,5708,5711,5714,5717,5720,5723,5726,5729],{"className":5704},[1143],[1109,5706],{"className":5707,"style":1498},[1147],[1109,5709,5611],{"className":5710},[1152],[1109,5712,1130],{"className":5713,"style":1154},[1152,1153],[1109,5715,1457],{"className":5716},[1508],[1109,5718,1268],{"className":5719},[1152,1153],[1109,5721,1462],{"className":5722},[1515],[1109,5724],{"className":5725,"style":1232},[1231],[1109,5727,1582],{"className":5728},[1236],[1109,5730],{"className":5731,"style":1232},[1231],[1109,5733,5735,5738],{"className":5734},[1143],[1109,5736],{"className":5737,"style":1246},[1147],[1109,5739,1210],{"className":5740},[1152],[1104,5742,5743],{},"并不能保证是全局最优点",",它可能只是局部最优点或鞍点。",[1096,5746,5747],{"id":5747},"凸函数",[1101,5749,5750,5752,5753,5877,5878,5955],{},[1104,5751,5747],{},":如果对任意 ",[1109,5754,5756,5782],{"className":5755},[1112],[1109,5757,5759],{"className":5758},[1116],[1118,5760,5761],{"xmlns":1120},[1122,5762,5763,5779],{},[1125,5764,5765,5771,5773],{},[1264,5766,5767,5769],{},[1128,5768,1268],{},[1208,5770,1685],{},[1204,5772,1910],{"separator":1139},[1264,5774,5775,5777],{},[1128,5776,1268],{},[1208,5778,1917],{},[1132,5780,5781],{"encoding":1134},"x_1, x_2",[1109,5783,5785],{"className":5784,"ariaHidden":1139},[1138],[1109,5786,5788,5791,5831,5834,5837],{"className":5787},[1143],[1109,5789],{"className":5790,"style":1182},[1147],[1109,5792,5794,5797],{"className":5793},[1152],[1109,5795,1268],{"className":5796},[1152,1153],[1109,5798,5800],{"className":5799},[1293],[1109,5801,5803,5823],{"className":5802},[1297,1298],[1109,5804,5806,5820],{"className":5805},[1302],[1109,5807,5809],{"className":5808,"style":1307},[1306],[1109,5810,5811,5814],{"style":1310},[1109,5812],{"className":5813,"style":1315},[1314],[1109,5815,5817],{"className":5816},[1319,1320,1321,1322],[1109,5818,1685],{"className":5819},[1152,1322],[1109,5821,1330],{"className":5822},[1329],[1109,5824,5826],{"className":5825},[1302],[1109,5827,5829],{"className":5828,"style":1337},[1306],[1109,5830],{},[1109,5832,1910],{"className":5833},[1975],[1109,5835],{"className":5836,"style":1540},[1231],[1109,5838,5840,5843],{"className":5839},[1152],[1109,5841,1268],{"className":5842},[1152,1153],[1109,5844,5846],{"className":5845},[1293],[1109,5847,5849,5869],{"className":5848},[1297,1298],[1109,5850,5852,5866],{"className":5851},[1302],[1109,5853,5855],{"className":5854,"style":1307},[1306],[1109,5856,5857,5860],{"style":1310},[1109,5858],{"className":5859,"style":1315},[1314],[1109,5861,5863],{"className":5862},[1319,1320,1321,1322],[1109,5864,1917],{"className":5865},[1152,1322],[1109,5867,1330],{"className":5868},[1329],[1109,5870,5872],{"className":5871},[1302],[1109,5873,5875],{"className":5874,"style":1337},[1306],[1109,5876],{}," 及 ",[1109,5879,5881,5909],{"className":5880},[1112],[1109,5882,5884],{"className":5883},[1116],[1118,5885,5886],{"xmlns":1120},[1122,5887,5888,5906],{},[1125,5889,5890,5893,5896,5898,5900,5902,5904],{},[1128,5891,5892],{},"λ",[1204,5894,5895],{},"∈",[1204,5897,5519],{"stretchy":1456},[1208,5899,1210],{},[1204,5901,1910],{"separator":1139},[1208,5903,1685],{},[1204,5905,5530],{"stretchy":1456},[1132,5907,5908],{"encoding":1134},"\\lambda \\in [0,1]",[1109,5910,5912,5931],{"className":5911,"ariaHidden":1139},[1138],[1109,5913,5915,5919,5922,5925,5928],{"className":5914},[1143],[1109,5916],{"className":5917,"style":5918},[1147],"height:0.7335em;vertical-align:-0.0391em;",[1109,5920,5892],{"className":5921},[1152,1153],[1109,5923],{"className":5924,"style":1232},[1231],[1109,5926,5895],{"className":5927},[1236],[1109,5929],{"className":5930,"style":1232},[1231],[1109,5932,5934,5937,5940,5943,5946,5949,5952],{"className":5933},[1143],[1109,5935],{"className":5936,"style":1498},[1147],[1109,5938,5519],{"className":5939},[1508],[1109,5941,1210],{"className":5942},[1152],[1109,5944,1910],{"className":5945},[1975],[1109,5947],{"className":5948,"style":1540},[1231],[1109,5950,1685],{"className":5951},[1152],[1109,5953,5530],{"className":5954},[1515],",有",[1101,5957,5958],{},[1109,5959,5961,6047],{"className":5960},[1112],[1109,5962,5964],{"className":5963},[1116],[1118,5965,5966],{"xmlns":1120},[1122,5967,5968,6044],{},[1125,5969,5970,5972,5975,5977,5983,5985,5987,5989,5992,5994,5996,6002,6004,6006,6008,6010,6012,6018,6020,6022,6024,6026,6028,6030,6032,6034,6036,6042],{},[1128,5971,1130],{},[1204,5973,1457],{"fence":1456,"stretchy":1139,"minsize":5974,"maxsize":5974},"1.2em",[1128,5976,5892],{},[1264,5978,5979,5981],{},[1128,5980,1268],{},[1208,5982,1685],{},[1204,5984,2342],{},[1204,5986,1457],{"stretchy":1456},[1208,5988,1685],{},[1204,5990,5991],{},"−",[1128,5993,5892],{},[1204,5995,1462],{"stretchy":1456},[1264,5997,5998,6000],{},[1128,5999,1268],{},[1208,6001,1917],{},[1204,6003,1462],{"fence":1456,"stretchy":1139,"minsize":5974,"maxsize":5974},[1204,6005,1467],{},[1128,6007,5892],{},[1128,6009,1130],{},[1204,6011,1457],{"stretchy":1456},[1264,6013,6014,6016],{},[1128,6015,1268],{},[1208,6017,1685],{},[1204,6019,1462],{"stretchy":1456},[1204,6021,2342],{},[1204,6023,1457],{"stretchy":1456},[1208,6025,1685],{},[1204,6027,5991],{},[1128,6029,5892],{},[1204,6031,1462],{"stretchy":1456},[1128,6033,1130],{},[1204,6035,1457],{"stretchy":1456},[1264,6037,6038,6040],{},[1128,6039,1268],{},[1208,6041,1917],{},[1204,6043,1462],{"stretchy":1456},[1132,6045,6046],{"encoding":1134},"f\\big(\\lambda x_1 + (1-\\lambda)x_2\\big) \\le \\lambda f(x_1) + (1-\\lambda)f(x_2)",[1109,6048,6050,6120,6141,6208,6275,6296],{"className":6049,"ariaHidden":1139},[1138],[1109,6051,6053,6057,6060,6068,6071,6111,6114,6117],{"className":6052},[1143],[1109,6054],{"className":6055,"style":6056},[1147],"height:1.2em;vertical-align:-0.35em;",[1109,6058,1130],{"className":6059,"style":1154},[1152,1153],[1109,6061,6063],{"className":6062},[1152],[1109,6064,1457],{"className":6065},[6066,6067],"delimsizing","size1",[1109,6069,5892],{"className":6070},[1152,1153],[1109,6072,6074,6077],{"className":6073},[1152],[1109,6075,1268],{"className":6076},[1152,1153],[1109,6078,6080],{"className":6079},[1293],[1109,6081,6083,6103],{"className":6082},[1297,1298],[1109,6084,6086,6100],{"className":6085},[1302],[1109,6087,6089],{"className":6088,"style":1307},[1306],[1109,6090,6091,6094],{"style":1310},[1109,6092],{"className":6093,"style":1315},[1314],[1109,6095,6097],{"className":6096},[1319,1320,1321,1322],[1109,6098,1685],{"className":6099},[1152,1322],[1109,6101,1330],{"className":6102},[1329],[1109,6104,6106],{"className":6105},[1302],[1109,6107,6109],{"className":6108,"style":1337},[1306],[1109,6110],{},[1109,6112],{"className":6113,"style":2497},[1231],[1109,6115,2342],{"className":6116},[2501],[1109,6118],{"className":6119,"style":2497},[1231],[1109,6121,6123,6126,6129,6132,6135,6138],{"className":6122},[1143],[1109,6124],{"className":6125,"style":1498},[1147],[1109,6127,1457],{"className":6128},[1508],[1109,6130,1685],{"className":6131},[1152],[1109,6133],{"className":6134,"style":2497},[1231],[1109,6136,5991],{"className":6137},[2501],[1109,6139],{"className":6140,"style":2497},[1231],[1109,6142,6144,6147,6150,6153,6193,6199,6202,6205],{"className":6143},[1143],[1109,6145],{"className":6146,"style":6056},[1147],[1109,6148,5892],{"className":6149},[1152,1153],[1109,6151,1462],{"className":6152},[1515],[1109,6154,6156,6159],{"className":6155},[1152],[1109,6157,1268],{"className":6158},[1152,1153],[1109,6160,6162],{"className":6161},[1293],[1109,6163,6165,6185],{"className":6164},[1297,1298],[1109,6166,6168,6182],{"className":6167},[1302],[1109,6169,6171],{"className":6170,"style":1307},[1306],[1109,6172,6173,6176],{"style":1310},[1109,6174],{"className":6175,"style":1315},[1314],[1109,6177,6179],{"className":6178},[1319,1320,1321,1322],[1109,6180,1917],{"className":6181},[1152,1322],[1109,6183,1330],{"className":6184},[1329],[1109,6186,6188],{"className":6187},[1302],[1109,6189,6191],{"className":6190,"style":1337},[1306],[1109,6192],{},[1109,6194,6196],{"className":6195},[1152],[1109,6197,1462],{"className":6198},[6066,6067],[1109,6200],{"className":6201,"style":1232},[1231],[1109,6203,1467],{"className":6204},[1236],[1109,6206],{"className":6207,"style":1232},[1231],[1109,6209,6211,6214,6217,6220,6223,6263,6266,6269,6272],{"className":6210},[1143],[1109,6212],{"className":6213,"style":1498},[1147],[1109,6215,5892],{"className":6216},[1152,1153],[1109,6218,1130],{"className":6219,"style":1154},[1152,1153],[1109,6221,1457],{"className":6222},[1508],[1109,6224,6226,6229],{"className":6225},[1152],[1109,6227,1268],{"className":6228},[1152,1153],[1109,6230,6232],{"className":6231},[1293],[1109,6233,6235,6255],{"className":6234},[1297,1298],[1109,6236,6238,6252],{"className":6237},[1302],[1109,6239,6241],{"className":6240,"style":1307},[1306],[1109,6242,6243,6246],{"style":1310},[1109,6244],{"className":6245,"style":1315},[1314],[1109,6247,6249],{"className":6248},[1319,1320,1321,1322],[1109,6250,1685],{"className":6251},[1152,1322],[1109,6253,1330],{"className":6254},[1329],[1109,6256,6258],{"className":6257},[1302],[1109,6259,6261],{"className":6260,"style":1337},[1306],[1109,6262],{},[1109,6264,1462],{"className":6265},[1515],[1109,6267],{"className":6268,"style":2497},[1231],[1109,6270,2342],{"className":6271},[2501],[1109,6273],{"className":6274,"style":2497},[1231],[1109,6276,6278,6281,6284,6287,6290,6293],{"className":6277},[1143],[1109,6279],{"className":6280,"style":1498},[1147],[1109,6282,1457],{"className":6283},[1508],[1109,6285,1685],{"className":6286},[1152],[1109,6288],{"className":6289,"style":2497},[1231],[1109,6291,5991],{"className":6292},[2501],[1109,6294],{"className":6295,"style":2497},[1231],[1109,6297,6299,6302,6305,6308,6311,6314,6354],{"className":6298},[1143],[1109,6300],{"className":6301,"style":1498},[1147],[1109,6303,5892],{"className":6304},[1152,1153],[1109,6306,1462],{"className":6307},[1515],[1109,6309,1130],{"className":6310,"style":1154},[1152,1153],[1109,6312,1457],{"className":6313},[1508],[1109,6315,6317,6320],{"className":6316},[1152],[1109,6318,1268],{"className":6319},[1152,1153],[1109,6321,6323],{"className":6322},[1293],[1109,6324,6326,6346],{"className":6325},[1297,1298],[1109,6327,6329,6343],{"className":6328},[1302],[1109,6330,6332],{"className":6331,"style":1307},[1306],[1109,6333,6334,6337],{"style":1310},[1109,6335],{"className":6336,"style":1315},[1314],[1109,6338,6340],{"className":6339},[1319,1320,1321,1322],[1109,6341,1917],{"className":6342},[1152,1322],[1109,6344,1330],{"className":6345},[1329],[1109,6347,6349],{"className":6348},[1302],[1109,6350,6352],{"className":6351,"style":1337},[1306],[1109,6353],{},[1109,6355,1462],{"className":6356},[1515],[1101,6358,6359,6360,6388],{},"那么 ",[1109,6361,6363,6376],{"className":6362},[1112],[1109,6364,6366],{"className":6365},[1116],[1118,6367,6368],{"xmlns":1120},[1122,6369,6370,6374],{},[1125,6371,6372],{},[1128,6373,1130],{},[1132,6375,1130],{"encoding":1134},[1109,6377,6379],{"className":6378,"ariaHidden":1139},[1138],[1109,6380,6382,6385],{"className":6381},[1143],[1109,6383],{"className":6384,"style":1148},[1147],[1109,6386,1130],{"className":6387,"style":1154},[1152,1153]," 被称为凸函数。",[1101,6390,6391,5752,6394,5877,6568,6642,6643,6672,6673,6702,6703,6731],{},[1104,6392,6393],{},"严格凸函数",[1109,6395,6397,6424],{"className":6396},[1112],[1109,6398,6400],{"className":6399},[1116],[1118,6401,6402],{"xmlns":1120},[1122,6403,6404,6421],{},[1125,6405,6406,6412,6415],{},[1264,6407,6408,6410],{},[1128,6409,1268],{},[1208,6411,1685],{},[1204,6413,6414],{"mathvariant":1450},"≠",[1264,6416,6417,6419],{},[1128,6418,1268],{},[1208,6420,1917],{},[1132,6422,6423],{"encoding":1134},"x_1 \\ne x_2",[1109,6425,6427,6522],{"className":6426,"ariaHidden":1139},[1138],[1109,6428,6430,6433,6473,6476,6519],{"className":6429},[1143],[1109,6431],{"className":6432,"style":1148},[1147],[1109,6434,6436,6439],{"className":6435},[1152],[1109,6437,1268],{"className":6438},[1152,1153],[1109,6440,6442],{"className":6441},[1293],[1109,6443,6445,6465],{"className":6444},[1297,1298],[1109,6446,6448,6462],{"className":6447},[1302],[1109,6449,6451],{"className":6450,"style":1307},[1306],[1109,6452,6453,6456],{"style":1310},[1109,6454],{"className":6455,"style":1315},[1314],[1109,6457,6459],{"className":6458},[1319,1320,1321,1322],[1109,6460,1685],{"className":6461},[1152,1322],[1109,6463,1330],{"className":6464},[1329],[1109,6466,6468],{"className":6467},[1302],[1109,6469,6471],{"className":6470,"style":1337},[1306],[1109,6472],{},[1109,6474],{"className":6475,"style":1232},[1231],[1109,6477,6479,6512,6516],{"className":6478},[1236],[1109,6480,6482],{"className":6481},[1236],[1109,6483,6486],{"className":6484},[1152,6485],"vbox",[1109,6487,6490],{"className":6488},[6489],"thinbox",[1109,6491,6494,6497,6508],{"className":6492},[6493],"rlap",[1109,6495],{"className":6496,"style":1148},[1147],[1109,6498,6501],{"className":6499},[6500],"inner",[1109,6502,6504],{"className":6503},[1152],[1109,6505,6507],{"className":6506},[1236],"",[1109,6509],{"className":6510},[6511],"fix",[1109,6513],{"className":6514},[1231,6515],"nobreak",[1109,6517,1582],{"className":6518},[1236],[1109,6520],{"className":6521,"style":1232},[1231],[1109,6523,6525,6528],{"className":6524},[1143],[1109,6526],{"className":6527,"style":1283},[1147],[1109,6529,6531,6534],{"className":6530},[1152],[1109,6532,1268],{"className":6533},[1152,1153],[1109,6535,6537],{"className":6536},[1293],[1109,6538,6540,6560],{"className":6539},[1297,1298],[1109,6541,6543,6557],{"className":6542},[1302],[1109,6544,6546],{"className":6545,"style":1307},[1306],[1109,6547,6548,6551],{"style":1310},[1109,6549],{"className":6550,"style":1315},[1314],[1109,6552,6554],{"className":6553},[1319,1320,1321,1322],[1109,6555,1917],{"className":6556},[1152,1322],[1109,6558,1330],{"className":6559},[1329],[1109,6561,6563],{"className":6562},[1302],[1109,6564,6566],{"className":6565,"style":1337},[1306],[1109,6567],{},[1109,6569,6571,6597],{"className":6570},[1112],[1109,6572,6574],{"className":6573},[1116],[1118,6575,6576],{"xmlns":1120},[1122,6577,6578,6594],{},[1125,6579,6580,6582,6584,6586,6588,6590,6592],{},[1128,6581,5892],{},[1204,6583,5895],{},[1204,6585,1457],{"stretchy":1456},[1208,6587,1210],{},[1204,6589,1910],{"separator":1139},[1208,6591,1685],{},[1204,6593,1462],{"stretchy":1456},[1132,6595,6596],{"encoding":1134},"\\lambda \\in (0,1)",[1109,6598,6600,6618],{"className":6599,"ariaHidden":1139},[1138],[1109,6601,6603,6606,6609,6612,6615],{"className":6602},[1143],[1109,6604],{"className":6605,"style":5918},[1147],[1109,6607,5892],{"className":6608},[1152,1153],[1109,6610],{"className":6611,"style":1232},[1231],[1109,6613,5895],{"className":6614},[1236],[1109,6616],{"className":6617,"style":1232},[1231],[1109,6619,6621,6624,6627,6630,6633,6636,6639],{"className":6620},[1143],[1109,6622],{"className":6623,"style":1498},[1147],[1109,6625,1457],{"className":6626},[1508],[1109,6628,1210],{"className":6629},[1152],[1109,6631,1910],{"className":6632},[1975],[1109,6634],{"className":6635,"style":1540},[1231],[1109,6637,1685],{"className":6638},[1152],[1109,6640,1462],{"className":6641},[1515],",把上式的 ",[1109,6644,6646,6660],{"className":6645},[1112],[1109,6647,6649],{"className":6648},[1116],[1118,6650,6651],{"xmlns":1120},[1122,6652,6653,6657],{},[1125,6654,6655],{},[1204,6656,1467],{},[1132,6658,6659],{"encoding":1134},"\\le",[1109,6661,6663],{"className":6662,"ariaHidden":1139},[1138],[1109,6664,6666,6669],{"className":6665},[1143],[1109,6667],{"className":6668,"style":1376},[1147],[1109,6670,1467],{"className":6671},[1236]," 换成严格的 ",[1109,6674,6676,6690],{"className":6675},[1112],[1109,6677,6679],{"className":6678},[1116],[1118,6680,6681],{"xmlns":1120},[1122,6682,6683,6688],{},[1125,6684,6685],{},[1204,6686,6687],{},"\u003C",[1132,6689,6687],{"encoding":1134},[1109,6691,6693],{"className":6692,"ariaHidden":1139},[1138],[1109,6694,6696,6699],{"className":6695},[1143],[1109,6697],{"className":6698,"style":4065},[1147],[1109,6700,6687],{"className":6701},[1236],",那么 ",[1109,6704,6706,6719],{"className":6705},[1112],[1109,6707,6709],{"className":6708},[1116],[1118,6710,6711],{"xmlns":1120},[1122,6712,6713,6717],{},[1125,6714,6715],{},[1128,6716,1130],{},[1132,6718,1130],{"encoding":1134},[1109,6720,6722],{"className":6721,"ariaHidden":1139},[1138],[1109,6723,6725,6728],{"className":6724},[1143],[1109,6726],{"className":6727,"style":1148},[1147],[1109,6729,1130],{"className":6730,"style":1154},[1152,1153]," 被称为严格凸函数。凸函数的重要意义在于:局部最优即全局最优。",[1096,6733,6734],{"id":6734},"中值定理",[1101,6736,6737,6738,6766,6767,6818,6819,6871,6872,6948],{},"假设连续函数 ",[1109,6739,6741,6754],{"className":6740},[1112],[1109,6742,6744],{"className":6743},[1116],[1118,6745,6746],{"xmlns":1120},[1122,6747,6748,6752],{},[1125,6749,6750],{},[1128,6751,1130],{},[1132,6753,1130],{"encoding":1134},[1109,6755,6757],{"className":6756,"ariaHidden":1139},[1138],[1109,6758,6760,6763],{"className":6759},[1143],[1109,6761],{"className":6762,"style":1148},[1147],[1109,6764,1130],{"className":6765,"style":1154},[1152,1153]," 取值在闭区间 ",[1109,6768,6770,6791],{"className":6769},[1112],[1109,6771,6773],{"className":6772},[1116],[1118,6774,6775],{"xmlns":1120},[1122,6776,6777,6789],{},[1125,6778,6779,6781,6783,6785,6787],{},[1204,6780,5519],{"stretchy":1456},[1128,6782,5522],{},[1204,6784,1910],{"separator":1139},[1128,6786,5527],{},[1204,6788,5530],{"stretchy":1456},[1132,6790,5533],{"encoding":1134},[1109,6792,6794],{"className":6793,"ariaHidden":1139},[1138],[1109,6795,6797,6800,6803,6806,6809,6812,6815],{"className":6796},[1143],[1109,6798],{"className":6799,"style":1498},[1147],[1109,6801,5519],{"className":6802},[1508],[1109,6804,5522],{"className":6805},[1152,1153],[1109,6807,1910],{"className":6808},[1975],[1109,6810],{"className":6811,"style":1540},[1231],[1109,6813,5527],{"className":6814},[1152,1153],[1109,6816,5530],{"className":6817},[1515]," 上,并且在 ",[1109,6820,6822,6844],{"className":6821},[1112],[1109,6823,6825],{"className":6824},[1116],[1118,6826,6827],{"xmlns":1120},[1122,6828,6829,6841],{},[1125,6830,6831,6833,6835,6837,6839],{},[1204,6832,1457],{"stretchy":1456},[1128,6834,5522],{},[1204,6836,1910],{"separator":1139},[1128,6838,5527],{},[1204,6840,1462],{"stretchy":1456},[1132,6842,6843],{"encoding":1134},"(a,b)",[1109,6845,6847],{"className":6846,"ariaHidden":1139},[1138],[1109,6848,6850,6853,6856,6859,6862,6865,6868],{"className":6849},[1143],[1109,6851],{"className":6852,"style":1498},[1147],[1109,6854,1457],{"className":6855},[1508],[1109,6857,5522],{"className":6858},[1152,1153],[1109,6860,1910],{"className":6861},[1975],[1109,6863],{"className":6864,"style":1540},[1231],[1109,6866,5527],{"className":6867},[1152,1153],[1109,6869,1462],{"className":6870},[1515]," 上可导,那么一定可以找到一点 ",[1109,6873,6875,6902],{"className":6874},[1112],[1109,6876,6878],{"className":6877},[1116],[1118,6879,6880],{"xmlns":1120},[1122,6881,6882,6899],{},[1125,6883,6884,6887,6889,6891,6893,6895,6897],{},[1128,6885,6886],{},"ξ",[1204,6888,5895],{},[1204,6890,1457],{"stretchy":1456},[1128,6892,5522],{},[1204,6894,1910],{"separator":1139},[1128,6896,5527],{},[1204,6898,1462],{"stretchy":1456},[1132,6900,6901],{"encoding":1134},"\\xi \\in 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使得",[1101,6950,6951],{},[1109,6952,6954,7013],{"className":6953},[1112],[1109,6955,6957],{"className":6956},[1116],[1118,6958,6959],{"xmlns":1120},[1122,6960,6961,7010],{},[1125,6962,6963,6972,6974,6976,6978,6980],{},[6964,6965,6966,6968],"msup",{},[1128,6967,1130],{},[1204,6969,6971],{"mathvariant":1450,"lspace":6970,"rspace":6970},"0em","′",[1204,6973,1457],{"stretchy":1456},[1128,6975,6886],{},[1204,6977,1462],{"stretchy":1456},[1204,6979,1582],{},[4490,6981,6982,7002],{},[1125,6983,6984,6986,6988,6990,6992,6994,6996,6998,7000],{},[1128,6985,1130],{},[1204,6987,1457],{"stretchy":1456},[1128,6989,5527],{},[1204,6991,1462],{"stretchy":1456},[1204,6993,5991],{},[1128,6995,1130],{},[1204,6997,1457],{"stretchy":1456},[1128,6999,5522],{},[1204,7001,1462],{"stretchy":1456},[1125,7003,7004,7006,7008],{},[1128,7005,5527],{},[1204,7007,5991],{},[1128,7009,5522],{},[1132,7011,7012],{"encoding":1134},"f'(\\xi) = \\frac{f(b) - f(a)}{b - a}",[1109,7014,7016,7075],{"className":7015,"ariaHidden":1139},[1138],[1109,7017,7019,7023,7057,7060,7063,7066,7069,7072],{"className":7018},[1143],[1109,7020],{"className":7021,"style":7022},[1147],"height:1.0019em;vertical-align:-0.25em;",[1109,7024,7026,7029],{"className":7025},[1152],[1109,7027,1130],{"className":7028,"style":1154},[1152,1153],[1109,7030,7032],{"className":7031},[1293],[1109,7033,7035],{"className":7034},[1297],[1109,7036,7038],{"className":7037},[1302],[1109,7039,7042],{"className":7040,"style":7041},[1306],"height:0.7519em;",[1109,7043,7045,7048],{"style":7044},"top:-3.063em;margin-right:0.05em;",[1109,7046],{"className":7047,"style":1315},[1314],[1109,7049,7051],{"className":7050},[1319,1320,1321,1322],[1109,7052,7054],{"className":7053},[1152,1322],[1109,7055,6971],{"className":7056},[1152,1322],[1109,7058,1457],{"className":7059},[1508],[1109,7061,6886],{"className":7062,"style":6914},[1152,1153],[1109,7064,1462],{"className":7065},[1515],[1109,7067],{"className":7068,"style":1232},[1231],[1109,7070,1582],{"className":7071},[1236],[1109,7073],{"className":7074,"style":1232},[1231],[1109,7076,7078,7082],{"className":7077},[1143],[1109,7079],{"className":7080,"style":7081},[1147],"height:1.4133em;vertical-align:-0.4033em;",[1109,7083,7085,7088,7178],{"className":7084},[1152],[1109,7086],{"className":7087},[1508,4589],[1109,7089,7091],{"className":7090},[4490],[1109,7092,7094,7169],{"className":7093},[1297,1298],[1109,7095,7097,7166],{"className":7096},[1302],[1109,7098,7100,7120,7128],{"className":7099,"style":4602},[1306],[1109,7101,7102,7105],{"style":4605},[1109,7103],{"className":7104,"style":4609},[1314],[1109,7106,7108],{"className":7107},[1319,1320,1321,1322],[1109,7109,7111,7114,7117],{"className":7110},[1152,1322],[1109,7112,5527],{"className":7113},[1152,1153,1322],[1109,7115,5991],{"className":7116},[2501,1322],[1109,7118,5522],{"className":7119},[1152,1153,1322],[1109,7121,7122,7125],{"style":4630},[1109,7123],{"className":7124,"style":4609},[1314],[1109,7126],{"className":7127,"style":4638},[4637],[1109,7129,7130,7133],{"style":4641},[1109,7131],{"className":7132,"style":4609},[1314],[1109,7134,7136],{"className":7135},[1319,1320,1321,1322],[1109,7137,7139,7142,7145,7148,7151,7154,7157,7160,7163],{"className":7138},[1152,1322],[1109,7140,1130],{"className":7141,"style":1154},[1152,1153,1322],[1109,7143,1457],{"className":7144},[1508,1322],[1109,7146,5527],{"className":7147},[1152,1153,1322],[1109,7149,1462],{"className":7150},[1515,1322],[1109,7152,5991],{"className":7153},[2501,1322],[1109,7155,1130],{"className":7156,"style":1154},[1152,1153,1322],[1109,7158,1457],{"className":7159},[1508,1322],[1109,7161,5522],{"className":7162},[1152,1153,1322],[1109,7164,1462],{"className":7165},[1515,1322],[1109,7167,1330],{"className":7168},[1329],[1109,7170,7172],{"className":7171},[1302],[1109,7173,7176],{"className":7174,"style":7175},[1306],"height:0.4033em;",[1109,7177],{},[1109,7179],{"className":7180},[1515,4589],[1096,7182,7183],{"id":7183},"泰勒展开",[1101,7185,7186,7187,7215,7216,7285,7286,7316,7317,7345,7346,7374,7375,7444],{},"假设 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附近的区间中 ",[1109,7287,7289,7303],{"className":7288},[1112],[1109,7290,7292],{"className":7291},[1116],[1118,7293,7294],{"xmlns":1120},[1122,7295,7296,7301],{},[1125,7297,7298],{},[1128,7299,7300],{},"n",[1132,7302,7300],{"encoding":1134},[1109,7304,7306],{"className":7305,"ariaHidden":1139},[1138],[1109,7307,7309,7313],{"className":7308},[1143],[1109,7310],{"className":7311,"style":7312},[1147],"height:0.4306em;",[1109,7314,7300],{"className":7315},[1152,1153]," 阶可导,那么对任意 ",[1109,7318,7320,7333],{"className":7319},[1112],[1109,7321,7323],{"className":7322},[1116],[1118,7324,7325],{"xmlns":1120},[1122,7326,7327,7331],{},[1125,7328,7329],{},[1128,7330,1268],{},[1132,7332,1268],{"encoding":1134},[1109,7334,7336],{"className":7335,"ariaHidden":1139},[1138],[1109,7337,7339,7342],{"className":7338},[1143],[1109,7340],{"className":7341,"style":7312},[1147],[1109,7343,1268],{"className":7344},[1152,1153],",都可以把 ",[1109,7347,7349,7362],{"className":7348},[1112],[1109,7350,7352],{"className":7351},[1116],[1118,7353,7354],{"xmlns":1120},[1122,7355,7356,7360],{},[1125,7357,7358],{},[1128,7359,1130],{},[1132,7361,1130],{"encoding":1134},[1109,7363,7365],{"className":7364,"ariaHidden":1139},[1138],[1109,7366,7368,7371],{"className":7367},[1143],[1109,7369],{"className":7370,"style":1148},[1147],[1109,7372,1130],{"className":7373,"style":1154},[1152,1153]," 在 ",[1109,7376,7378,7395],{"className":7377},[1112],[1109,7379,7381],{"className":7380},[1116],[1118,7382,7383],{"xmlns":1120},[1122,7384,7385,7393],{},[1125,7386,7387],{},[1264,7388,7389,7391],{},[1128,7390,1268],{},[1208,7392,1210],{},[1132,7394,1273],{"encoding":1134},[1109,7396,7398],{"className":7397,"ariaHidden":1139},[1138],[1109,7399,7401,7404],{"className":7400},[1143],[1109,7402],{"className":7403,"style":1283},[1147],[1109,7405,7407,7410],{"className":7406},[1152],[1109,7408,1268],{"className":7409},[1152,1153],[1109,7411,7413],{"className":7412},[1293],[1109,7414,7416,7436],{"className":7415},[1297,1298],[1109,7417,7419,7433],{"className":7418},[1302],[1109,7420,7422],{"className":7421,"style":1307},[1306],[1109,7423,7424,7427],{"style":1310},[1109,7425],{"className":7426,"style":1315},[1314],[1109,7428,7430],{"className":7429},[1319,1320,1321,1322],[1109,7431,1210],{"className":7432},[1152,1322],[1109,7434,1330],{"className":7435},[1329],[1109,7437,7439],{"className":7438},[1302],[1109,7440,7442],{"className":7441,"style":1337},[1306],[1109,7443],{}," 处泰勒展开:",[1101,7446,7447],{},[1109,7448,7450,7650],{"className":7449},[1112],[1109,7451,7453],{"className":7452},[1116],[1118,7454,7455],{"xmlns":1120},[1122,7456,7457,7647],{},[1125,7458,7459,7461,7463,7465,7467,7469,7471,7473,7479,7481,7483,7489,7491,7497,7499,7501,7503,7505,7511,7513,7515,7546,7548,7550,7552,7558,7564,7566,7569,7571,7603,7605,7607,7609,7615,7621,7623,7625,7627,7629,7631,7633,7639,7645],{},[1128,7460,1130],{},[1204,7462,1457],{"stretchy":1456},[1128,7464,1268],{},[1204,7466,1462],{"stretchy":1456},[1204,7468,1582],{},[1128,7470,1130],{},[1204,7472,1457],{"stretchy":1456},[1264,7474,7475,7477],{},[1128,7476,1268],{},[1208,7478,1210],{},[1204,7480,1462],{"stretchy":1456},[1204,7482,2342],{},[6964,7484,7485,7487],{},[1128,7486,1130],{},[1204,7488,6971],{"mathvariant":1450,"lspace":6970,"rspace":6970},[1204,7490,1457],{"stretchy":1456},[1264,7492,7493,7495],{},[1128,7494,1268],{},[1208,7496,1210],{},[1204,7498,1462],{"stretchy":1456},[1204,7500,1457],{"stretchy":1456},[1128,7502,1268],{},[1204,7504,5991],{},[1264,7506,7507,7509],{},[1128,7508,1268],{},[1208,7510,1210],{},[1204,7512,1462],{"stretchy":1456},[1204,7514,2342],{},[4490,7516,7517,7539],{},[1125,7518,7519,7529,7531,7537],{},[6964,7520,7521,7523],{},[1128,7522,1130],{},[1125,7524,7525,7527],{},[1204,7526,6971],{"mathvariant":1450},[1204,7528,6971],{"mathvariant":1450},[1204,7530,1457],{"stretchy":1456},[1264,7532,7533,7535],{},[1128,7534,1268],{},[1208,7536,1210],{},[1204,7538,1462],{"stretchy":1456},[1125,7540,7541,7543],{},[1208,7542,1917],{},[1204,7544,7545],{"stretchy":1456},"!",[1204,7547,1457],{"stretchy":1456},[1128,7549,1268],{},[1204,7551,5991],{},[1264,7553,7554,7556],{},[1128,7555,1268],{},[1208,7557,1210],{},[6964,7559,7560,7562],{},[1204,7561,1462],{"stretchy":1456},[1208,7563,1917],{},[1204,7565,2342],{},[1204,7567,7568],{},"⋯",[1204,7570,2342],{},[4490,7572,7573,7597],{},[1125,7574,7575,7587,7589,7595],{},[6964,7576,7577,7579],{},[1128,7578,1130],{},[1125,7580,7581,7583,7585],{},[1204,7582,1457],{"stretchy":1456},[1128,7584,7300],{},[1204,7586,1462],{"stretchy":1456},[1204,7588,1457],{"stretchy":1456},[1264,7590,7591,7593],{},[1128,7592,1268],{},[1208,7594,1210],{},[1204,7596,1462],{"stretchy":1456},[1125,7598,7599,7601],{},[1128,7600,7300],{},[1204,7602,7545],{"stretchy":1456},[1204,7604,1457],{"stretchy":1456},[1128,7606,1268],{},[1204,7608,5991],{},[1264,7610,7611,7613],{},[1128,7612,1268],{},[1208,7614,1210],{},[6964,7616,7617,7619],{},[1204,7618,1462],{"stretchy":1456},[1128,7620,7300],{},[1204,7622,2342],{},[1128,7624,4385],{},[1204,7626,1457],{"fence":1456,"stretchy":1139,"minsize":5974,"maxsize":5974},[1204,7628,1457],{"stretchy":1456},[1128,7630,1268],{},[1204,7632,5991],{},[1264,7634,7635,7637],{},[1128,7636,1268],{},[1208,7638,1210],{},[6964,7640,7641,7643],{},[1204,7642,1462],{"stretchy":1456},[1128,7644,7300],{},[1204,7646,1462],{"fence":1456,"stretchy":1139,"minsize":5974,"maxsize":5974},[1132,7648,7649],{"encoding":1134},"f(x) = f(x_0) + f'(x_0)(x-x_0) + \\frac{f''(x_0)}{2!}(x-x_0)^2 + \\cdots + \\frac{f^{(n)}(x_0)}{n!}(x-x_0)^n + 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收敛到 ",[1109,9182,9184,9202],{"className":9183},[1112],[1109,9185,9187],{"className":9186},[1116],[1118,9188,9189],{"xmlns":1120},[1122,9190,9191,9199],{},[1125,9192,9193],{},[6964,9194,9195,9197],{},[1128,9196,1268],{},[1204,9198,8652],{},[1132,9200,9201],{"encoding":1134},"x^*",[1109,9203,9205],{"className":9204,"ariaHidden":1139},[1138],[1109,9206,9208,9212],{"className":9207},[1143],[1109,9209],{"className":9210,"style":9211},[1147],"height:0.6887em;",[1109,9213,9215,9218],{"className":9214},[1152],[1109,9216,1268],{"className":9217},[1152,1153],[1109,9219,9221],{"className":9220},[1293],[1109,9222,9224],{"className":9223},[1297],[1109,9225,9227],{"className":9226},[1302],[1109,9228,9230],{"className":9229,"style":9211},[1306],[1109,9231,9232,9235],{"style":7044},[1109,9233],{"className":9234,"style":1315},[1314],[1109,9236,9238],{"className":9237},[1319,1320,1321,1322],[1109,9239,8652],{"className":9240},[2501,1322]," 的速率为 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矩阵",[1101,9386,9387,9390],{},[1104,9388,9389],{},"梯度向量","(一阶偏导组成的列向量):",[1101,9392,9393],{},[1109,9394,9396,9475],{"className":9395},[1112],[1109,9397,9399],{"className":9398},[1116],[1118,9400,9401],{"xmlns":1120},[1122,9402,9403,9472],{},[1125,9404,9405,9407,9409,9411,9413,9415,9417],{},[1128,9406,5611],{"mathvariant":1450},[1128,9408,1130],{},[1204,9410,1457],{"stretchy":1456},[1128,9412,1268],{},[1204,9414,1462],{"stretchy":1456},[1204,9416,1582],{},[6964,9418,9419,9469],{},[1125,9420,9421,9423,9442,9444,9447,9449,9467],{},[1204,9422,1457],{"fence":1139},[4490,9424,9425,9432],{},[1125,9426,9427,9430],{},[1128,9428,9429],{"mathvariant":1450},"∂",[1128,9431,1130],{},[1125,9433,9434,9436],{},[1128,9435,9429],{"mathvariant":1450},[1264,9437,9438,9440],{},[1128,9439,1268],{},[1208,9441,1685],{},[1204,9443,1910],{"separator":1139},[1204,9445,9446],{},"…",[1204,9448,1910],{"separator":1139},[4490,9450,9451,9457],{},[1125,9452,9453,9455],{},[1128,9454,9429],{"mathvariant":1450},[1128,9456,1130],{},[1125,9458,9459,9461],{},[1128,9460,9429],{"mathvariant":1450},[1264,9462,9463,9465],{},[1128,9464,1268],{},[1128,9466,1101],{},[1204,9468,1462],{"fence":1139},[1128,9470,9471],{"mathvariant":1450},"⊤",[1132,9473,9474],{"encoding":1134},"\\nabla f(x) = \\left(\\frac{\\partial f}{\\partial x_1}, \\dots, \\frac{\\partial f}{\\partial x_p}\\right)^{\\top}",[1109,9476,9478,9508],{"className":9477,"ariaHidden":1139},[1138],[1109,9479,9481,9484,9487,9490,9493,9496,9499,9502,9505],{"className":9480},[1143],[1109,9482],{"className":9483,"style":1498},[1147],[1109,9485,5611],{"className":9486},[1152],[1109,9488,1130],{"className":9489,"style":1154},[1152,1153],[1109,9491,1457],{"className":9492},[1508],[1109,9494,1268],{"className":9495},[1152,1153],[1109,9497,1462],{"className":9498},[1515],[1109,9500],{"className":9501,"style":1232},[1231],[1109,9503,1582],{"className":9504},[1236],[1109,9506],{"className":9507,"style":1232},[1231],[1109,9509,9511,9515],{"className":9510},[1143],[1109,9512],{"className":9513,"style":9514},[1147],"height:2.039em;vertical-align:-0.65em;",[1109,9516,9518,9783],{"className":9517},[8175],[1109,9519,9521,9530,9645,9648,9651,9654,9657,9660,9663,9777],{"className":9520},[8175],[1109,9522,9526],{"className":9523,"style":9525},[1508,9524],"delimcenter","top:0em;",[1109,9527,1457],{"className":9528},[6066,9529],"size2",[1109,9531,9533,9536,9642],{"className":9532},[1152],[1109,9534],{"className":9535},[1508,4589],[1109,9537,9539],{"className":9538},[4490],[1109,9540,9542,9633],{"className":9541},[1297,1298],[1109,9543,9545,9630],{"className":9544},[1302],[1109,9546,9549,9604,9612],{"className":9547,"style":9548},[1306],"height:0.9322em;",[1109,9550,9551,9554],{"style":4605},[1109,9552],{"className":9553,"style":4609},[1314],[1109,9555,9557],{"className":9556},[1319,1320,1321,1322],[1109,9558,9560,9564],{"className":9559},[1152,1322],[1109,9561,9429],{"className":9562,"style":9563},[1152,1322],"margin-right:0.0556em;",[1109,9565,9567,9570],{"className":9566},[1152,1322],[1109,9568,1268],{"className":9569},[1152,1153,1322],[1109,9571,9573],{"className":9572},[1293],[1109,9574,9576,9596],{"className":9575},[1297,1298],[1109,9577,9579,9593],{"className":9578},[1302],[1109,9580,9582],{"className":9581,"style":8021},[1306],[1109,9583,9584,9587],{"style":8024},[1109,9585],{"className":9586,"style":7988},[1314],[1109,9588,9590],{"className":9589},[1319,7992,6067,1322],[1109,9591,1685],{"className":9592},[1152,1322],[1109,9594,1330],{"className":9595},[1329],[1109,9597,9599],{"className":9598},[1302],[1109,9600,9602],{"className":9601,"style":8043},[1306],[1109,9603],{},[1109,9605,9606,9609],{"style":4630},[1109,9607],{"className":9608,"style":4609},[1314],[1109,9610],{"className":9611,"style":4638},[4637],[1109,9613,9615,9618],{"style":9614},"top:-3.4461em;",[1109,9616],{"className":9617,"style":4609},[1314],[1109,9619,9621],{"className":9620},[1319,1320,1321,1322],[1109,9622,9624,9627],{"className":9623},[1152,1322],[1109,9625,9429],{"className":9626,"style":9563},[1152,1322],[1109,9628,1130],{"className":9629,"style":1154},[1152,1153,1322],[1109,9631,1330],{"className":9632},[1329],[1109,9634,9636],{"className":9635},[1302],[1109,9637,9640],{"className":9638,"style":9639},[1306],"height:0.4451em;",[1109,9641],{},[1109,9643],{"className":9644},[1515,4589],[1109,9646,1910],{"className":9647},[1975],[1109,9649],{"className":9650,"style":1540},[1231],[1109,9652,9446],{"className":9653},[8175],[1109,9655],{"className":9656,"style":1540},[1231],[1109,9658,1910],{"className":9659},[1975],[1109,9661],{"className":9662,"style":1540},[1231],[1109,9664,9666,9669,9774],{"className":9665},[1152],[1109,9667],{"className":9668},[1508,4589],[1109,9670,9672],{"className":9671},[4490],[1109,9673,9675,9765],{"className":9674},[1297,1298],[1109,9676,9678,9762],{"className":9677},[1302],[1109,9679,9681,9737,9745],{"className":9680,"style":9548},[1306],[1109,9682,9683,9686],{"style":4605},[1109,9684],{"className":9685,"style":4609},[1314],[1109,9687,9689],{"className":9688},[1319,1320,1321,1322],[1109,9690,9692,9695],{"className":9691},[1152,1322],[1109,9693,9429],{"className":9694,"style":9563},[1152,1322],[1109,9696,9698,9701],{"className":9697},[1152,1322],[1109,9699,1268],{"className":9700},[1152,1153,1322],[1109,9702,9704],{"className":9703},[1293],[1109,9705,9707,9728],{"className":9706},[1297,1298],[1109,9708,9710,9725],{"className":9709},[1302],[1109,9711,9714],{"className":9712,"style":9713},[1306],"height:0.1645em;",[1109,9715,9716,9719],{"style":8024},[1109,9717],{"className":9718,"style":7988},[1314],[1109,9720,9722],{"className":9721},[1319,7992,6067,1322],[1109,9723,1101],{"className":9724},[1152,1153,1322],[1109,9726,1330],{"className":9727},[1329],[1109,9729,9731],{"className":9730},[1302],[1109,9732,9735],{"className":9733,"style":9734},[1306],"height:0.2819em;",[1109,9736],{},[1109,9738,9739,9742],{"style":4630},[1109,9740],{"className":9741,"style":4609},[1314],[1109,9743],{"className":9744,"style":4638},[4637],[1109,9746,9747,9750],{"style":9614},[1109,9748],{"className":9749,"style":4609},[1314],[1109,9751,9753],{"className":9752},[1319,1320,1321,1322],[1109,9754,9756,9759],{"className":9755},[1152,1322],[1109,9757,9429],{"className":9758,"style":9563},[1152,1322],[1109,9760,1130],{"className":9761,"style":1154},[1152,1153,1322],[1109,9763,1330],{"className":9764},[1329],[1109,9766,9768],{"className":9767},[1302],[1109,9769,9772],{"className":9770,"style":9771},[1306],"height:0.5423em;",[1109,9773],{},[1109,9775],{"className":9776},[1515,4589],[1109,9778,9780],{"className":9779,"style":9525},[1515,9524],[1109,9781,1462],{"className":9782},[6066,9529],[1109,9784,9786],{"className":9785},[1293],[1109,9787,9789],{"className":9788},[1297],[1109,9790,9792],{"className":9791},[1302],[1109,9793,9796],{"className":9794,"style":9795},[1306],"height:1.389em;",[1109,9797,9799,9802],{"style":9798},"top:-3.6029em;margin-right:0.05em;",[1109,9800],{"className":9801,"style":1315},[1314],[1109,9803,9805],{"className":9804},[1319,1320,1321,1322],[1109,9806,9808],{"className":9807},[1152,1322],[1109,9809,9471],{"className":9810},[1152,1322],[1101,9812,9813,9816],{},[1104,9814,9815],{},"Hessian 矩阵","(二阶偏导组成的对称方阵):",[1101,9818,9819],{},[1109,9820,9822,9899],{"className":9821},[1112],[1109,9823,9825],{"className":9824},[1116],[1118,9826,9827],{"xmlns":1120},[1122,9828,9829,9896],{},[1125,9830,9831,9837,9839,9841,9843,9845,9847],{},[6964,9832,9833,9835],{},[1128,9834,5611],{"mathvariant":1450},[1208,9836,1917],{},[1128,9838,1130],{},[1204,9840,1457],{"stretchy":1456},[1128,9842,1268],{},[1204,9844,1462],{"stretchy":1456},[1204,9846,1582],{},[1264,9848,9849,9887],{},[1125,9850,9851,9853,9885],{},[1204,9852,5519],{"fence":1139},[4490,9854,9855,9865],{},[1125,9856,9857,9863],{},[6964,9858,9859,9861],{},[1128,9860,9429],{"mathvariant":1450},[1208,9862,1917],{},[1128,9864,1130],{},[1125,9866,9867,9869,9876,9878],{},[1128,9868,9429],{"mathvariant":1450},[1264,9870,9871,9873],{},[1128,9872,1268],{},[1128,9874,9875],{},"i",[1128,9877,9429],{"mathvariant":1450},[1264,9879,9880,9882],{},[1128,9881,1268],{},[1128,9883,9884],{},"j",[1204,9886,5530],{"fence":1139},[1125,9888,9889,9891,9894],{},[1128,9890,1101],{},[1204,9892,9893],{},"×",[1128,9895,1101],{},[1132,9897,9898],{"encoding":1134},"\\nabla^2 f(x) = \\left[\\frac{\\partial^2 f}{\\partial x_i \\partial x_j}\\right]_{p\\times p}",[1109,9900,9902,9958],{"className":9901,"ariaHidden":1139},[1138],[1109,9903,9905,9908,9937,9940,9943,9946,9949,9952,9955],{"className":9904},[1143],[1109,9906],{"className":9907,"style":8085},[1147],[1109,9909,9911,9914],{"className":9910},[1152],[1109,9912,5611],{"className":9913},[1152],[1109,9915,9917],{"className":9916},[1293],[1109,9918,9920],{"className":9919},[1297],[1109,9921,9923],{"className":9922},[1302],[1109,9924,9926],{"className":9925,"style":8144},[1306],[1109,9927,9928,9931],{"style":7044},[1109,9929],{"className":9930,"style":1315},[1314],[1109,9932,9934],{"className":9933},[1319,1320,1321,1322],[1109,9935,1917],{"className":9936},[1152,1322],[1109,9938,1130],{"className":9939,"style":1154},[1152,1153],[1109,9941,1457],{"className":9942},[1508],[1109,9944,1268],{"className":9945},[1152,1153],[1109,9947,1462],{"className":9948},[1515],[1109,9950],{"className":9951,"style":1232},[1231],[1109,9953,1582],{"className":9954},[1236],[1109,9956],{"className":9957,"style":1232},[1231],[1109,9959,9961,9965],{"className":9960},[1143],[1109,9962],{"className":9963,"style":9964},[1147],"height:1.9858em;vertical-align:-0.8358em;",[1109,9966,9968,10167],{"className":9967},[8175],[1109,9969,9971,9977,10161],{"className":9970},[8175],[1109,9972,9974],{"className":9973,"style":9525},[1508,9524],[1109,9975,5519],{"className":9976},[6066,9529],[1109,9978,9980,9983,10158],{"className":9979},[1152],[1109,9981],{"className":9982},[1508,4589],[1109,9984,9986],{"className":9985},[4490],[1109,9987,9989,10150],{"className":9988},[1297,1298],[1109,9990,9992,10147],{"className":9991},[1302],[1109,9993,9996,10095,10103],{"className":9994,"style":9995},[1306],"height:1.07em;",[1109,9997,9998,10001],{"style":4605},[1109,9999],{"className":10000,"style":4609},[1314],[1109,10002,10004],{"className":10003},[1319,1320,1321,1322],[1109,10005,10007,10010,10051,10054],{"className":10006},[1152,1322],[1109,10008,9429],{"className":10009,"style":9563},[1152,1322],[1109,10011,10013,10016],{"className":10012},[1152,1322],[1109,10014,1268],{"className":10015},[1152,1153,1322],[1109,10017,10019],{"className":10018},[1293],[1109,10020,10022,10043],{"className":10021},[1297,1298],[1109,10023,10025,10040],{"className":10024},[1302],[1109,10026,10029],{"className":10027,"style":10028},[1306],"height:0.3281em;",[1109,10030,10031,10034],{"style":8024},[1109,10032],{"className":10033,"style":7988},[1314],[1109,10035,10037],{"className":10036},[1319,7992,6067,1322],[1109,10038,9875],{"className":10039},[1152,1153,1322],[1109,10041,1330],{"className":10042},[1329],[1109,10044,10046],{"className":10045},[1302],[1109,10047,10049],{"className":10048,"style":8043},[1306],[1109,10050],{},[1109,10052,9429],{"className":10053,"style":9563},[1152,1322],[1109,10055,10057,10060],{"className":10056},[1152,1322],[1109,10058,1268],{"className":10059},[1152,1153,1322],[1109,10061,10063],{"className":10062},[1293],[1109,10064,10066,10087],{"className":10065},[1297,1298],[1109,10067,10069,10084],{"className":10068},[1302],[1109,10070,10072],{"className":10071,"style":10028},[1306],[1109,10073,10074,10077],{"style":8024},[1109,10075],{"className":10076,"style":7988},[1314],[1109,10078,10080],{"className":10079},[1319,7992,6067,1322],[1109,10081,9884],{"className":10082,"style":10083},[1152,1153,1322],"margin-right:0.0572em;",[1109,10085,1330],{"className":10086},[1329],[1109,10088,10090],{"className":10089},[1302],[1109,10091,10093],{"className":10092,"style":9734},[1306],[1109,10094],{},[1109,10096,10097,10100],{"style":4630},[1109,10098],{"className":10099,"style":4609},[1314],[1109,10101],{"className":10102,"style":4638},[4637],[1109,10104,10105,10108],{"style":9614},[1109,10106],{"className":10107,"style":4609},[1314],[1109,10109,10111],{"className":10110},[1319,1320,1321,1322],[1109,10112,10114,10144],{"className":10113},[1152,1322],[1109,10115,10117,10120],{"className":10116},[1152,1322],[1109,10118,9429],{"className":10119,"style":9563},[1152,1322],[1109,10121,10123],{"className":10122},[1293],[1109,10124,10126],{"className":10125},[1297],[1109,10127,10129],{"className":10128},[1302],[1109,10130,10133],{"className":10131,"style":10132},[1306],"height:0.8913em;",[1109,10134,10135,10138],{"style":7984},[1109,10136],{"className":10137,"style":7988},[1314],[1109,10139,10141],{"className":10140},[1319,7992,6067,1322],[1109,10142,1917],{"className":10143},[1152,1322],[1109,10145,1130],{"className":10146,"style":1154},[1152,1153,1322],[1109,10148,1330],{"className":10149},[1329],[1109,10151,10153],{"className":10152},[1302],[1109,10154,10156],{"className":10155,"style":9771},[1306],[1109,10157],{},[1109,10159],{"className":10160},[1515,4589],[1109,10162,10164],{"className":10163,"style":9525},[1515,9524],[1109,10165,5530],{"className":10166},[6066,9529],[1109,10168,10170],{"className":10169},[1293],[1109,10171,10173,10204],{"className":10172},[1297,1298],[1109,10174,10176,10201],{"className":10175},[1302],[1109,10177,10180],{"className":10178,"style":10179},[1306],"height:-0.2914em;",[1109,10181,10183,10186],{"style":10182},"top:-2.0003em;margin-right:0.05em;",[1109,10184],{"className":10185,"style":1315},[1314],[1109,10187,10189],{"className":10188},[1319,1320,1321,1322],[1109,10190,10192,10195,10198],{"className":10191},[1152,1322],[1109,10193,1101],{"className":10194},[1152,1153,1322],[1109,10196,9893],{"className":10197},[2501,1322],[1109,10199,1101],{"className":10200},[1152,1153,1322],[1109,10202,1330],{"className":10203},[1329],[1109,10205,10207],{"className":10206},[1302],[1109,10208,10211],{"className":10209,"style":10210},[1306],"height:0.8358em;",[1109,10212],{},[10214,10215],"hr",{},[1091,10217,10219],{"id":10218},"二牛顿迭代法newtons-method","二、牛顿迭代法(Newton's method)",[1101,10221,10222,10223,10321,10322,10350,10351,10354,10355,10424,10425,7374,10453,10522],{},"对于优化问题 ",[1109,10224,10226,10257],{"className":10225},[1112],[1109,10227,10229],{"className":10228},[1116],[1118,10230,10231],{"xmlns":1120},[1122,10232,10233,10254],{},[1125,10234,10235,10246,10248,10250,10252],{},[1264,10236,10237,10244],{},[1125,10238,10239,10242],{},[1128,10240,10241],{},"min",[1204,10243,2412],{},[1128,10245,1268],{},[1128,10247,1130],{},[1204,10249,1457],{"stretchy":1456},[1128,10251,1268],{},[1204,10253,1462],{"stretchy":1456},[1132,10255,10256],{"encoding":1134},"\\min_x f(x)",[1109,10258,10260],{"className":10259,"ariaHidden":1139},[1138],[1109,10261,10263,10266,10306,10309,10312,10315,10318],{"className":10262},[1143],[1109,10264],{"className":10265,"style":1498},[1147],[1109,10267,10269,10272],{"className":10268},[2878],[1109,10270,10241],{"className":10271},[2878],[1109,10273,10275],{"className":10274},[1293],[1109,10276,10278,10298],{"className":10277},[1297,1298],[1109,10279,10281,10295],{"className":10280},[1302],[1109,10282,10284],{"className":10283,"style":4547},[1306],[1109,10285,10286,10289],{"style":4550},[1109,10287],{"className":10288,"style":1315},[1314],[1109,10290,10292],{"className":10291},[1319,1320,1321,1322],[1109,10293,1268],{"className":10294},[1152,1153,1322],[1109,10296,1330],{"className":10297},[1329],[1109,10299,10301],{"className":10300},[1302],[1109,10302,10304],{"className":10303,"style":1337},[1306],[1109,10305],{},[1109,10307],{"className":10308,"style":1540},[1231],[1109,10310,1130],{"className":10311,"style":1154},[1152,1153],[1109,10313,1457],{"className":10314},[1508],[1109,10316,1268],{"className":10317},[1152,1153],[1109,10319,1462],{"className":10320},[1515],",这里要求 ",[1109,10323,10325,10338],{"className":10324},[1112],[1109,10326,10328],{"className":10327},[1116],[1118,10329,10330],{"xmlns":1120},[1122,10331,10332,10336],{},[1125,10333,10334],{},[1128,10335,1130],{},[1132,10337,1130],{"encoding":1134},[1109,10339,10341],{"className":10340,"ariaHidden":1139},[1138],[1109,10342,10344,10347],{"className":10343},[1143],[1109,10345],{"className":10346,"style":1148},[1147],[1109,10348,1130],{"className":10349,"style":1154},[1152,1153]," 为",[1104,10352,10353],{},"二阶可导的凸函数","。在给定当前估计 ",[1109,10356,10358,10375],{"className":10357},[1112],[1109,10359,10361],{"className":10360},[1116],[1118,10362,10363],{"xmlns":1120},[1122,10364,10365,10373],{},[1125,10366,10367],{},[1264,10368,10369,10371],{},[1128,10370,1268],{},[1128,10372,8620],{},[1132,10374,9130],{"encoding":1134},[1109,10376,10378],{"className":10377,"ariaHidden":1139},[1138],[1109,10379,10381,10384],{"className":10380},[1143],[1109,10382],{"className":10383,"style":1283},[1147],[1109,10385,10387,10390],{"className":10386},[1152],[1109,10388,1268],{"className":10389},[1152,1153],[1109,10391,10393],{"className":10392},[1293],[1109,10394,10396,10416],{"className":10395},[1297,1298],[1109,10397,10399,10413],{"className":10398},[1302],[1109,10400,10402],{"className":10401,"style":8735},[1306],[1109,10403,10404,10407],{"style":1310},[1109,10405],{"className":10406,"style":1315},[1314],[1109,10408,10410],{"className":10409},[1319,1320,1321,1322],[1109,10411,8620],{"className":10412,"style":8750},[1152,1153,1322],[1109,10414,1330],{"className":10415},[1329],[1109,10417,10419],{"className":10418},[1302],[1109,10420,10422],{"className":10421,"style":1337},[1306],[1109,10423],{}," 时,把 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处做二阶泰勒展开并对其极小化,可得迭代式:",[1101,10524,10525],{},[1109,10526,10528,10606],{"className":10527},[1112],[1109,10529,10531],{"className":10530},[1116],[1118,10532,10533],{"xmlns":1120},[1122,10534,10535,10603],{},[1125,10536,10537,10549,10551,10557,10559,10561,10567,10569,10571,10577,10579,10589,10591,10593,10595,10601],{},[1264,10538,10539,10541],{},[1128,10540,1268],{},[1125,10542,10543,10545,10547],{},[1128,10544,8620],{},[1204,10546,2342],{},[1208,10548,1685],{},[1204,10550,1582],{},[1264,10552,10553,10555],{},[1128,10554,1268],{},[1128,10556,8620],{},[1204,10558,5991],{},[1204,10560,5519],{"fence":1456,"stretchy":1139,"minsize":5974,"maxsize":5974},[6964,10562,10563,10565],{},[1128,10564,5611],{"mathvariant":1450},[1208,10566,1917],{},[1128,10568,1130],{},[1204,10570,1457],{"stretchy":1456},[1264,10572,10573,10575],{},[1128,10574,1268],{},[1128,10576,8620],{},[1204,10578,1462],{"stretchy":1456},[6964,10580,10581,10583],{},[1204,10582,5530],{"fence":1456,"stretchy":1139,"minsize":5974,"maxsize":5974},[1125,10584,10585,10587],{},[1204,10586,5991],{},[1208,10588,1685],{},[1128,10590,5611],{"mathvariant":1450},[1128,10592,1130],{},[1204,10594,1457],{"stretchy":1456},[1264,10596,10597,10599],{},[1128,10598,1268],{},[1128,10600,8620],{},[1204,10602,1462],{"stretchy":1456},[1132,10604,10605],{"encoding":1134},"x_{k+1} = x_k - \\big[\\nabla^2 f(x_k)\\big]^{-1}\\nabla 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f(x_k)+\\nabla f(x_k)^{\\top}(x-x_k)+\\frac{1}{2}(x-x_k)^{\\top}\\nabla^2 f(x_k)(x-x_k)",[1109,11051,11053,11080,11144,11248,11306,11398,11569],{"className":11052,"ariaHidden":1139},[1138],[1109,11054,11056,11059,11062,11065,11068,11071,11074,11077],{"className":11055},[1143],[1109,11057],{"className":11058,"style":1498},[1147],[1109,11060,1130],{"className":11061,"style":1154},[1152,1153],[1109,11063,1457],{"className":11064},[1508],[1109,11066,1268],{"className":11067},[1152,1153],[1109,11069,1462],{"className":11070},[1515],[1109,11072],{"className":11073,"style":1232},[1231],[1109,11075,10942],{"className":11076},[1236],[1109,11078],{"className":11079,"style":1232},[1231],[1109,11081,11083,11086,11089,11092,11132,11135,11138,11141],{"className":11082},[1143],[1109,11084],{"className":11085,"style":1498},[1147],[1109,11087,1130],{"className":11088,"style":1154},[1152,1153],[1109,11090,1457],{"className":11091},[1508],[1109,11093,11095,11098],{"className":11094},[1152],[1109,11096,1268],{"className":11097},[1152,1153],[1109,11099,11101],{"className":11100},[1293],[1109,11102,11104,11124],{"className":11103},[1297,1298],[1109,11105,11107,11121],{"className":11106},[1302],[1109,11108,11110],{"className":11109,"style":8735},[1306],[1109,11111,11112,11115],{"style":1310},[1109,11113],{"className":11114,"style":1315},[1314],[1109,11116,11118],{"className":11117},[1319,1320,1321,1322],[1109,11119,8620],{"className":11120,"style":8750},[1152,1153,1322],[1109,11122,1330],{"className":11123},[1329],[1109,11125,11127],{"className":11126},[1302],[1109,11128,11130],{"className":11129,"style":1337},[1306],[1109,11131],{},[1109,11133,1462],{"className":11134},[1515],[1109,11136],{"className":11137,"style":2497},[1231],[1109,11139,2342],{"className":11140},[2501],[1109,11142],{"className":11143,"style":2497},[1231],[1109,11145,11147,11151,11154,11157,11160,11200,11233,11236,11239,11242,11245],{"className":11146},[1143],[1109,11148],{"className":11149,"style":11150},[1147],"height:1.0991em;vertical-align:-0.25em;",[1109,11152,5611],{"className":11153},[1152],[1109,11155,1130],{"className":11156,"style":1154},[1152,1153],[1109,11158,1457],{"className":11159},[1508],[1109,11161,11163,11166],{"className":11162},[1152],[1109,11164,1268],{"className":11165},[1152,1153],[1109,11167,11169],{"className":11168},[1293],[1109,11170,11172,11192],{"className":11171},[1297,1298],[1109,11173,11175,11189],{"className":11174},[1302],[1109,11176,11178],{"className":11177,"style":8735},[1306],[1109,11179,11180,11183],{"style":1310},[1109,11181],{"className":11182,"style":1315},[1314],[1109,11184,11186],{"className":11185},[1319,1320,1321,1322],[1109,11187,8620],{"className":11188,"style":8750},[1152,1153,1322],[1109,11190,1330],{"className":11191},[1329],[1109,11193,11195],{"className":11194},[1302],[1109,11196,11198],{"className":11197,"style":1337},[1306],[1109,11199],{},[1109,11201,11203,11206],{"className":11202},[1515],[1109,11204,1462],{"className":11205},[1515],[1109,11207,11209],{"className":11208},[1293],[1109,11210,11212],{"className":11211},[1297],[1109,11213,11215],{"className":11214},[1302],[1109,11216,11219],{"className":11217,"style":11218},[1306],"height:0.8491em;",[1109,11220,11221,11224],{"style":7044},[1109,11222],{"className":11223,"style":1315},[1314],[1109,11225,11227],{"className":11226},[1319,1320,1321,1322],[1109,11228,11230],{"className":11229},[1152,1322],[1109,11231,9471],{"className":11232},[1152,1322],[1109,11234,1457],{"className":11235},[1508],[1109,11237,1268],{"className":11238},[1152,1153],[1109,11240],{"className":11241,"style":2497},[1231],[1109,11243,5991],{"className":11244},[2501],[1109,11246],{"className":11247,"style":2497},[1231],[1109,11249,11251,11254,11294,11297,11300,11303],{"className":11250},[1143],[1109,11252],{"className":11253,"style":1498},[1147],[1109,11255,11257,11260],{"className":11256},[1152],[1109,11258,1268],{"className":11259},[1152,1153],[1109,11261,11263],{"className":11262},[1293],[1109,11264,11266,11286],{"className":11265},[1297,1298],[1109,11267,11269,11283],{"className":11268},[1302],[1109,11270,11272],{"className":11271,"style":8735},[1306],[1109,11273,11274,11277],{"style":1310},[1109,11275],{"className":11276,"style":1315},[1314],[1109,11278,11280],{"className":11279},[1319,1320,1321,1322],[1109,11281,8620],{"className":11282,"style":8750},[1152,1153,1322],[1109,11284,1330],{"className":11285},[1329],[1109,11287,11289],{"className":11288},[1302],[1109,11290,11292],{"className":11291,"style":1337},[1306],[1109,11293],{},[1109,11295,1462],{"className":11296},[1515],[1109,11298],{"className":11299,"style":2497},[1231],[1109,11301,2342],{"className":11302},[2501],[1109,11304],{"className":11305,"style":2497},[1231],[1109,11307,11309,11313,11383,11386,11389,11392,11395],{"className":11308},[1143],[1109,11310],{"className":11311,"style":11312},[1147],"height:1.1901em;vertical-align:-0.345em;",[1109,11314,11316,11319,11380],{"className":11315},[1152],[1109,11317],{"className":11318},[1508,4589],[1109,11320,11322],{"className":11321},[4490],[1109,11323,11325,11372],{"className":11324},[1297,1298],[1109,11326,11328,11369],{"className":11327},[1302],[1109,11329,11332,11346,11354],{"className":11330,"style":11331},[1306],"height:0.8451em;",[1109,11333,11334,11337],{"style":4605},[1109,11335],{"className":11336,"style":4609},[1314],[1109,11338,11340],{"className":11339},[1319,1320,1321,1322],[1109,11341,11343],{"className":11342},[1152,1322],[1109,11344,1917],{"className":11345},[1152,1322],[1109,11347,11348,11351],{"style":4630},[1109,11349],{"className":11350,"style":4609},[1314],[1109,11352],{"className":11353,"style":4638},[4637],[1109,11355,11357,11360],{"style":11356},"top:-3.394em;",[1109,11358],{"className":11359,"style":4609},[1314],[1109,11361,11363],{"className":11362},[1319,1320,1321,1322],[1109,11364,11366],{"className":11365},[1152,1322],[1109,11367,1685],{"className":11368},[1152,1322],[1109,11370,1330],{"className":11371},[1329],[1109,11373,11375],{"className":11374},[1302],[1109,11376,11378],{"className":11377,"style":8058},[1306],[1109,11379],{},[1109,11381],{"className":11382},[1515,4589],[1109,11384,1457],{"className":11385},[1508],[1109,11387,1268],{"className":11388},[1152,1153],[1109,11390],{"className":11391,"style":2497},[1231],[1109,11393,5991],{"className":11394},[2501],[1109,11396],{"className":11397,"style":2497},[1231],[1109,11399,11401,11404,11444,11476,11505,11508,11511,11551,11554,11557,11560,11563,11566],{"className":11400},[1143],[1109,11402],{"className":11403,"style":11150},[1147],[1109,11405,11407,11410],{"className":11406},[1152],[1109,11408,1268],{"className":11409},[1152,1153],[1109,11411,11413],{"className":11412},[1293],[1109,11414,11416,11436],{"className":11415},[1297,1298],[1109,11417,11419,11433],{"className":11418},[1302],[1109,11420,11422],{"className":11421,"style":8735},[1306],[1109,11423,11424,11427],{"style":1310},[1109,11425],{"className":11426,"style":1315},[1314],[1109,11428,11430],{"className":11429},[1319,1320,1321,1322],[1109,11431,8620],{"className":11432,"style":8750},[1152,1153,1322],[1109,11434,1330],{"className":11435},[1329],[1109,11437,11439],{"className":11438},[1302],[1109,11440,11442],{"className":11441,"style":1337},[1306],[1109,11443],{},[1109,11445,11447,11450],{"className":11446},[1515],[1109,11448,1462],{"className":11449},[1515],[1109,11451,11453],{"className":11452},[1293],[1109,11454,11456],{"className":11455},[1297],[1109,11457,11459],{"className":11458},[1302],[1109,11460,11462],{"className":11461,"style":11218},[1306],[1109,11463,11464,11467],{"style":7044},[1109,11465],{"className":11466,"style":1315},[1314],[1109,11468,11470],{"className":11469},[1319,1320,1321,1322],[1109,11471,11473],{"className":11472},[1152,1322],[1109,11474,9471],{"className":11475},[1152,1322],[1109,11477,11479,11482],{"className":11478},[1152],[1109,11480,5611],{"className":11481},[1152],[1109,11483,11485],{"className":11484},[1293],[1109,11486,11488],{"className":11487},[1297],[1109,11489,11491],{"className":11490},[1302],[1109,11492,11494],{"className":11493,"style":8144},[1306],[1109,11495,11496,11499],{"style":7044},[1109,11497],{"className":11498,"style":1315},[1314],[1109,11500,11502],{"className":11501},[1319,1320,1321,1322],[1109,11503,1917],{"className":11504},[1152,1322],[1109,11506,1130],{"className":11507,"style":1154},[1152,1153],[1109,11509,1457],{"className":11510},[1508],[1109,11512,11514,11517],{"className":11513},[1152],[1109,11515,1268],{"className":11516},[1152,1153],[1109,11518,11520],{"className":11519},[1293],[1109,11521,11523,11543],{"className":11522},[1297,1298],[1109,11524,11526,11540],{"className":11525},[1302],[1109,11527,11529],{"className":11528,"style":8735},[1306],[1109,11530,11531,11534],{"style":1310},[1109,11532],{"className":11533,"style":1315},[1314],[1109,11535,11537],{"className":11536},[1319,1320,1321,1322],[1109,11538,8620],{"className":11539,"style":8750},[1152,1153,1322],[1109,11541,1330],{"className":11542},[1329],[1109,11544,11546],{"className":11545},[1302],[1109,11547,11549],{"className":11548,"style":1337},[1306],[1109,11550],{},[1109,11552,1462],{"className":11553},[1515],[1109,11555,1457],{"className":11556},[1508],[1109,11558,1268],{"className":11559},[1152,1153],[1109,11561],{"className":11562,"style":2497},[1231],[1109,11564,5991],{"className":11565},[2501],[1109,11567],{"className":11568,"style":2497},[1231],[1109,11570,11572,11575,11615],{"className":11571},[1143],[1109,11573],{"className":11574,"style":1498},[1147],[1109,11576,11578,11581],{"className":11577},[1152],[1109,11579,1268],{"className":11580},[1152,1153],[1109,11582,11584],{"className":11583},[1293],[1109,11585,11587,11607],{"className":11586},[1297,1298],[1109,11588,11590,11604],{"className":11589},[1302],[1109,11591,11593],{"className":11592,"style":8735},[1306],[1109,11594,11595,11598],{"style":1310},[1109,11596],{"className":11597,"style":1315},[1314],[1109,11599,11601],{"className":11600},[1319,1320,1321,1322],[1109,11602,8620],{"className":11603,"style":8750},[1152,1153,1322],[1109,11605,1330],{"className":11606},[1329],[1109,11608,11610],{"className":11609},[1302],[1109,11611,11613],{"className":11612,"style":1337},[1306],[1109,11614],{},[1109,11616,1462],{"className":11617},[1515]," 求导并令导数为 0,得 ",[1109,11620,11622,11686],{"className":11621},[1112],[1109,11623,11625],{"className":11624},[1116],[1118,11626,11627],{"xmlns":1120},[1122,11628,11629,11683],{},[1125,11630,11631,11633,11635,11637,11643,11645,11647,11653,11655,11657,11663,11665,11667,11669,11671,11677,11679,11681],{},[1128,11632,5611],{"mathvariant":1450},[1128,11634,1130],{},[1204,11636,1457],{"stretchy":1456},[1264,11638,11639,11641],{},[1128,11640,1268],{},[1128,11642,8620],{},[1204,11644,1462],{"stretchy":1456},[1204,11646,2342],{},[6964,11648,11649,11651],{},[1128,11650,5611],{"mathvariant":1450},[1208,11652,1917],{},[1128,11654,1130],{},[1204,11656,1457],{"stretchy":1456},[1264,11658,11659,11661],{},[1128,11660,1268],{},[1128,11662,8620],{},[1204,11664,1462],{"stretchy":1456},[1204,11666,1457],{"stretchy":1456},[1128,11668,1268],{},[1204,11670,5991],{},[1264,11672,11673,11675],{},[1128,11674,1268],{},[1128,11676,8620],{},[1204,11678,1462],{"stretchy":1456},[1204,11680,1582],{},[1208,11682,1210],{},[1132,11684,11685],{"encoding":1134},"\\nabla f(x_k) + \\nabla^2 f(x_k)(x - x_k) = 0",[1109,11687,11689,11756,11855,11913],{"className":11688,"ariaHidden":1139},[1138],[1109,11690,11692,11695,11698,11701,11704,11744,11747,11750,11753],{"className":11691},[1143],[1109,11693],{"className":11694,"style":1498},[1147],[1109,11696,5611],{"className":11697},[1152],[1109,11699,1130],{"className":11700,"style":1154},[1152,1153],[1109,11702,1457],{"className":11703},[1508],[1109,11705,11707,11710],{"className":11706},[1152],[1109,11708,1268],{"className":11709},[1152,1153],[1109,11711,11713],{"className":11712},[1293],[1109,11714,11716,11736],{"className":11715},[1297,1298],[1109,11717,11719,11733],{"className":11718},[1302],[1109,11720,11722],{"className":11721,"style":8735},[1306],[1109,11723,11724,11727],{"style":1310},[1109,11725],{"className":11726,"style":1315},[1314],[1109,11728,11730],{"className":11729},[1319,1320,1321,1322],[1109,11731,8620],{"className":11732,"style":8750},[1152,1153,1322],[1109,11734,1330],{"className":11735},[1329],[1109,11737,11739],{"className":11738},[1302],[1109,11740,11742],{"className":11741,"style":1337},[1306],[1109,11743],{},[1109,11745,1462],{"className":11746},[1515],[1109,11748],{"className":11749,"style":2497},[1231],[1109,11751,2342],{"className":11752},[2501],[1109,11754],{"className":11755,"style":2497},[1231],[1109,11757,11759,11762,11791,11794,11797,11837,11840,11843,11846,11849,11852],{"className":11758},[1143],[1109,11760],{"className":11761,"style":8085},[1147],[1109,11763,11765,11768],{"className":11764},[1152],[1109,11766,5611],{"className":11767},[1152],[1109,11769,11771],{"className":11770},[1293],[1109,11772,11774],{"className":11773},[1297],[1109,11775,11777],{"className":11776},[1302],[1109,11778,11780],{"className":11779,"style":8144},[1306],[1109,11781,11782,11785],{"style":7044},[1109,11783],{"className":11784,"style":1315},[1314],[1109,11786,11788],{"className":11787},[1319,1320,1321,1322],[1109,11789,1917],{"className":11790},[1152,1322],[1109,11792,1130],{"className":11793,"style":1154},[1152,1153],[1109,11795,1457],{"className":11796},[1508],[1109,11798,11800,11803],{"className":11799},[1152],[1109,11801,1268],{"className":11802},[1152,1153],[1109,11804,11806],{"className":11805},[1293],[1109,11807,11809,11829],{"className":11808},[1297,1298],[1109,11810,11812,11826],{"className":11811},[1302],[1109,11813,11815],{"className":11814,"style":8735},[1306],[1109,11816,11817,11820],{"style":1310},[1109,11818],{"className":11819,"style":1315},[1314],[1109,11821,11823],{"className":11822},[1319,1320,1321,1322],[1109,11824,8620],{"className":11825,"style":8750},[1152,1153,1322],[1109,11827,1330],{"className":11828},[1329],[1109,11830,11832],{"className":11831},[1302],[1109,11833,11835],{"className":11834,"style":1337},[1306],[1109,11836],{},[1109,11838,1462],{"className":11839},[1515],[1109,11841,1457],{"className":11842},[1508],[1109,11844,1268],{"className":11845},[1152,1153],[1109,11847],{"className":11848,"style":2497},[1231],[1109,11850,5991],{"className":11851},[2501],[1109,11853],{"className":11854,"style":2497},[1231],[1109,11856,11858,11861,11901,11904,11907,11910],{"className":11857},[1143],[1109,11859],{"className":11860,"style":1498},[1147],[1109,11862,11864,11867],{"className":11863},[1152],[1109,11865,1268],{"className":11866},[1152,1153],[1109,11868,11870],{"className":11869},[1293],[1109,11871,11873,11893],{"className":11872},[1297,1298],[1109,11874,11876,11890],{"className":11875},[1302],[1109,11877,11879],{"className":11878,"style":8735},[1306],[1109,11880,11881,11884],{"style":1310},[1109,11882],{"className":11883,"style":1315},[1314],[1109,11885,11887],{"className":11886},[1319,1320,1321,1322],[1109,11888,8620],{"className":11889,"style":8750},[1152,1153,1322],[1109,11891,1330],{"className":11892},[1329],[1109,11894,11896],{"className":11895},[1302],[1109,11897,11899],{"className":11898,"style":1337},[1306],[1109,11900],{},[1109,11902,1462],{"className":11903},[1515],[1109,11905],{"className":11906,"style":1232},[1231],[1109,11908,1582],{"className":11909},[1236],[1109,11911],{"className":11912,"style":1232},[1231],[1109,11914,11916,11919],{"className":11915},[1143],[1109,11917],{"className":11918,"style":1246},[1147],[1109,11920,1210],{"className":11921},[1152],",解出 ",[1109,11924,11926,11994],{"className":11925},[1112],[1109,11927,11929],{"className":11928},[1116],[1118,11930,11931],{"xmlns":1120},[1122,11932,11933,11991],{},[1125,11934,11935,11937,11939,11945,11947,11949,11955,11957,11959,11965,11967,11977,11979,11981,11983,11989],{},[1128,11936,1268],{},[1204,11938,1582],{},[1264,11940,11941,11943],{},[1128,11942,1268],{},[1128,11944,8620],{},[1204,11946,5991],{},[1204,11948,5519],{"stretchy":1456},[6964,11950,11951,11953],{},[1128,11952,5611],{"mathvariant":1450},[1208,11954,1917],{},[1128,11956,1130],{},[1204,11958,1457],{"stretchy":1456},[1264,11960,11961,11963],{},[1128,11962,1268],{},[1128,11964,8620],{},[1204,11966,1462],{"stretchy":1456},[6964,11968,11969,11971],{},[1204,11970,5530],{"stretchy":1456},[1125,11972,11973,11975],{},[1204,11974,5991],{},[1208,11976,1685],{},[1128,11978,5611],{"mathvariant":1450},[1128,11980,1130],{},[1204,11982,1457],{"stretchy":1456},[1264,11984,11985,11987],{},[1128,11986,1268],{},[1128,11988,8620],{},[1204,11990,1462],{"stretchy":1456},[1132,11992,11993],{"encoding":1134},"x = x_k - [\\nabla^2 f(x_k)]^{-1}\\nabla f(x_k)",[1109,11995,11997,12015,12070],{"className":11996,"ariaHidden":1139},[1138],[1109,11998,12000,12003,12006,12009,12012],{"className":11999},[1143],[1109,12001],{"className":12002,"style":7312},[1147],[1109,12004,1268],{"className":12005},[1152,1153],[1109,12007],{"className":12008,"style":1232},[1231],[1109,12010,1582],{"className":12011},[1236],[1109,12013],{"className":12014,"style":1232},[1231],[1109,12016,12018,12021,12061,12064,12067],{"className":12017},[1143],[1109,12019],{"className":12020,"style":10681},[1147],[1109,12022,12024,12027],{"className":12023},[1152],[1109,12025,1268],{"className":12026},[1152,1153],[1109,12028,12030],{"className":12029},[1293],[1109,12031,12033,12053],{"className":12032},[1297,1298],[1109,12034,12036,12050],{"className":12035},[1302],[1109,12037,12039],{"className":12038,"style":8735},[1306],[1109,12040,12041,12044],{"style":1310},[1109,12042],{"className":12043,"style":1315},[1314],[1109,12045,12047],{"className":12046},[1319,1320,1321,1322],[1109,12048,8620],{"className":12049,"style":8750},[1152,1153,1322],[1109,12051,1330],{"className":12052},[1329],[1109,12054,12056],{"className":12055},[1302],[1109,12057,12059],{"className":12058,"style":1337},[1306],[1109,12060],{},[1109,12062],{"className":12063,"style":2497},[1231],[1109,12065,5991],{"className":12066},[2501],[1109,12068],{"className":12069,"style":2497},[1231],[1109,12071,12073,12076,12079,12108,12111,12114,12154,12157,12192,12195,12198,12201,12241],{"className":12072},[1143],[1109,12074],{"className":12075,"style":8085},[1147],[1109,12077,5519],{"className":12078},[1508],[1109,12080,12082,12085],{"className":12081},[1152],[1109,12083,5611],{"className":12084},[1152],[1109,12086,12088],{"className":12087},[1293],[1109,12089,12091],{"className":12090},[1297],[1109,12092,12094],{"className":12093},[1302],[1109,12095,12097],{"className":12096,"style":8144},[1306],[1109,12098,12099,12102],{"style":7044},[1109,12100],{"className":12101,"style":1315},[1314],[1109,12103,12105],{"className":12104},[1319,1320,1321,1322],[1109,12106,1917],{"className":12107},[1152,1322],[1109,12109,1130],{"className":12110,"style":1154},[1152,1153],[1109,12112,1457],{"className":12113},[1508],[1109,12115,12117,12120],{"className":12116},[1152],[1109,12118,1268],{"className":12119},[1152,1153],[1109,12121,12123],{"className":12122},[1293],[1109,12124,12126,12146],{"className":12125},[1297,1298],[1109,12127,12129,12143],{"className":12128},[1302],[1109,12130,12132],{"className":12131,"style":8735},[1306],[1109,12133,12134,12137],{"style":1310},[1109,12135],{"className":12136,"style":1315},[1314],[1109,12138,12140],{"className":12139},[1319,1320,1321,1322],[1109,12141,8620],{"className":12142,"style":8750},[1152,1153,1322],[1109,12144,1330],{"className":12145},[1329],[1109,12147,12149],{"className":12148},[1302],[1109,12150,12152],{"className":12151,"style":1337},[1306],[1109,12153],{},[1109,12155,1462],{"className":12156},[1515],[1109,12158,12160,12163],{"className":12159},[1515],[1109,12161,5530],{"className":12162},[1515],[1109,12164,12166],{"className":12165},[1293],[1109,12167,12169],{"className":12168},[1297],[1109,12170,12172],{"className":12171},[1302],[1109,12173,12175],{"className":12174,"style":8144},[1306],[1109,12176,12177,12180],{"style":7044},[1109,12178],{"className":12179,"style":1315},[1314],[1109,12181,12183],{"className":12182},[1319,1320,1321,1322],[1109,12184,12186,12189],{"className":12185},[1152,1322],[1109,12187,5991],{"className":12188},[1152,1322],[1109,12190,1685],{"className":12191},[1152,1322],[1109,12193,5611],{"className":12194},[1152],[1109,12196,1130],{"className":12197,"style":1154},[1152,1153],[1109,12199,1457],{"className":12200},[1508],[1109,12202,12204,12207],{"className":12203},[1152],[1109,12205,1268],{"className":12206},[1152,1153],[1109,12208,12210],{"className":12209},[1293],[1109,12211,12213,12233],{"className":12212},[1297,1298],[1109,12214,12216,12230],{"className":12215},[1302],[1109,12217,12219],{"className":12218,"style":8735},[1306],[1109,12220,12221,12224],{"style":1310},[1109,12222],{"className":12223,"style":1315},[1314],[1109,12225,12227],{"className":12226},[1319,1320,1321,1322],[1109,12228,8620],{"className":12229,"style":8750},[1152,1153,1322],[1109,12231,1330],{"className":12232},[1329],[1109,12234,12236],{"className":12235},[1302],[1109,12237,12239],{"className":12238,"style":1337},[1306],[1109,12240],{},[1109,12242,1462],{"className":12243},[1515],[1096,12245,12247],{"id":12246},"nr-算法的一些问题","NR 算法的一些问题",[12249,12250,12251,12262,12354],"ul",{},[12252,12253,12254,12257,12258,12261],"li",{},[1104,12255,12256],{},"Hessian 矩阵求逆的计算量","在某些情况下过大,并且 Hessian ",[1104,12259,12260],{},"可能不可逆",";",[12252,12263,12264,12265,4729,12350,12353],{},"在 NR 算法中,函数值 ",[1109,12266,12268,12292],{"className":12267},[1112],[1109,12269,12271],{"className":12270},[1116],[1118,12272,12273],{"xmlns":1120},[1122,12274,12275,12289],{},[1125,12276,12277,12279,12281,12287],{},[1128,12278,1130],{},[1204,12280,1457],{"stretchy":1456},[1264,12282,12283,12285],{},[1128,12284,1268],{},[1128,12286,8620],{},[1204,12288,1462],{"stretchy":1456},[1132,12290,12291],{"encoding":1134},"f(x_k)",[1109,12293,12295],{"className":12294,"ariaHidden":1139},[1138],[1109,12296,12298,12301,12304,12307,12347],{"className":12297},[1143],[1109,12299],{"className":12300,"style":1498},[1147],[1109,12302,1130],{"className":12303,"style":1154},[1152,1153],[1109,12305,1457],{"className":12306},[1508],[1109,12308,12310,12313],{"className":12309},[1152],[1109,12311,1268],{"className":12312},[1152,1153],[1109,12314,12316],{"className":12315},[1293],[1109,12317,12319,12339],{"className":12318},[1297,1298],[1109,12320,12322,12336],{"className":12321},[1302],[1109,12323,12325],{"className":12324,"style":8735},[1306],[1109,12326,12327,12330],{"style":1310},[1109,12328],{"className":12329,"style":1315},[1314],[1109,12331,12333],{"className":12332},[1319,1320,1321,1322],[1109,12334,8620],{"className":12335,"style":8750},[1152,1153,1322],[1109,12337,1330],{"className":12338},[1329],[1109,12340,12342],{"className":12341},[1302],[1109,12343,12345],{"className":12344,"style":1337},[1306],[1109,12346],{},[1109,12348,1462],{"className":12349},[1515],[1104,12351,12352],{},"并不保证随每次迭代递减","(步长可能太大冲过头);",[12252,12355,12356,12357,12360],{},"NR 算法对",[1104,12358,12359],{},"初值的依赖程度较高",",初值不好可能不收敛。",[1096,12362,12364],{"id":12363},"回溯线搜索backtracking-line-search","回溯线搜索(Backtracking line search)",[1101,12366,12367,12370,12371,12399],{},[1104,12368,12369],{},"提出背景",":为了解决 NR 算法在某些情况下迭代步长太大、导致不收敛的问题,用回溯线搜索筛选出既足够长、又能保证 ",[1109,12372,12374,12387],{"className":12373},[1112],[1109,12375,12377],{"className":12376},[1116],[1118,12378,12379],{"xmlns":1120},[1122,12380,12381,12385],{},[1125,12382,12383],{},[1128,12384,1130],{},[1132,12386,1130],{"encoding":1134},[1109,12388,12390],{"className":12389,"ariaHidden":1139},[1138],[1109,12391,12393,12396],{"className":12392},[1143],[1109,12394],{"className":12395,"style":1148},[1147],[1109,12397,1130],{"className":12398,"style":1154},[1152,1153]," 递减的步长。",[1101,12401,12402,1662],{},[1104,12403,12404],{},"算法步骤",[12406,12407,12408,12618],"ol",{},[12252,12409,12410,12411,2170,12488,12564,12565,12261],{},"给定参数 ",[1109,12412,12414,12442],{"className":12413},[1112],[1109,12415,12417],{"className":12416},[1116],[1118,12418,12419],{"xmlns":1120},[1122,12420,12421,12439],{},[1125,12422,12423,12426,12428,12430,12432,12434,12437],{},[1128,12424,12425],{},"α",[1204,12427,5895],{},[1204,12429,1457],{"stretchy":1456},[1208,12431,1210],{},[1204,12433,1910],{"separator":1139},[1208,12435,12436],{},"0.5",[1204,12438,1462],{"stretchy":1456},[1132,12440,12441],{"encoding":1134},"\\alpha \\in (0, 0.5)",[1109,12443,12445,12464],{"className":12444,"ariaHidden":1139},[1138],[1109,12446,12448,12451,12455,12458,12461],{"className":12447},[1143],[1109,12449],{"className":12450,"style":4065},[1147],[1109,12452,12425],{"className":12453,"style":12454},[1152,1153],"margin-right:0.0037em;",[1109,12456],{"className":12457,"style":1232},[1231],[1109,12459,5895],{"className":12460},[1236],[1109,12462],{"className":12463,"style":1232},[1231],[1109,12465,12467,12470,12473,12476,12479,12482,12485],{"className":12466},[1143],[1109,12468],{"className":12469,"style":1498},[1147],[1109,12471,1457],{"className":12472},[1508],[1109,12474,1210],{"className":12475},[1152],[1109,12477,1910],{"className":12478},[1975],[1109,12480],{"className":12481,"style":1540},[1231],[1109,12483,12436],{"className":12484},[1152],[1109,12486,1462],{"className":12487},[1515],[1109,12489,12491,12518],{"className":12490},[1112],[1109,12492,12494],{"className":12493},[1116],[1118,12495,12496],{"xmlns":1120},[1122,12497,12498,12515],{},[1125,12499,12500,12503,12505,12507,12509,12511,12513],{},[1128,12501,12502],{},"β",[1204,12504,5895],{},[1204,12506,1457],{"stretchy":1456},[1208,12508,1210],{},[1204,12510,1910],{"separator":1139},[1208,12512,1685],{},[1204,12514,1462],{"stretchy":1456},[1132,12516,12517],{"encoding":1134},"\\beta \\in (0,1)",[1109,12519,12521,12540],{"className":12520,"ariaHidden":1139},[1138],[1109,12522,12524,12527,12531,12534,12537],{"className":12523},[1143],[1109,12525],{"className":12526,"style":1148},[1147],[1109,12528,12502],{"className":12529,"style":12530},[1152,1153],"margin-right:0.0528em;",[1109,12532],{"className":12533,"style":1232},[1231],[1109,12535,5895],{"className":12536},[1236],[1109,12538],{"className":12539,"style":1232},[1231],[1109,12541,12543,12546,12549,12552,12555,12558,12561],{"className":12542},[1143],[1109,12544],{"className":12545,"style":1498},[1147],[1109,12547,1457],{"className":12548},[1508],[1109,12550,1210],{"className":12551},[1152],[1109,12553,1910],{"className":12554},[1975],[1109,12556],{"className":12557,"style":1540},[1231],[1109,12559,1685],{"className":12560},[1152],[1109,12562,1462],{"className":12563},[1515],",令步长 ",[1109,12566,12568,12587],{"className":12567},[1112],[1109,12569,12571],{"className":12570},[1116],[1118,12572,12573],{"xmlns":1120},[1122,12574,12575,12584],{},[1125,12576,12577,12580,12582],{},[1128,12578,12579],{},"t",[1204,12581,1582],{},[1208,12583,1685],{},[1132,12585,12586],{"encoding":1134},"t = 1",[1109,12588,12590,12609],{"className":12589,"ariaHidden":1139},[1138],[1109,12591,12593,12597,12600,12603,12606],{"className":12592},[1143],[1109,12594],{"className":12595,"style":12596},[1147],"height:0.6151em;",[1109,12598,12579],{"className":12599},[1152,1153],[1109,12601],{"className":12602,"style":1232},[1231],[1109,12604,1582],{"className":12605},[1236],[1109,12607],{"className":12608,"style":1232},[1231],[1109,12610,12612,12615],{"className":12611},[1143],[1109,12613],{"className":12614,"style":1246},[1147],[1109,12616,1685],{"className":12617},[1152],[12252,12619,12620,12621,12624],{},"判断下面的",[1104,12622,12623],{},"充分下降条件","是否成立:",[1101,12626,12627],{},[1109,12628,12630,12699],{"className":12629},[1112],[1109,12631,12633],{"className":12632},[1116],[1118,12634,12635],{"xmlns":1120},[1122,12636,12637,12696],{},[1125,12638,12639,12641,12643,12645,12647,12649,12651,12654,12656,12658,12660,12662,12664,12666,12668,12670,12672,12674,12676,12678,12680,12682,12684,12686,12692,12694],{},[1128,12640,1130],{},[1204,12642,1457],{"stretchy":1456},[1128,12644,1268],{},[1204,12646,2342],{},[1128,12648,12579],{},[1471,12650,1473],{},[1128,12652,12653],{"mathvariant":1450},"Δ",[1128,12655,1268],{},[1204,12657,1462],{"stretchy":1456},[1204,12659,1467],{},[1128,12661,1130],{},[1204,12663,1457],{"stretchy":1456},[1128,12665,1268],{},[1204,12667,1462],{"stretchy":1456},[1204,12669,2342],{},[1128,12671,12425],{},[1471,12673,1473],{},[1128,12675,12579],{},[1471,12677,1473],{},[1128,12679,5611],{"mathvariant":1450},[1128,12681,1130],{},[1204,12683,1457],{"stretchy":1456},[1128,12685,1268],{},[6964,12687,12688,12690],{},[1204,12689,1462],{"stretchy":1456},[1128,12691,9471],{"mathvariant":1450},[1128,12693,12653],{"mathvariant":1450},[1128,12695,1268],{},[1132,12697,12698],{"encoding":1134},"f(x + t\\,\\Delta x) \\le f(x) + \\alpha\\,t\\,\\nabla f(x)^{\\top}\\Delta x",[1109,12700,12702,12726,12756,12783],{"className":12701,"ariaHidden":1139},[1138],[1109,12703,12705,12708,12711,12714,12717,12720,12723],{"className":12704},[1143],[1109,12706],{"className":12707,"style":1498},[1147],[1109,12709,1130],{"className":12710,"style":1154},[1152,1153],[1109,12712,1457],{"className":12713},[1508],[1109,12715,1268],{"className":12716},[1152,1153],[1109,12718],{"className":12719,"style":2497},[1231],[1109,12721,2342],{"className":12722},[2501],[1109,12724],{"className":12725,"style":2497},[1231],[1109,12727,12729,12732,12735,12738,12741,12744,12747,12750,12753],{"className":12728},[1143],[1109,12730],{"className":12731,"style":1498},[1147],[1109,12733,12579],{"className":12734},[1152,1153],[1109,12736],{"className":12737,"style":1540},[1231],[1109,12739,12653],{"className":12740},[1152],[1109,12742,1268],{"className":12743},[1152,1153],[1109,12745,1462],{"className":12746},[1515],[1109,12748],{"className":12749,"style":1232},[1231],[1109,12751,1467],{"className":12752},[1236],[1109,12754],{"className":12755,"style":1232},[1231],[1109,12757,12759,12762,12765,12768,12771,12774,12777,12780],{"className":12758},[1143],[1109,12760],{"className":12761,"style":1498},[1147],[1109,12763,1130],{"className":12764,"style":1154},[1152,1153],[1109,12766,1457],{"className":12767},[1508],[1109,12769,1268],{"className":12770},[1152,1153],[1109,12772,1462],{"className":12773},[1515],[1109,12775],{"className":12776,"style":2497},[1231],[1109,12778,2342],{"className":12779},[2501],[1109,12781],{"className":12782,"style":2497},[1231],[1109,12784,12786,12789,12792,12795,12798,12801,12804,12807,12810,12813,12845,12848],{"className":12785},[1143],[1109,12787],{"className":12788,"style":11150},[1147],[1109,12790,12425],{"className":12791,"style":12454},[1152,1153],[1109,12793],{"className":12794,"style":1540},[1231],[1109,12796,12579],{"className":12797},[1152,1153],[1109,12799],{"className":12800,"style":1540},[1231],[1109,12802,5611],{"className":12803},[1152],[1109,12805,1130],{"className":12806,"style":1154},[1152,1153],[1109,12808,1457],{"className":12809},[1508],[1109,12811,1268],{"className":12812},[1152,1153],[1109,12814,12816,12819],{"className":12815},[1515],[1109,12817,1462],{"className":12818},[1515],[1109,12820,12822],{"className":12821},[1293],[1109,12823,12825],{"className":12824},[1297],[1109,12826,12828],{"className":12827},[1302],[1109,12829,12831],{"className":12830,"style":11218},[1306],[1109,12832,12833,12836],{"style":7044},[1109,12834],{"className":12835,"style":1315},[1314],[1109,12837,12839],{"className":12838},[1319,1320,1321,1322],[1109,12840,12842],{"className":12841},[1152,1322],[1109,12843,9471],{"className":12844},[1152,1322],[1109,12846,12653],{"className":12847},[1152],[1109,12849,1268],{"className":12850},[1152,1153],[1101,12852,12853,12854,12882],{},"若为真则进行第 3 步(接受这个 ",[1109,12855,12857,12870],{"className":12856},[1112],[1109,12858,12860],{"className":12859},[1116],[1118,12861,12862],{"xmlns":1120},[1122,12863,12864,12868],{},[1125,12865,12866],{},[1128,12867,12579],{},[1132,12869,12579],{"encoding":1134},[1109,12871,12873],{"className":12872,"ariaHidden":1139},[1138],[1109,12874,12876,12879],{"className":12875},[1143],[1109,12877],{"className":12878,"style":12596},[1147],[1109,12880,12579],{"className":12881},[1152,1153],"),否则进行第 4 步;",[12406,12884,12886,12973],{"start":12885},3,[12252,12887,12888,12889,12261],{},"接受步长,更新 ",[1109,12890,12892,12919],{"className":12891},[1112],[1109,12893,12895],{"className":12894},[1116],[1118,12896,12897],{"xmlns":1120},[1122,12898,12899,12916],{},[1125,12900,12901,12903,12906,12908,12910,12912,12914],{},[1128,12902,1268],{},[1204,12904,12905],{},"←",[1128,12907,1268],{},[1204,12909,2342],{},[1128,12911,12579],{},[1128,12913,12653],{"mathvariant":1450},[1128,12915,1268],{},[1132,12917,12918],{"encoding":1134},"x \\leftarrow x + t\\Delta x",[1109,12920,12922,12940,12958],{"className":12921,"ariaHidden":1139},[1138],[1109,12923,12925,12928,12931,12934,12937],{"className":12924},[1143],[1109,12926],{"className":12927,"style":7312},[1147],[1109,12929,1268],{"className":12930},[1152,1153],[1109,12932],{"className":12933,"style":1232},[1231],[1109,12935,12905],{"className":12936},[1236],[1109,12938],{"className":12939,"style":1232},[1231],[1109,12941,12943,12946,12949,12952,12955],{"className":12942},[1143],[1109,12944],{"className":12945,"style":8171},[1147],[1109,12947,1268],{"className":12948},[1152,1153],[1109,12950],{"className":12951,"style":2497},[1231],[1109,12953,2342],{"className":12954},[2501],[1109,12956],{"className":12957,"style":2497},[1231],[1109,12959,12961,12964,12967,12970],{"className":12960},[1143],[1109,12962],{"className":12963,"style":3026},[1147],[1109,12965,12579],{"className":12966},[1152,1153],[1109,12968,12653],{"className":12969},[1152],[1109,12971,1268],{"className":12972},[1152,1153],[12252,12974,12975,12976,13032],{},"缩小步长 ",[1109,12977,12979,12999],{"className":12978},[1112],[1109,12980,12982],{"className":12981},[1116],[1118,12983,12984],{"xmlns":1120},[1122,12985,12986,12996],{},[1125,12987,12988,12990,12992,12994],{},[1128,12989,12579],{},[1204,12991,12905],{},[1128,12993,12502],{},[1128,12995,12579],{},[1132,12997,12998],{"encoding":1134},"t \\leftarrow \\beta t",[1109,13000,13002,13020],{"className":13001,"ariaHidden":1139},[1138],[1109,13003,13005,13008,13011,13014,13017],{"className":13004},[1143],[1109,13006],{"className":13007,"style":12596},[1147],[1109,13009,12579],{"className":13010},[1152,1153],[1109,13012],{"className":13013,"style":1232},[1231],[1109,13015,12905],{"className":13016},[1236],[1109,13018],{"className":13019,"style":1232},[1231],[1109,13021,13023,13026,13029],{"className":13022},[1143],[1109,13024],{"className":13025,"style":1148},[1147],[1109,13027,12502],{"className":13028,"style":12530},[1152,1153],[1109,13030,12579],{"className":13031},[1152,1153],",并重新回到第 2 步",[1101,13034,13035,13038,13039,1155,13067,13128,13129,13157,13158,13187],{},[1104,13036,13037],{},"解释",":充分下降条件成立时,说明 ",[1109,13040,13042,13055],{"className":13041},[1112],[1109,13043,13045],{"className":13044},[1116],[1118,13046,13047],{"xmlns":1120},[1122,13048,13049,13053],{},[1125,13050,13051],{},[1128,13052,1268],{},[1132,13054,1268],{"encoding":1134},[1109,13056,13058],{"className":13057,"ariaHidden":1139},[1138],[1109,13059,13061,13064],{"className":13060},[1143],[1109,13062],{"className":13063,"style":7312},[1147],[1109,13065,1268],{"className":13066},[1152,1153],[1109,13068,13070,13092],{"className":13069},[1112],[1109,13071,13073],{"className":13072},[1116],[1118,13074,13075],{"xmlns":1120},[1122,13076,13077,13089],{},[1125,13078,13079,13081,13083,13085,13087],{},[1128,13080,1268],{},[1204,13082,2342],{},[1128,13084,12579],{},[1128,13086,12653],{"mathvariant":1450},[1128,13088,1268],{},[1132,13090,13091],{"encoding":1134},"x+t\\Delta x",[1109,13093,13095,13113],{"className":13094,"ariaHidden":1139},[1138],[1109,13096,13098,13101,13104,13107,13110],{"className":13097},[1143],[1109,13099],{"className":13100,"style":8171},[1147],[1109,13102,1268],{"className":13103},[1152,1153],[1109,13105],{"className":13106,"style":2497},[1231],[1109,13108,2342],{"className":13109},[2501],[1109,13111],{"className":13112,"style":2497},[1231],[1109,13114,13116,13119,13122,13125],{"className":13115},[1143],[1109,13117],{"className":13118,"style":3026},[1147],[1109,13120,12579],{"className":13121},[1152,1153],[1109,13123,12653],{"className":13124},[1152],[1109,13126,1268],{"className":13127},[1152,1153]," 之间连线的斜率,比 ",[1109,13130,13132,13145],{"className":13131},[1112],[1109,13133,13135],{"className":13134},[1116],[1118,13136,13137],{"xmlns":1120},[1122,13138,13139,13143],{},[1125,13140,13141],{},[1128,13142,1268],{},[1132,13144,1268],{"encoding":1134},[1109,13146,13148],{"className":13147,"ariaHidden":1139},[1138],[1109,13149,13151,13154],{"className":13150},[1143],[1109,13152],{"className":13153,"style":7312},[1147],[1109,13155,1268],{"className":13156},[1152,1153]," 点处斜率的 ",[1109,13159,13161,13175],{"className":13160},[1112],[1109,13162,13164],{"className":13163},[1116],[1118,13165,13166],{"xmlns":1120},[1122,13167,13168,13172],{},[1125,13169,13170],{},[1128,13171,12425],{},[1132,13173,13174],{"encoding":1134},"\\alpha",[1109,13176,13178],{"className":13177,"ariaHidden":1139},[1138],[1109,13179,13181,13184],{"className":13180},[1143],[1109,13182],{"className":13183,"style":7312},[1147],[1109,13185,12425],{"className":13186,"style":12454},[1152,1153]," 倍还要平缓——若不满足就说明步子迈太大,需要缩小步长。",[1096,13189,13190],{"id":13190},"收敛标准",[12249,13192,13193,13378,13683],{},[12252,13194,13195,1662,13198,12261],{},[1104,13196,13197],{},"绝对收敛标准",[1109,13199,13201,13241],{"className":13200},[1112],[1109,13202,13204],{"className":13203},[1116],[1118,13205,13206],{"xmlns":1120},[1122,13207,13208,13238],{},[1125,13209,13210,13212,13224,13226,13232,13234,13236],{},[1128,13211,8631],{"mathvariant":1450},[1264,13213,13214,13216],{},[1128,13215,1268],{},[1125,13217,13218,13220,13222],{},[1128,13219,8620],{},[1204,13221,2342],{},[1208,13223,1685],{},[1204,13225,5991],{},[1264,13227,13228,13230],{},[1128,13229,1268],{},[1128,13231,8620],{},[1128,13233,8631],{"mathvariant":1450},[1204,13235,6687],{},[1128,13237,4048],{},[1132,13239,13240],{"encoding":1134},"\\|x_{k+1} - x_k\\| \u003C \\epsilon",[1109,13242,13244,13311,13369],{"className":13243,"ariaHidden":1139},[1138],[1109,13245,13247,13250,13253,13302,13305,13308],{"className":13246},[1143],[1109,13248],{"className":13249,"style":1498},[1147],[1109,13251,8631],{"className":13252},[1152],[1109,13254,13256,13259],{"className":13255},[1152],[1109,13257,1268],{"className":13258},[1152,1153],[1109,13260,13262],{"className":13261},[1293],[1109,13263,13265,13294],{"className":13264},[1297,1298],[1109,13266,13268,13291],{"className":13267},[1302],[1109,13269,13271],{"className":13270,"style":8735},[1306],[1109,13272,13273,13276],{"style":1310},[1109,13274],{"className":13275,"style":1315},[1314],[1109,13277,13279],{"className":13278},[1319,1320,1321,1322],[1109,13280,13282,13285,13288],{"className":13281},[1152,1322],[1109,13283,8620],{"className":13284,"style":8750},[1152,1153,1322],[1109,13286,2342],{"className":13287},[2501,1322],[1109,13289,1685],{"className":13290},[1152,1322],[1109,13292,1330],{"className":13293},[1329],[1109,13295,13297],{"className":13296},[1302],[1109,13298,13300],{"className":13299,"style":10663},[1306],[1109,13301],{},[1109,13303],{"className":13304,"style":2497},[1231],[1109,13306,5991],{"className":13307},[2501],[1109,13309],{"className":13310,"style":2497},[1231],[1109,13312,13314,13317,13357,13360,13363,13366],{"className":13313},[1143],[1109,13315],{"className":13316,"style":1498},[1147],[1109,13318,13320,13323],{"className":13319},[1152],[1109,13321,1268],{"className":13322},[1152,1153],[1109,13324,13326],{"className":13325},[1293],[1109,13327,13329,13349],{"className":13328},[1297,1298],[1109,13330,13332,13346],{"className":13331},[1302],[1109,13333,13335],{"className":13334,"style":8735},[1306],[1109,13336,13337,13340],{"style":1310},[1109,13338],{"className":13339,"style":1315},[1314],[1109,13341,13343],{"className":13342},[1319,1320,1321,1322],[1109,13344,8620],{"className":13345,"style":8750},[1152,1153,1322],[1109,13347,1330],{"className":13348},[1329],[1109,13350,13352],{"className":13351},[1302],[1109,13353,13355],{"className":13354,"style":1337},[1306],[1109,13356],{},[1109,13358,8631],{"className":13359},[1152],[1109,13361],{"className":13362,"style":1232},[1231],[1109,13364,6687],{"className":13365},[1236],[1109,13367],{"className":13368,"style":1232},[1231],[1109,13370,13372,13375],{"className":13371},[1143],[1109,13373],{"className":13374,"style":7312},[1147],[1109,13376,4048],{"className":13377},[1152,1153],[12252,13379,13380,1662,13383,12261],{},[1104,13381,13382],{},"相对收敛标准",[1109,13384,13386,13445],{"className":13385},[1112],[1109,13387,13389],{"className":13388},[1116],[1118,13390,13391],{"xmlns":1120},[1122,13392,13393,13442],{},[1125,13394,13395,13438,13440],{},[13396,13397,13398],"mstyle",{"scriptlevel":1210,"displaystyle":1139},[4490,13399,13400,13426],{},[1125,13401,13402,13404,13416,13418,13424],{},[1128,13403,8631],{"mathvariant":1450},[1264,13405,13406,13408],{},[1128,13407,1268],{},[1125,13409,13410,13412,13414],{},[1128,13411,8620],{},[1204,13413,2342],{},[1208,13415,1685],{},[1204,13417,5991],{},[1264,13419,13420,13422],{},[1128,13421,1268],{},[1128,13423,8620],{},[1128,13425,8631],{"mathvariant":1450},[1125,13427,13428,13430,13436],{},[1128,13429,8631],{"mathvariant":1450},[1264,13431,13432,13434],{},[1128,13433,1268],{},[1128,13435,8620],{},[1128,13437,8631],{"mathvariant":1450},[1204,13439,6687],{},[1128,13441,4048],{},[1132,13443,13444],{"encoding":1134},"\\dfrac{\\|x_{k+1} - x_k\\|}{\\|x_k\\|} \u003C \\epsilon",[1109,13446,13448,13674],{"className":13447,"ariaHidden":1139},[1138],[1109,13449,13451,13455,13665,13668,13671],{"className":13450},[1143],[1109,13452],{"className":13453,"style":13454},[1147],"height:2.363em;vertical-align:-0.936em;",[1109,13456,13458,13461,13662],{"className":13457},[1152],[1109,13459],{"className":13460},[1508,4589],[1109,13462,13464],{"className":13463},[4490],[1109,13465,13467,13653],{"className":13466},[1297,1298],[1109,13468,13470,13650],{"className":13469},[1302],[1109,13471,13474,13529,13537],{"className":13472,"style":13473},[1306],"height:1.427em;",[1109,13475,13477,13480],{"style":13476},"top:-2.314em;",[1109,13478],{"className":13479,"style":4609},[1314],[1109,13481,13483,13486,13526],{"className":13482},[1152],[1109,13484,8631],{"className":13485},[1152],[1109,13487,13489,13492],{"className":13488},[1152],[1109,13490,1268],{"className":13491},[1152,1153],[1109,13493,13495],{"className":13494},[1293],[1109,13496,13498,13518],{"className":13497},[1297,1298],[1109,13499,13501,13515],{"className":13500},[1302],[1109,13502,13504],{"className":13503,"style":8735},[1306],[1109,13505,13506,13509],{"style":1310},[1109,13507],{"className":13508,"style":1315},[1314],[1109,13510,13512],{"className":13511},[1319,1320,1321,1322],[1109,13513,8620],{"className":13514,"style":8750},[1152,1153,1322],[1109,13516,1330],{"className":13517},[1329],[1109,13519,13521],{"className":13520},[1302],[1109,13522,13524],{"className":13523,"style":1337},[1306],[1109,13525],{},[1109,13527,8631],{"className":13528},[1152],[1109,13530,13531,13534],{"style":4630},[1109,13532],{"className":13533,"style":4609},[1314],[1109,13535],{"className":13536,"style":4638},[4637],[1109,13538,13540,13543],{"style":13539},"top:-3.677em;",[1109,13541],{"className":13542,"style":4609},[1314],[1109,13544,13546,13549,13598,13601,13604,13607,13647],{"className":13545},[1152],[1109,13547,8631],{"className":13548},[1152],[1109,13550,13552,13555],{"className":13551},[1152],[1109,13553,1268],{"className":13554},[1152,1153],[1109,13556,13558],{"className":13557},[1293],[1109,13559,13561,13590],{"className":13560},[1297,1298],[1109,13562,13564,13587],{"className":13563},[1302],[1109,13565,13567],{"className":13566,"style":8735},[1306],[1109,13568,13569,13572],{"style":1310},[1109,13570],{"className":13571,"style":1315},[1314],[1109,13573,13575],{"className":13574},[1319,1320,1321,1322],[1109,13576,13578,13581,13584],{"className":13577},[1152,1322],[1109,13579,8620],{"className":13580,"style":8750},[1152,1153,1322],[1109,13582,2342],{"className":13583},[2501,1322],[1109,13585,1685],{"className":13586},[1152,1322],[1109,13588,1330],{"className":13589},[1329],[1109,13591,13593],{"className":13592},[1302],[1109,13594,13596],{"className":13595,"style":10663},[1306],[1109,13597],{},[1109,13599],{"className":13600,"style":2497},[1231],[1109,13602,5991],{"className":13603},[2501],[1109,13605],{"className":13606,"style":2497},[1231],[1109,13608,13610,13613],{"className":13609},[1152],[1109,13611,1268],{"className":13612},[1152,1153],[1109,13614,13616],{"className":13615},[1293],[1109,13617,13619,13639],{"className":13618},[1297,1298],[1109,13620,13622,13636],{"className":13621},[1302],[1109,13623,13625],{"className":13624,"style":8735},[1306],[1109,13626,13627,13630],{"style":1310},[1109,13628],{"className":13629,"style":1315},[1314],[1109,13631,13633],{"className":13632},[1319,1320,1321,1322],[1109,13634,8620],{"className":13635,"style":8750},[1152,1153,1322],[1109,13637,1330],{"className":13638},[1329],[1109,13640,13642],{"className":13641},[1302],[1109,13643,13645],{"className":13644,"style":1337},[1306],[1109,13646],{},[1109,13648,8631],{"className":13649},[1152],[1109,13651,1330],{"className":13652},[1329],[1109,13654,13656],{"className":13655},[1302],[1109,13657,13660],{"className":13658,"style":13659},[1306],"height:0.936em;",[1109,13661],{},[1109,13663],{"className":13664},[1515,4589],[1109,13666],{"className":13667,"style":1232},[1231],[1109,13669,6687],{"className":13670},[1236],[1109,13672],{"className":13673,"style":1232},[1231],[1109,13675,13677,13680],{"className":13676},[1143],[1109,13678],{"className":13679,"style":7312},[1147],[1109,13681,4048],{"className":13682},[1152,1153],[12252,13684,13685,1662,13688,13998,13999,14079],{},[1104,13686,13687],{},"调整的相对收敛标准",[1109,13689,13691,13753],{"className":13690},[1112],[1109,13692,13694],{"className":13693},[1116],[1118,13695,13696],{"xmlns":1120},[1122,13697,13698,13750],{},[1125,13699,13700,13746,13748],{},[13396,13701,13702],{"scriptlevel":1210,"displaystyle":1139},[4490,13703,13704,13730],{},[1125,13705,13706,13708,13720,13722,13728],{},[1128,13707,8631],{"mathvariant":1450},[1264,13709,13710,13712],{},[1128,13711,1268],{},[1125,13713,13714,13716,13718],{},[1128,13715,8620],{},[1204,13717,2342],{},[1208,13719,1685],{},[1204,13721,5991],{},[1264,13723,13724,13726],{},[1128,13725,1268],{},[1128,13727,8620],{},[1128,13729,8631],{"mathvariant":1450},[1125,13731,13732,13734,13740,13742,13744],{},[1128,13733,8631],{"mathvariant":1450},[1264,13735,13736,13738],{},[1128,13737,1268],{},[1128,13739,8620],{},[1128,13741,8631],{"mathvariant":1450},[1204,13743,2342],{},[1208,13745,1685],{},[1204,13747,6687],{},[1128,13749,4048],{},[1132,13751,13752],{"encoding":1134},"\\dfrac{\\|x_{k+1} - x_k\\|}{\\|x_k\\| + 1} \u003C \\epsilon",[1109,13754,13756,13989],{"className":13755,"ariaHidden":1139},[1138],[1109,13757,13759,13762,13980,13983,13986],{"className":13758},[1143],[1109,13760],{"className":13761,"style":13454},[1147],[1109,13763,13765,13768,13977],{"className":13764},[1152],[1109,13766],{"className":13767},[1508,4589],[1109,13769,13771],{"className":13770},[4490],[1109,13772,13774,13969],{"className":13773},[1297,1298],[1109,13775,13777,13966],{"className":13776},[1302],[1109,13778,13780,13846,13854],{"className":13779,"style":13473},[1306],[1109,13781,13782,13785],{"style":13476},[1109,13783],{"className":13784,"style":4609},[1314],[1109,13786,13788,13791,13831,13834,13837,13840,13843],{"className":13787},[1152],[1109,13789,8631],{"className":13790},[1152],[1109,13792,13794,13797],{"className":13793},[1152],[1109,13795,1268],{"className":13796},[1152,1153],[1109,13798,13800],{"className":13799},[1293],[1109,13801,13803,13823],{"className":13802},[1297,1298],[1109,13804,13806,13820],{"className":13805},[1302],[1109,13807,13809],{"className":13808,"style":8735},[1306],[1109,13810,13811,13814],{"style":1310},[1109,13812],{"className":13813,"style":1315},[1314],[1109,13815,13817],{"className":13816},[1319,1320,1321,1322],[1109,13818,8620],{"className":13819,"style":8750},[1152,1153,1322],[1109,13821,1330],{"className":13822},[1329],[1109,13824,13826],{"className":13825},[1302],[1109,13827,13829],{"className":13828,"style":1337},[1306],[1109,13830],{},[1109,13832,8631],{"className":13833},[1152],[1109,13835],{"className":13836,"style":2497},[1231],[1109,13838,2342],{"className":13839},[2501],[1109,13841],{"className":13842,"style":2497},[1231],[1109,13844,1685],{"className":13845},[1152],[1109,13847,13848,13851],{"style":4630},[1109,13849],{"className":13850,"style":4609},[1314],[1109,13852],{"className":13853,"style":4638},[4637],[1109,13855,13856,13859],{"style":13539},[1109,13857],{"className":13858,"style":4609},[1314],[1109,13860,13862,13865,13914,13917,13920,13923,13963],{"className":13861},[1152],[1109,13863,8631],{"className":13864},[1152],[1109,13866,13868,13871],{"className":13867},[1152],[1109,13869,1268],{"className":13870},[1152,1153],[1109,13872,13874],{"className":13873},[1293],[1109,13875,13877,13906],{"className":13876},[1297,1298],[1109,13878,13880,13903],{"className":13879},[1302],[1109,13881,13883],{"className":13882,"style":8735},[1306],[1109,13884,13885,13888],{"style":1310},[1109,13886],{"className":13887,"style":1315},[1314],[1109,13889,13891],{"className":13890},[1319,1320,1321,1322],[1109,13892,13894,13897,13900],{"className":13893},[1152,1322],[1109,13895,8620],{"className":13896,"style":8750},[1152,1153,1322],[1109,13898,2342],{"className":13899},[2501,1322],[1109,13901,1685],{"className":13902},[1152,1322],[1109,13904,1330],{"className":13905},[1329],[1109,13907,13909],{"className":13908},[1302],[1109,13910,13912],{"className":13911,"style":10663},[1306],[1109,13913],{},[1109,13915],{"className":13916,"style":2497},[1231],[1109,13918,5991],{"className":13919},[2501],[1109,13921],{"className":13922,"style":2497},[1231],[1109,13924,13926,13929],{"className":13925},[1152],[1109,13927,1268],{"className":13928},[1152,1153],[1109,13930,13932],{"className":13931},[1293],[1109,13933,13935,13955],{"className":13934},[1297,1298],[1109,13936,13938,13952],{"className":13937},[1302],[1109,13939,13941],{"className":13940,"style":8735},[1306],[1109,13942,13943,13946],{"style":1310},[1109,13944],{"className":13945,"style":1315},[1314],[1109,13947,13949],{"className":13948},[1319,1320,1321,1322],[1109,13950,8620],{"className":13951,"style":8750},[1152,1153,1322],[1109,13953,1330],{"className":13954},[1329],[1109,13956,13958],{"className":13957},[1302],[1109,13959,13961],{"className":13960,"style":1337},[1306],[1109,13962],{},[1109,13964,8631],{"className":13965},[1152],[1109,13967,1330],{"className":13968},[1329],[1109,13970,13972],{"className":13971},[1302],[1109,13973,13975],{"className":13974,"style":13659},[1306],[1109,13976],{},[1109,13978],{"className":13979},[1515,4589],[1109,13981],{"className":13982,"style":1232},[1231],[1109,13984,6687],{"className":13985},[1236],[1109,13987],{"className":13988,"style":1232},[1231],[1109,13990,13992,13995],{"className":13991},[1143],[1109,13993],{"className":13994,"style":7312},[1147],[1109,13996,4048],{"className":13997},[1152,1153],"(避免 ",[1109,14000,14002,14024],{"className":14001},[1112],[1109,14003,14005],{"className":14004},[1116],[1118,14006,14007],{"xmlns":1120},[1122,14008,14009,14021],{},[1125,14010,14011,14013,14019],{},[1128,14012,8631],{"mathvariant":1450},[1264,14014,14015,14017],{},[1128,14016,1268],{},[1128,14018,8620],{},[1128,14020,8631],{"mathvariant":1450},[1132,14022,14023],{"encoding":1134},"\\|x_k\\|",[1109,14025,14027],{"className":14026,"ariaHidden":1139},[1138],[1109,14028,14030,14033,14036,14076],{"className":14029},[1143],[1109,14031],{"className":14032,"style":1498},[1147],[1109,14034,8631],{"className":14035},[1152],[1109,14037,14039,14042],{"className":14038},[1152],[1109,14040,1268],{"className":14041},[1152,1153],[1109,14043,14045],{"className":14044},[1293],[1109,14046,14048,14068],{"className":14047},[1297,1298],[1109,14049,14051,14065],{"className":14050},[1302],[1109,14052,14054],{"className":14053,"style":8735},[1306],[1109,14055,14056,14059],{"style":1310},[1109,14057],{"className":14058,"style":1315},[1314],[1109,14060,14062],{"className":14061},[1319,1320,1321,1322],[1109,14063,8620],{"className":14064,"style":8750},[1152,1153,1322],[1109,14066,1330],{"className":14067},[1329],[1109,14069,14071],{"className":14070},[1302],[1109,14072,14074],{"className":14073,"style":1337},[1306],[1109,14075],{},[1109,14077,8631],{"className":14078},[1152]," 太小时分母失稳)。",[1096,14081,14083],{"id":14082},"例子逻辑回归","例子:逻辑回归",[1101,14085,14086,14087,14223,14224,14506],{},"假设有一组样本 ",[1109,14088,14090,14121],{"className":14089},[1112],[1109,14091,14093],{"className":14092},[1116],[1118,14094,14095],{"xmlns":1120},[1122,14096,14097,14118],{},[1125,14098,14099,14101,14107,14109,14116],{},[1204,14100,1457],{"stretchy":1456},[1264,14102,14103,14105],{},[1128,14104,1268],{},[1128,14106,9875],{},[1204,14108,1910],{"separator":1139},[1264,14110,14111,14114],{},[1128,14112,14113],{},"y",[1128,14115,9875],{},[1204,14117,1462],{"stretchy":1456},[1132,14119,14120],{"encoding":1134},"(x_i, y_i)",[1109,14122,14124],{"className":14123,"ariaHidden":1139},[1138],[1109,14125,14127,14130,14133,14174,14177,14180,14220],{"className":14126},[1143],[1109,14128],{"className":14129,"style":1498},[1147],[1109,14131,1457],{"className":14132},[1508],[1109,14134,14136,14139],{"className":14135},[1152],[1109,14137,1268],{"className":14138},[1152,1153],[1109,14140,14142],{"className":14141},[1293],[1109,14143,14145,14166],{"className":14144},[1297,1298],[1109,14146,14148,14163],{"className":14147},[1302],[1109,14149,14152],{"className":14150,"style":14151},[1306],"height:0.3117em;",[1109,14153,14154,14157],{"style":1310},[1109,14155],{"className":14156,"style":1315},[1314],[1109,14158,14160],{"className":14159},[1319,1320,1321,1322],[1109,14161,9875],{"className":14162},[1152,1153,1322],[1109,14164,1330],{"className":14165},[1329],[1109,14167,14169],{"className":14168},[1302],[1109,14170,14172],{"className":14171,"style":1337},[1306],[1109,14173],{},[1109,14175,1910],{"className":14176},[1975],[1109,14178],{"className":14179,"style":1540},[1231],[1109,14181,14183,14186],{"className":14182},[1152],[1109,14184,14113],{"className":14185,"style":1186},[1152,1153],[1109,14187,14189],{"className":14188},[1293],[1109,14190,14192,14212],{"className":14191},[1297,1298],[1109,14193,14195,14209],{"className":14194},[1302],[1109,14196,14198],{"className":14197,"style":14151},[1306],[1109,14199,14200,14203],{"style":2146},[1109,14201],{"className":14202,"style":1315},[1314],[1109,14204,14206],{"className":14205},[1319,1320,1321,1322],[1109,14207,9875],{"className":14208},[1152,1153,1322],[1109,14210,1330],{"className":14211},[1329],[1109,14213,14215],{"className":14214},[1302],[1109,14216,14218],{"className":14217,"style":1337},[1306],[1109,14219],{},[1109,14221,1462],{"className":14222},[1515],",考虑逻辑回归 ",[1109,14225,14227,14277],{"className":14226},[1112],[1109,14228,14230],{"className":14229},[1116],[1118,14231,14232],{"xmlns":1120},[1122,14233,14234,14274],{},[1125,14235,14236,14242,14244],{},[1264,14237,14238,14240],{},[1128,14239,1101],{},[1128,14241,9875],{},[1204,14243,1582],{},[4490,14245,14246,14248],{},[1208,14247,1685],{},[1125,14249,14250,14252,14254],{},[1208,14251,1685],{},[1204,14253,2342],{},[6964,14255,14256,14259],{},[1128,14257,14258],{},"e",[1125,14260,14261,14263,14272],{},[1204,14262,5991],{},[14264,14265,14266,14268,14270],"msubsup",{},[1128,14267,1268],{},[1128,14269,9875],{},[1128,14271,9471],{"mathvariant":1450},[1128,14273,12502],{},[1132,14275,14276],{"encoding":1134},"p_i = \\frac{1}{1+e^{-x_i^{\\top}\\beta}}",[1109,14278,14280,14335],{"className":14279,"ariaHidden":1139},[1138],[1109,14281,14283,14286,14326,14329,14332],{"className":14282},[1143],[1109,14284],{"className":14285,"style":1182},[1147],[1109,14287,14289,14292],{"className":14288},[1152],[1109,14290,1101],{"className":14291},[1152,1153],[1109,14293,14295],{"className":14294},[1293],[1109,14296,14298,14318],{"className":14297},[1297,1298],[1109,14299,14301,14315],{"className":14300},[1302],[1109,14302,14304],{"className":14303,"style":14151},[1306],[1109,14305,14306,14309],{"style":1310},[1109,14307],{"className":14308,"style":1315},[1314],[1109,14310,14312],{"className":14311},[1319,1320,1321,1322],[1109,14313,9875],{"className":14314},[1152,1153,1322],[1109,14316,1330],{"className":14317},[1329],[1109,14319,14321],{"className":14320},[1302],[1109,14322,14324],{"className":14323,"style":1337},[1306],[1109,14325],{},[1109,14327],{"className":14328,"style":1232},[1231],[1109,14330,1582],{"className":14331},[1236],[1109,14333],{"className":14334,"style":1232},[1231],[1109,14336,14338,14342],{"className":14337},[1143],[1109,14339],{"className":14340,"style":14341},[1147],"height:1.5638em;vertical-align:-0.7187em;",[1109,14343,14345,14348,14503],{"className":14344},[1152],[1109,14346],{"className":14347},[1508,4589],[1109,14349,14351],{"className":14350},[4490],[1109,14352,14354,14494],{"className":14353},[1297,1298],[1109,14355,14357,14491],{"className":14356},[1302],[1109,14358,14360,14469,14477],{"className":14359,"style":11331},[1306],[1109,14361,14363,14366],{"style":14362},"top:-2.3396em;",[1109,14364],{"className":14365,"style":4609},[1314],[1109,14367,14369],{"className":14368},[1319,1320,1321,1322],[1109,14370,14372,14375,14378],{"className":14371},[1152,1322],[1109,14373,1685],{"className":14374},[1152,1322],[1109,14376,2342],{"className":14377},[2501,1322],[1109,14379,14381,14384],{"className":14380},[1152,1322],[1109,14382,14258],{"className":14383},[1152,1153,1322],[1109,14385,14387],{"className":14386},[1293],[1109,14388,14390],{"className":14389},[1297],[1109,14391,14393],{"className":14392},[1302],[1109,14394,14397],{"className":14395,"style":14396},[1306],"height:1.2149em;",[1109,14398,14400,14404],{"style":14399},"top:-3.2149em;margin-right:0.0714em;",[1109,14401],{"className":14402,"style":14403},[1314],"height:2.7423em;",[1109,14405,14407],{"className":14406},[1319,7992,6067,1322],[1109,14408,14410,14413,14466],{"className":14409},[1152,1322],[1109,14411,5991],{"className":14412},[1152,1322],[1109,14414,14416,14419],{"className":14415},[1152,1322],[1109,14417,1268],{"className":14418},[1152,1153,1322],[1109,14420,14422],{"className":14421},[1293],[1109,14423,14425,14457],{"className":14424},[1297,1298],[1109,14426,14428,14454],{"className":14427},[1302],[1109,14429,14432,14442],{"className":14430,"style":14431},[1306],"height:1.0392em;",[1109,14433,14435,14439],{"style":14434},"top:-2.1837em;margin-left:0em;margin-right:0.1em;",[1109,14436],{"className":14437,"style":14438},[1314],"height:2.6944em;",[1109,14440,9875],{"className":14441},[1152,1153,1322],[1109,14443,14445,14448],{"style":14444},"top:-3.0392em;margin-right:0.1em;",[1109,14446],{"className":14447,"style":14438},[1314],[1109,14449,14451],{"className":14450},[1152,1322],[1109,14452,9471],{"className":14453},[1152,1322],[1109,14455,1330],{"className":14456},[1329],[1109,14458,14460],{"className":14459},[1302],[1109,14461,14464],{"className":14462,"style":14463},[1306],"height:0.5107em;",[1109,14465],{},[1109,14467,12502],{"className":14468,"style":12530},[1152,1153,1322],[1109,14470,14471,14474],{"style":4630},[1109,14472],{"className":14473,"style":4609},[1314],[1109,14475],{"className":14476,"style":4638},[4637],[1109,14478,14479,14482],{"style":11356},[1109,14480],{"className":14481,"style":4609},[1314],[1109,14483,14485],{"className":14484},[1319,1320,1321,1322],[1109,14486,14488],{"className":14487},[1152,1322],[1109,14489,1685],{"className":14490},[1152,1322],[1109,14492,1330],{"className":14493},[1329],[1109,14495,14497],{"className":14496},[1302],[1109,14498,14501],{"className":14499,"style":14500},[1306],"height:0.7187em;",[1109,14502],{},[1109,14504],{"className":14505},[1515,4589],"。似然函数与对数似然分别为",[1101,14508,14509],{},[1109,14510,14512,14660],{"className":14511},[1112],[1109,14513,14515],{"className":14514},[1116],[1118,14516,14517],{"xmlns":1120},[1122,14518,14519,14657],{},[1125,14520,14521,14524,14526,14528,14530,14532,14539,14551,14553,14555,14557,14563,14579,14581,14584,14587,14589,14591,14593,14595,14602,14604,14610,14613,14615,14621,14623,14625,14627,14629,14635,14637,14639,14641,14643,14645,14647,14653,14655],{},[1128,14522,14523],{},"L",[1204,14525,1457],{"stretchy":1456},[1128,14527,12502],{},[1204,14529,1462],{"stretchy":1456},[1204,14531,1582],{},[1264,14533,14534,14537],{},[1204,14535,14536],{},"∏",[1128,14538,9875],{},[14264,14540,14541,14543,14545],{},[1128,14542,1101],{},[1128,14544,9875],{},[1264,14546,14547,14549],{},[1128,14548,14113],{},[1128,14550,9875],{},[1204,14552,1457],{"stretchy":1456},[1208,14554,1685],{},[1204,14556,5991],{},[1264,14558,14559,14561],{},[1128,14560,1101],{},[1128,14562,9875],{},[6964,14564,14565,14567],{},[1204,14566,1462],{"stretchy":1456},[1125,14568,14569,14571,14573],{},[1208,14570,1685],{},[1204,14572,5991],{},[1264,14574,14575,14577],{},[1128,14576,14113],{},[1128,14578,9875],{},[1204,14580,1910],{"separator":1139},[1231,14582],{"width":14583},"2em",[1128,14585,14586],{"mathvariant":1450},"ℓ",[1204,14588,1457],{"stretchy":1456},[1128,14590,12502],{},[1204,14592,1462],{"stretchy":1456},[1204,14594,1582],{},[1264,14596,14597,14600],{},[1204,14598,14599],{},"∑",[1128,14601,9875],{},[1204,14603,5519],{"fence":1456,"stretchy":1139,"minsize":5974,"maxsize":5974},[1264,14605,14606,14608],{},[1128,14607,14113],{},[1128,14609,9875],{},[1128,14611,14612],{},"log",[1204,14614,2412],{},[1264,14616,14617,14619],{},[1128,14618,1101],{},[1128,14620,9875],{},[1204,14622,2342],{},[1204,14624,1457],{"stretchy":1456},[1208,14626,1685],{},[1204,14628,5991],{},[1264,14630,14631,14633],{},[1128,14632,14113],{},[1128,14634,9875],{},[1204,14636,1462],{"stretchy":1456},[1128,14638,14612],{},[1204,14640,2412],{},[1204,14642,1457],{"stretchy":1456},[1208,14644,1685],{},[1204,14646,5991],{},[1264,14648,14649,14651],{},[1128,14650,1101],{},[1128,14652,9875],{},[1204,14654,1462],{"stretchy":1456},[1204,14656,5530],{"fence":1456,"stretchy":1139,"minsize":5974,"maxsize":5974},[1132,14658,14659],{"encoding":1134},"L(\\beta) = \\prod_i p_i^{y_i}(1-p_i)^{1-y_i}, \\qquad \\ell(\\beta) = \\sum_i\\big[y_i\\log p_i + (1-y_i)\\log(1-p_i)\\big]",[1109,14661,14663,14690,14857,15009,15166,15187,15259],{"className":14662,"ariaHidden":1139},[1138],[1109,14664,14666,14669,14672,14675,14678,14681,14684,14687],{"className":14665},[1143],[1109,14667],{"className":14668,"style":1498},[1147],[1109,14670,14523],{"className":14671},[1152,1153],[1109,14673,1457],{"className":14674},[1508],[1109,14676,12502],{"className":14677,"style":12530},[1152,1153],[1109,14679,1462],{"className":14680},[1515],[1109,14682],{"className":14683,"style":1232},[1231],[1109,14685,1582],{"className":14686},[1236],[1109,14688],{"className":14689,"style":1232},[1231],[1109,14691,14693,14697,14743,14746,14842,14845,14848,14851,14854],{"className":14692},[1143],[1109,14694],{"className":14695,"style":14696},[1147],"height:1.082em;vertical-align:-0.2997em;",[1109,14698,14700,14706],{"className":14699},[2878],[1109,14701,14536],{"className":14702,"style":14705},[2878,14703,14704],"op-symbol","small-op","position:relative;top:0em;",[1109,14707,14709],{"className":14708},[1293],[1109,14710,14712,14734],{"className":14711},[1297,1298],[1109,14713,14715,14731],{"className":14714},[1302],[1109,14716,14719],{"className":14717,"style":14718},[1306],"height:0.162em;",[1109,14720,14722,14725],{"style":14721},"top:-2.4003em;margin-left:0em;margin-right:0.05em;",[1109,14723],{"className":14724,"style":1315},[1314],[1109,14726,14728],{"className":14727},[1319,1320,1321,1322],[1109,14729,9875],{"className":14730},[1152,1153,1322],[1109,14732,1330],{"className":14733},[1329],[1109,14735,14737],{"className":14736},[1302],[1109,14738,14741],{"className":14739,"style":14740},[1306],"height:0.2997em;",[1109,14742],{},[1109,14744],{"className":14745,"style":1540},[1231],[1109,14747,14749,14752],{"className":14748},[1152],[1109,14750,1101],{"className":14751},[1152,1153],[1109,14753,14755],{"className":14754},[1293],[1109,14756,14758,14833],{"className":14757},[1297,1298],[1109,14759,14761,14830],{"className":14760},[1302],[1109,14762,14765,14777],{"className":14763,"style":14764},[1306],"height:0.7823em;",[1109,14766,14768,14771],{"style":14767},"top:-2.4231em;margin-left:0em;margin-right:0.05em;",[1109,14769],{"className":14770,"style":1315},[1314],[1109,14772,14774],{"className":14773},[1319,1320,1321,1322],[1109,14775,9875],{"className":14776},[1152,1153,1322],[1109,14778,14780,14783],{"style":14779},"top:-3.1809em;margin-right:0.05em;",[1109,14781],{"className":14782,"style":1315},[1314],[1109,14784,14786],{"className":14785},[1319,1320,1321,1322],[1109,14787,14789],{"className":14788},[1152,1322],[1109,14790,14792,14795],{"className":14791},[1152,1322],[1109,14793,14113],{"className":14794,"style":1186},[1152,1153,1322],[1109,14796,14798],{"className":14797},[1293],[1109,14799,14801,14822],{"className":14800},[1297,1298],[1109,14802,14804,14819],{"className":14803},[1302],[1109,14805,14807],{"className":14806,"style":10028},[1306],[1109,14808,14810,14813],{"style":14809},"top:-2.357em;margin-left:-0.0359em;margin-right:0.0714em;",[1109,14811],{"className":14812,"style":7988},[1314],[1109,14814,14816],{"className":14815},[1319,7992,6067,1322],[1109,14817,9875],{"className":14818},[1152,1153,1322],[1109,14820,1330],{"className":14821},[1329],[1109,14823,14825],{"className":14824},[1302],[1109,14826,14828],{"className":14827,"style":8043},[1306],[1109,14829],{},[1109,14831,1330],{"className":14832},[1329],[1109,14834,14836],{"className":14835},[1302],[1109,14837,14840],{"className":14838,"style":14839},[1306],"height:0.2769em;",[1109,14841],{},[1109,14843,1457],{"className":14844},[1508],[1109,14846,1685],{"className":14847},[1152],[1109,14849],{"className":14850,"style":2497},[1231],[1109,14852,5991],{"className":14853},[2501],[1109,14855],{"className":14856,"style":2497},[1231],[1109,14858,14860,14863,14903,14978,14981,14985,14988,14991,14994,14997,15000,15003,15006],{"className":14859},[1143],[1109,14861],{"className":14862,"style":8085},[1147],[1109,14864,14866,14869],{"className":14865},[1152],[1109,14867,1101],{"className":14868},[1152,1153],[1109,14870,14872],{"className":14871},[1293],[1109,14873,14875,14895],{"className":14874},[1297,1298],[1109,14876,14878,14892],{"className":14877},[1302],[1109,14879,14881],{"className":14880,"style":14151},[1306],[1109,14882,14883,14886],{"style":1310},[1109,14884],{"className":14885,"style":1315},[1314],[1109,14887,14889],{"className":14888},[1319,1320,1321,1322],[1109,14890,9875],{"className":14891},[1152,1153,1322],[1109,14893,1330],{"className":14894},[1329],[1109,14896,14898],{"className":14897},[1302],[1109,14899,14901],{"className":14900,"style":1337},[1306],[1109,14902],{},[1109,14904,14906,14909],{"className":14905},[1515],[1109,14907,1462],{"className":14908},[1515],[1109,14910,14912],{"className":14911},[1293],[1109,14913,14915],{"className":14914},[1297],[1109,14916,14918],{"className":14917},[1302],[1109,14919,14921],{"className":14920,"style":8144},[1306],[1109,14922,14923,14926],{"style":7044},[1109,14924],{"className":14925,"style":1315},[1314],[1109,14927,14929],{"className":14928},[1319,1320,1321,1322],[1109,14930,14932,14935,14938],{"className":14931},[1152,1322],[1109,14933,1685],{"className":14934},[1152,1322],[1109,14936,5991],{"className":14937},[2501,1322],[1109,14939,14941,14944],{"className":14940},[1152,1322],[1109,14942,14113],{"className":14943,"style":1186},[1152,1153,1322],[1109,14945,14947],{"className":14946},[1293],[1109,14948,14950,14970],{"className":14949},[1297,1298],[1109,14951,14953,14967],{"className":14952},[1302],[1109,14954,14956],{"className":14955,"style":10028},[1306],[1109,14957,14958,14961],{"style":14809},[1109,14959],{"className":14960,"style":7988},[1314],[1109,14962,14964],{"className":14963},[1319,7992,6067,1322],[1109,14965,9875],{"className":14966},[1152,1153,1322],[1109,14968,1330],{"className":14969},[1329],[1109,14971,14973],{"className":14972},[1302],[1109,14974,14976],{"className":14975,"style":8043},[1306],[1109,14977],{},[1109,14979,1910],{"className":14980},[1975],[1109,14982],{"className":14983,"style":14984},[1231],"margin-right:2em;",[1109,14986],{"className":14987,"style":1540},[1231],[1109,14989,14586],{"className":14990},[1152],[1109,14992,1457],{"className":14993},[1508],[1109,14995,12502],{"className":14996,"style":12530},[1152,1153],[1109,14998,1462],{"className":14999},[1515],[1109,15001],{"className":15002,"style":1232},[1231],[1109,15004,1582],{"className":15005},[1236],[1109,15007],{"className":15008,"style":1232},[1231],[1109,15010,15012,15015,15055,15058,15064,15104,15107,15114,15117,15157,15160,15163],{"className":15011},[1143],[1109,15013],{"className":15014,"style":6056},[1147],[1109,15016,15018,15021],{"className":15017},[2878],[1109,15019,14599],{"className":15020,"style":14705},[2878,14703,14704],[1109,15022,15024],{"className":15023},[1293],[1109,15025,15027,15047],{"className":15026},[1297,1298],[1109,15028,15030,15044],{"className":15029},[1302],[1109,15031,15033],{"className":15032,"style":14718},[1306],[1109,15034,15035,15038],{"style":14721},[1109,15036],{"className":15037,"style":1315},[1314],[1109,15039,15041],{"className":15040},[1319,1320,1321,1322],[1109,15042,9875],{"className":15043},[1152,1153,1322],[1109,15045,1330],{"className":15046},[1329],[1109,15048,15050],{"className":15049},[1302],[1109,15051,15053],{"className":15052,"style":14740},[1306],[1109,15054],{},[1109,15056],{"className":15057,"style":1540},[1231],[1109,15059,15061],{"className":15060},[1152],[1109,15062,5519],{"className":15063},[6066,6067],[1109,15065,15067,15070],{"className":15066},[1152],[1109,15068,14113],{"className":15069,"style":1186},[1152,1153],[1109,15071,15073],{"className":15072},[1293],[1109,15074,15076,15096],{"className":15075},[1297,1298],[1109,15077,15079,15093],{"className":15078},[1302],[1109,15080,15082],{"className":15081,"style":14151},[1306],[1109,15083,15084,15087],{"style":2146},[1109,15085],{"className":15086,"style":1315},[1314],[1109,15088,15090],{"className":15089},[1319,1320,1321,1322],[1109,15091,9875],{"className":15092},[1152,1153,1322],[1109,15094,1330],{"className":15095},[1329],[1109,15097,15099],{"className":15098},[1302],[1109,15100,15102],{"className":15101,"style":1337},[1306],[1109,15103],{},[1109,15105],{"className":15106,"style":1540},[1231],[1109,15108,15110,15111],{"className":15109},[2878],"lo",[1109,15112,1170],{"style":15113},"margin-right:0.0139em;",[1109,15115],{"className":15116,"style":1540},[1231],[1109,15118,15120,15123],{"className":15119},[1152],[1109,15121,1101],{"className":15122},[1152,1153],[1109,15124,15126],{"className":15125},[1293],[1109,15127,15129,15149],{"className":15128},[1297,1298],[1109,15130,15132,15146],{"className":15131},[1302],[1109,15133,15135],{"className":15134,"style":14151},[1306],[1109,15136,15137,15140],{"style":1310},[1109,15138],{"className":15139,"style":1315},[1314],[1109,15141,15143],{"className":15142},[1319,1320,1321,1322],[1109,15144,9875],{"className":15145},[1152,1153,1322],[1109,15147,1330],{"className":15148},[1329],[1109,15150,15152],{"className":15151},[1302],[1109,15153,15155],{"className":15154,"style":1337},[1306],[1109,15156],{},[1109,15158],{"className":15159,"style":2497},[1231],[1109,15161,2342],{"className":15162},[2501],[1109,15164],{"className":15165,"style":2497},[1231],[1109,15167,15169,15172,15175,15178,15181,15184],{"className":15168},[1143],[1109,15170],{"className":15171,"style":1498},[1147],[1109,15173,1457],{"className":15174},[1508],[1109,15176,1685],{"className":15177},[1152],[1109,15179],{"className":15180,"style":2497},[1231],[1109,15182,5991],{"className":15183},[2501],[1109,15185],{"className":15186,"style":2497},[1231],[1109,15188,15190,15193,15233,15236,15239,15244,15247,15250,15253,15256],{"className":15189},[1143],[1109,15191],{"className":15192,"style":1498},[1147],[1109,15194,15196,15199],{"className":15195},[1152],[1109,15197,14113],{"className":15198,"style":1186},[1152,1153],[1109,15200,15202],{"className":15201},[1293],[1109,15203,15205,15225],{"className":15204},[1297,1298],[1109,15206,15208,15222],{"className":15207},[1302],[1109,15209,15211],{"className":15210,"style":14151},[1306],[1109,15212,15213,15216],{"style":2146},[1109,15214],{"className":15215,"style":1315},[1314],[1109,15217,15219],{"className":15218},[1319,1320,1321,1322],[1109,15220,9875],{"className":15221},[1152,1153,1322],[1109,15223,1330],{"className":15224},[1329],[1109,15226,15228],{"className":15227},[1302],[1109,15229,15231],{"className":15230,"style":1337},[1306],[1109,15232],{},[1109,15234,1462],{"className":15235},[1515],[1109,15237],{"className":15238,"style":1540},[1231],[1109,15240,15110,15242],{"className":15241},[2878],[1109,15243,1170],{"style":15113},[1109,15245,1457],{"className":15246},[1508],[1109,15248,1685],{"className":15249},[1152],[1109,15251],{"className":15252,"style":2497},[1231],[1109,15254,5991],{"className":15255},[2501],[1109,15257],{"className":15258,"style":2497},[1231],[1109,15260,15262,15265,15305,15308],{"className":15261},[1143],[1109,15263],{"className":15264,"style":6056},[1147],[1109,15266,15268,15271],{"className":15267},[1152],[1109,15269,1101],{"className":15270},[1152,1153],[1109,15272,15274],{"className":15273},[1293],[1109,15275,15277,15297],{"className":15276},[1297,1298],[1109,15278,15280,15294],{"className":15279},[1302],[1109,15281,15283],{"className":15282,"style":14151},[1306],[1109,15284,15285,15288],{"style":1310},[1109,15286],{"className":15287,"style":1315},[1314],[1109,15289,15291],{"className":15290},[1319,1320,1321,1322],[1109,15292,9875],{"className":15293},[1152,1153,1322],[1109,15295,1330],{"className":15296},[1329],[1109,15298,15300],{"className":15299},[1302],[1109,15301,15303],{"className":15302,"style":1337},[1306],[1109,15304],{},[1109,15306,1462],{"className":15307},[1515],[1109,15309,15311],{"className":15310},[1152],[1109,15312,5530],{"className":15313},[6066,6067],[1101,15315,15316,15317,15346],{},"对 ",[1109,15318,15320,15334],{"className":15319},[1112],[1109,15321,15323],{"className":15322},[1116],[1118,15324,15325],{"xmlns":1120},[1122,15326,15327,15331],{},[1125,15328,15329],{},[1128,15330,12502],{},[1132,15332,15333],{"encoding":1134},"\\beta",[1109,15335,15337],{"className":15336,"ariaHidden":1139},[1138],[1109,15338,15340,15343],{"className":15339},[1143],[1109,15341],{"className":15342,"style":1148},[1147],[1109,15344,12502],{"className":15345,"style":12530},[1152,1153]," 分别求一阶导(梯度)、二阶导(Hessian),得到",[1101,15348,15349],{},[1109,15350,15352,15426],{"className":15351},[1112],[1109,15353,15355],{"className":15354},[1116],[1118,15356,15357],{"xmlns":1120},[1122,15358,15359,15423],{},[1125,15360,15361,15363,15365,15367,15369,15371,15373,15380,15382,15384,15386,15388,15390,15392,15394,15400,15402,15404,15406,15408,15410,15412,15418,15421],{},[1128,15362,5611],{"mathvariant":1450},[1128,15364,14586],{"mathvariant":1450},[1204,15366,1457],{"stretchy":1456},[1128,15368,12502],{},[1204,15370,1462],{"stretchy":1456},[1204,15372,1582],{},[6964,15374,15375,15378],{},[1128,15376,15377],{},"X",[1128,15379,9471],{"mathvariant":1450},[1204,15381,1457],{"stretchy":1456},[1128,15383,14113],{},[1204,15385,5991],{},[1128,15387,1101],{},[1204,15389,1462],{"stretchy":1456},[1204,15391,1910],{"separator":1139},[1231,15393],{"width":14583},[6964,15395,15396,15398],{},[1128,15397,5611],{"mathvariant":1450},[1208,15399,1917],{},[1128,15401,14586],{"mathvariant":1450},[1204,15403,1457],{"stretchy":1456},[1128,15405,12502],{},[1204,15407,1462],{"stretchy":1456},[1204,15409,1582],{},[1204,15411,5991],{},[6964,15413,15414,15416],{},[1128,15415,15377],{},[1128,15417,9471],{"mathvariant":1450},[1128,15419,15420],{},"W",[1128,15422,15377],{},[1132,15424,15425],{"encoding":1134},"\\nabla\\ell(\\beta) = X^{\\top}(y - p), \\qquad \\nabla^2\\ell(\\beta) = -X^{\\top}W X",[1109,15427,15429,15459,15513,15584],{"className":15428,"ariaHidden":1139},[1138],[1109,15430,15432,15435,15438,15441,15444,15447,15450,15453,15456],{"className":15431},[1143],[1109,15433],{"className":15434,"style":1498},[1147],[1109,15436,5611],{"className":15437},[1152],[1109,15439,14586],{"className":15440},[1152],[1109,15442,1457],{"className":15443},[1508],[1109,15445,12502],{"className":15446,"style":12530},[1152,1153],[1109,15448,1462],{"className":15449},[1515],[1109,15451],{"className":15452,"style":1232},[1231],[1109,15454,1582],{"className":15455},[1236],[1109,15457],{"className":15458,"style":1232},[1231],[1109,15460,15462,15465,15498,15501,15504,15507,15510],{"className":15461},[1143],[1109,15463],{"className":15464,"style":11150},[1147],[1109,15466,15468,15472],{"className":15467},[1152],[1109,15469,15377],{"className":15470,"style":15471},[1152,1153],"margin-right:0.0785em;",[1109,15473,15475],{"className":15474},[1293],[1109,15476,15478],{"className":15477},[1297],[1109,15479,15481],{"className":15480},[1302],[1109,15482,15484],{"className":15483,"style":11218},[1306],[1109,15485,15486,15489],{"style":7044},[1109,15487],{"className":15488,"style":1315},[1314],[1109,15490,15492],{"className":15491},[1319,1320,1321,1322],[1109,15493,15495],{"className":15494},[1152,1322],[1109,15496,9471],{"className":15497},[1152,1322],[1109,15499,1457],{"className":15500},[1508],[1109,15502,14113],{"className":15503,"style":1186},[1152,1153],[1109,15505],{"className":15506,"style":2497},[1231],[1109,15508,5991],{"className":15509},[2501],[1109,15511],{"className":15512,"style":2497},[1231],[1109,15514,15516,15519,15522,15525,15528,15531,15534,15563,15566,15569,15572,15575,15578,15581],{"className":15515},[1143],[1109,15517],{"className":15518,"style":8085},[1147],[1109,15520,1101],{"className":15521},[1152,1153],[1109,15523,1462],{"className":15524},[1515],[1109,15526,1910],{"className":15527},[1975],[1109,15529],{"className":15530,"style":14984},[1231],[1109,15532],{"className":15533,"style":1540},[1231],[1109,15535,15537,15540],{"className":15536},[1152],[1109,15538,5611],{"className":15539},[1152],[1109,15541,15543],{"className":15542},[1293],[1109,15544,15546],{"className":15545},[1297],[1109,15547,15549],{"className":15548},[1302],[1109,15550,15552],{"className":15551,"style":8144},[1306],[1109,15553,15554,15557],{"style":7044},[1109,15555],{"className":15556,"style":1315},[1314],[1109,15558,15560],{"className":15559},[1319,1320,1321,1322],[1109,15561,1917],{"className":15562},[1152,1322],[1109,15564,14586],{"className":15565},[1152],[1109,15567,1457],{"className":15568},[1508],[1109,15570,12502],{"className":15571,"style":12530},[1152,1153],[1109,15573,1462],{"className":15574},[1515],[1109,15576],{"className":15577,"style":1232},[1231],[1109,15579,1582],{"className":15580},[1236],[1109,15582],{"className":15583,"style":1232},[1231],[1109,15585,15587,15591,15594,15626,15630],{"className":15586},[1143],[1109,15588],{"className":15589,"style":15590},[1147],"height:0.9324em;vertical-align:-0.0833em;",[1109,15592,5991],{"className":15593},[1152],[1109,15595,15597,15600],{"className":15596},[1152],[1109,15598,15377],{"className":15599,"style":15471},[1152,1153],[1109,15601,15603],{"className":15602},[1293],[1109,15604,15606],{"className":15605},[1297],[1109,15607,15609],{"className":15608},[1302],[1109,15610,15612],{"className":15611,"style":11218},[1306],[1109,15613,15614,15617],{"style":7044},[1109,15615],{"className":15616,"style":1315},[1314],[1109,15618,15620],{"className":15619},[1319,1320,1321,1322],[1109,15621,15623],{"className":15622},[1152,1322],[1109,15624,9471],{"className":15625},[1152,1322],[1109,15627,15420],{"className":15628,"style":15629},[1152,1153],"margin-right:0.1389em;",[1109,15631,15377],{"className":15632,"style":15471},[1152,1153],[1101,15634,15635,15636,15832,15833,15861],{},"其中 ",[1109,15637,15639,15682],{"className":15638},[1112],[1109,15640,15642],{"className":15641},[1116],[1118,15643,15644],{"xmlns":1120},[1122,15645,15646,15679],{},[1125,15647,15648,15650,15652,15655,15657,15663,15665,15667,15669,15675,15677],{},[1128,15649,15420],{},[1204,15651,1582],{},[1471,15653,15654],{},"diag",[1204,15656,1457],{"fence":1456,"stretchy":1139,"minsize":5974,"maxsize":5974},[1264,15658,15659,15661],{},[1128,15660,1101],{},[1128,15662,9875],{},[1204,15664,1457],{"stretchy":1456},[1208,15666,1685],{},[1204,15668,5991],{},[1264,15670,15671,15673],{},[1128,15672,1101],{},[1128,15674,9875],{},[1204,15676,1462],{"stretchy":1456},[1204,15678,1462],{"fence":1456,"stretchy":1139,"minsize":5974,"maxsize":5974},[1132,15680,15681],{"encoding":1134},"W = \\text{diag}\\big(p_i(1-p_i)\\big)",[1109,15683,15685,15703,15777],{"className":15684,"ariaHidden":1139},[1138],[1109,15686,15688,15691,15694,15697,15700],{"className":15687},[1143],[1109,15689],{"className":15690,"style":3026},[1147],[1109,15692,15420],{"className":15693,"style":15629},[1152,1153],[1109,15695],{"className":15696,"style":1232},[1231],[1109,15698,1582],{"className":15699},[1236],[1109,15701],{"className":15702,"style":1232},[1231],[1109,15704,15706,15709,15716,15722,15762,15765,15768,15771,15774],{"className":15705},[1143],[1109,15707],{"className":15708,"style":6056},[1147],[1109,15710,15713],{"className":15711},[1152,15712],"text",[1109,15714,15654],{"className":15715},[1152],[1109,15717,15719],{"className":15718},[1152],[1109,15720,1457],{"className":15721},[6066,6067],[1109,15723,15725,15728],{"className":15724},[1152],[1109,15726,1101],{"className":15727},[1152,1153],[1109,15729,15731],{"className":15730},[1293],[1109,15732,15734,15754],{"className":15733},[1297,1298],[1109,15735,15737,15751],{"className":15736},[1302],[1109,15738,15740],{"className":15739,"style":14151},[1306],[1109,15741,15742,15745],{"style":1310},[1109,15743],{"className":15744,"style":1315},[1314],[1109,15746,15748],{"className":15747},[1319,1320,1321,1322],[1109,15749,9875],{"className":15750},[1152,1153,1322],[1109,15752,1330],{"className":15753},[1329],[1109,15755,15757],{"className":15756},[1302],[1109,15758,15760],{"className":15759,"style":1337},[1306],[1109,15761],{},[1109,15763,1457],{"className":15764},[1508],[1109,15766,1685],{"className":15767},[1152],[1109,15769],{"className":15770,"style":2497},[1231],[1109,15772,5991],{"className":15773},[2501],[1109,15775],{"className":15776,"style":2497},[1231],[1109,15778,15780,15783,15823,15826],{"className":15779},[1143],[1109,15781],{"className":15782,"style":6056},[1147],[1109,15784,15786,15789],{"className":15785},[1152],[1109,15787,1101],{"className":15788},[1152,1153],[1109,15790,15792],{"className":15791},[1293],[1109,15793,15795,15815],{"className":15794},[1297,1298],[1109,15796,15798,15812],{"className":15797},[1302],[1109,15799,15801],{"className":15800,"style":14151},[1306],[1109,15802,15803,15806],{"style":1310},[1109,15804],{"className":15805,"style":1315},[1314],[1109,15807,15809],{"className":15808},[1319,1320,1321,1322],[1109,15810,9875],{"className":15811},[1152,1153,1322],[1109,15813,1330],{"className":15814},[1329],[1109,15816,15818],{"className":15817},[1302],[1109,15819,15821],{"className":15820,"style":1337},[1306],[1109,15822],{},[1109,15824,1462],{"className":15825},[1515],[1109,15827,15829],{"className":15828},[1152],[1109,15830,1462],{"className":15831},[6066,6067],"。因而 ",[1109,15834,15836,15849],{"className":15835},[1112],[1109,15837,15839],{"className":15838},[1116],[1118,15840,15841],{"xmlns":1120},[1122,15842,15843,15847],{},[1125,15844,15845],{},[1128,15846,12502],{},[1132,15848,15333],{"encoding":1134},[1109,15850,15852],{"className":15851,"ariaHidden":1139},[1138],[1109,15853,15855,15858],{"className":15854},[1143],[1109,15856],{"className":15857,"style":1148},[1147],[1109,15859,12502],{"className":15860,"style":12530},[1152,1153]," 可以通过下面的迭代(牛顿法 \u002F IRLS)来估计",[1101,15863,15864],{},[1109,15865,15867,15939],{"className":15866},[1112],[1109,15868,15870],{"className":15869},[1116],[1118,15871,15872],{"xmlns":1120},[1122,15873,15874,15936],{},[1125,15875,15876,15888,15890,15896,15898,15900,15906,15908,15910,15920,15926,15928,15930,15932,15934],{},[1264,15877,15878,15880],{},[1128,15879,12502],{},[1125,15881,15882,15884,15886],{},[1128,15883,8620],{},[1204,15885,2342],{},[1208,15887,1685],{},[1204,15889,1582],{},[1264,15891,15892,15894],{},[1128,15893,12502],{},[1128,15895,8620],{},[1204,15897,2342],{},[1204,15899,1457],{"stretchy":1456},[6964,15901,15902,15904],{},[1128,15903,15377],{},[1128,15905,9471],{"mathvariant":1450},[1128,15907,15420],{},[1128,15909,15377],{},[6964,15911,15912,15914],{},[1204,15913,1462],{"stretchy":1456},[1125,15915,15916,15918],{},[1204,15917,5991],{},[1208,15919,1685],{},[6964,15921,15922,15924],{},[1128,15923,15377],{},[1128,15925,9471],{"mathvariant":1450},[1204,15927,1457],{"stretchy":1456},[1128,15929,14113],{},[1204,15931,5991],{},[1128,15933,1101],{},[1204,15935,1462],{"stretchy":1456},[1132,15937,15938],{"encoding":1134},"\\beta_{k+1} = \\beta_k + (X^{\\top}W X)^{-1}X^{\\top}(y - p)",[1109,15940,15942,16008,16063,16192],{"className":15941,"ariaHidden":1139},[1138],[1109,15943,15945,15949,15999,16002,16005],{"className":15944},[1143],[1109,15946],{"className":15947,"style":15948},[1147],"height:0.9028em;vertical-align:-0.2083em;",[1109,15950,15952,15955],{"className":15951},[1152],[1109,15953,12502],{"className":15954,"style":12530},[1152,1153],[1109,15956,15958],{"className":15957},[1293],[1109,15959,15961,15991],{"className":15960},[1297,1298],[1109,15962,15964,15988],{"className":15963},[1302],[1109,15965,15967],{"className":15966,"style":8735},[1306],[1109,15968,15970,15973],{"style":15969},"top:-2.55em;margin-left:-0.0528em;margin-right:0.05em;",[1109,15971],{"className":15972,"style":1315},[1314],[1109,15974,15976],{"className":15975},[1319,1320,1321,1322],[1109,15977,15979,15982,15985],{"className":15978},[1152,1322],[1109,15980,8620],{"className":15981,"style":8750},[1152,1153,1322],[1109,15983,2342],{"className":15984},[2501,1322],[1109,15986,1685],{"className":15987},[1152,1322],[1109,15989,1330],{"className":15990},[1329],[1109,15992,15994],{"className":15993},[1302],[1109,15995,15997],{"className":15996,"style":10663},[1306],[1109,15998],{},[1109,16000],{"className":16001,"style":1232},[1231],[1109,16003,1582],{"className":16004},[1236],[1109,16006],{"className":16007,"style":1232},[1231],[1109,16009,16011,16014,16054,16057,16060],{"className":16010},[1143],[1109,16012],{"className":16013,"style":1148},[1147],[1109,16015,16017,16020],{"className":16016},[1152],[1109,16018,12502],{"className":16019,"style":12530},[1152,1153],[1109,16021,16023],{"className":16022},[1293],[1109,16024,16026,16046],{"className":16025},[1297,1298],[1109,16027,16029,16043],{"className":16028},[1302],[1109,16030,16032],{"className":16031,"style":8735},[1306],[1109,16033,16034,16037],{"style":15969},[1109,16035],{"className":16036,"style":1315},[1314],[1109,16038,16040],{"className":16039},[1319,1320,1321,1322],[1109,16041,8620],{"className":16042,"style":8750},[1152,1153,1322],[1109,16044,1330],{"className":16045},[1329],[1109,16047,16049],{"className":16048},[1302],[1109,16050,16052],{"className":16051,"style":1337},[1306],[1109,16053],{},[1109,16055],{"className":16056,"style":2497},[1231],[1109,16058,2342],{"className":16059},[2501],[1109,16061],{"className":16062,"style":2497},[1231],[1109,16064,16066,16069,16072,16104,16107,16110,16145,16177,16180,16183,16186,16189],{"className":16065},[1143],[1109,16067],{"className":16068,"style":11150},[1147],[1109,16070,1457],{"className":16071},[1508],[1109,16073,16075,16078],{"className":16074},[1152],[1109,16076,15377],{"className":16077,"style":15471},[1152,1153],[1109,16079,16081],{"className":16080},[1293],[1109,16082,16084],{"className":16083},[1297],[1109,16085,16087],{"className":16086},[1302],[1109,16088,16090],{"className":16089,"style":11218},[1306],[1109,16091,16092,16095],{"style":7044},[1109,16093],{"className":16094,"style":1315},[1314],[1109,16096,16098],{"className":16097},[1319,1320,1321,1322],[1109,16099,16101],{"className":16100},[1152,1322],[1109,16102,9471],{"className":16103},[1152,1322],[1109,16105,15420],{"className":16106,"style":15629},[1152,1153],[1109,16108,15377],{"className":16109,"style":15471},[1152,1153],[1109,16111,16113,16116],{"className":16112},[1515],[1109,16114,1462],{"className":16115},[1515],[1109,16117,16119],{"className":16118},[1293],[1109,16120,16122],{"className":16121},[1297],[1109,16123,16125],{"className":16124},[1302],[1109,16126,16128],{"className":16127,"style":8144},[1306],[1109,16129,16130,16133],{"style":7044},[1109,16131],{"className":16132,"style":1315},[1314],[1109,16134,16136],{"className":16135},[1319,1320,1321,1322],[1109,16137,16139,16142],{"className":16138},[1152,1322],[1109,16140,5991],{"className":16141},[1152,1322],[1109,16143,1685],{"className":16144},[1152,1322],[1109,16146,16148,16151],{"className":16147},[1152],[1109,16149,15377],{"className":16150,"style":15471},[1152,1153],[1109,16152,16154],{"className":16153},[1293],[1109,16155,16157],{"className":16156},[1297],[1109,16158,16160],{"className":16159},[1302],[1109,16161,16163],{"className":16162,"style":11218},[1306],[1109,16164,16165,16168],{"style":7044},[1109,16166],{"className":16167,"style":1315},[1314],[1109,16169,16171],{"className":16170},[1319,1320,1321,1322],[1109,16172,16174],{"className":16173},[1152,1322],[1109,16175,9471],{"className":16176},[1152,1322],[1109,16178,1457],{"className":16179},[1508],[1109,16181,14113],{"className":16182,"style":1186},[1152,1153],[1109,16184],{"className":16185,"style":2497},[1231],[1109,16187,5991],{"className":16188},[2501],[1109,16190],{"className":16191,"style":2497},[1231],[1109,16193,16195,16198,16201],{"className":16194},[1143],[1109,16196],{"className":16197,"style":1498},[1147],[1109,16199,1101],{"className":16200},[1152,1153],[1109,16202,1462],{"className":16203},[1515],[1101,16205,16206],{},"每次迭代后计算变化量并与给定阈值比较,判断是否收敛。",[10214,16208],{},[1091,16210,16212],{"id":16211},"三交替方向乘子法admm","三、交替方向乘子法(ADMM)",[1101,16214,16215,16216,16369,16370,16458,16459,1662],{},"考虑带等式约束的问题 ",[1109,16217,16219,16266],{"className":16218},[1112],[1109,16220,16222],{"className":16221},[1116],[1118,16223,16224],{"xmlns":1120},[1122,16225,16226,16263],{},[1125,16227,16228,16245,16247,16249,16251,16253,16255,16257,16259,16261],{},[1264,16229,16230,16236],{},[1125,16231,16232,16234],{},[1128,16233,10241],{},[1204,16235,2412],{},[1125,16237,16238,16240,16242],{},[1128,16239,1268],{},[1204,16241,1910],{"separator":1139},[1128,16243,16244],{},"z",[1128,16246,1130],{},[1204,16248,1457],{"stretchy":1456},[1128,16250,1268],{},[1204,16252,1462],{"stretchy":1456},[1204,16254,2342],{},[1128,16256,1170],{},[1204,16258,1457],{"stretchy":1456},[1128,16260,16244],{},[1204,16262,1462],{"stretchy":1456},[1132,16264,16265],{"encoding":1134},"\\min_{x,z} f(x) + g(z)",[1109,16267,16269,16351],{"className":16268,"ariaHidden":1139},[1138],[1109,16270,16272,16276,16327,16330,16333,16336,16339,16342,16345,16348],{"className":16271},[1143],[1109,16273],{"className":16274,"style":16275},[1147],"height:1.0361em;vertical-align:-0.2861em;",[1109,16277,16279,16282],{"className":16278},[2878],[1109,16280,10241],{"className":16281},[2878],[1109,16283,16285],{"className":16284},[1293],[1109,16286,16288,16318],{"className":16287},[1297,1298],[1109,16289,16291,16315],{"className":16290},[1302],[1109,16292,16294],{"className":16293,"style":4547},[1306],[1109,16295,16296,16299],{"style":4550},[1109,16297],{"className":16298,"style":1315},[1314],[1109,16300,16302],{"className":16301},[1319,1320,1321,1322],[1109,16303,16305,16308,16311],{"className":16304},[1152,1322],[1109,16306,1268],{"className":16307},[1152,1153,1322],[1109,16309,1910],{"className":16310},[1975,1322],[1109,16312,16244],{"className":16313,"style":16314},[1152,1153,1322],"margin-right:0.044em;",[1109,16316,1330],{"className":16317},[1329],[1109,16319,16321],{"className":16320},[1302],[1109,16322,16325],{"className":16323,"style":16324},[1306],"height:0.2861em;",[1109,16326],{},[1109,16328],{"className":16329,"style":1540},[1231],[1109,16331,1130],{"className":16332,"style":1154},[1152,1153],[1109,16334,1457],{"className":16335},[1508],[1109,16337,1268],{"className":16338},[1152,1153],[1109,16340,1462],{"className":16341},[1515],[1109,16343],{"className":16344,"style":2497},[1231],[1109,16346,2342],{"className":16347},[2501],[1109,16349],{"className":16350,"style":2497},[1231],[1109,16352,16354,16357,16360,16363,16366],{"className":16353},[1143],[1109,16355],{"className":16356,"style":1498},[1147],[1109,16358,1170],{"className":16359,"style":1186},[1152,1153],[1109,16361,1457],{"className":16362},[1508],[1109,16364,16244],{"className":16365,"style":16314},[1152,1153],[1109,16367,1462],{"className":16368},[1515]," s.t. ",[1109,16371,16373,16402],{"className":16372},[1112],[1109,16374,16376],{"className":16375},[1116],[1118,16377,16378],{"xmlns":1120},[1122,16379,16380,16399],{},[1125,16381,16382,16385,16387,16389,16392,16394,16396],{},[1128,16383,16384],{},"A",[1128,16386,1268],{},[1204,16388,2342],{},[1128,16390,16391],{},"B",[1128,16393,16244],{},[1204,16395,1582],{},[1128,16397,16398],{},"c",[1132,16400,16401],{"encoding":1134},"Ax + Bz = c",[1109,16403,16405,16427,16449],{"className":16404,"ariaHidden":1139},[1138],[1109,16406,16408,16412,16415,16418,16421,16424],{"className":16407},[1143],[1109,16409],{"className":16410,"style":16411},[1147],"height:0.7667em;vertical-align:-0.0833em;",[1109,16413,16384],{"className":16414},[1152,1153],[1109,16416,1268],{"className":16417},[1152,1153],[1109,16419],{"className":16420,"style":2497},[1231],[1109,16422,2342],{"className":16423},[2501],[1109,16425],{"className":16426,"style":2497},[1231],[1109,16428,16430,16433,16437,16440,16443,16446],{"className":16429},[1143],[1109,16431],{"className":16432,"style":3026},[1147],[1109,16434,16391],{"className":16435,"style":16436},[1152,1153],"margin-right:0.0502em;",[1109,16438,16244],{"className":16439,"style":16314},[1152,1153],[1109,16441],{"className":16442,"style":1232},[1231],[1109,16444,1582],{"className":16445},[1236],[1109,16447],{"className":16448,"style":1232},[1231],[1109,16450,16452,16455],{"className":16451},[1143],[1109,16453],{"className":16454,"style":7312},[1147],[1109,16456,16398],{"className":16457},[1152,1153],"。定义",[1104,16460,16461],{},"增广拉格朗日函数",[1101,16463,16464],{},[1109,16465,16467,16576],{"className":16466},[1112],[1109,16468,16470],{"className":16469},[1116],[1118,16471,16472],{"xmlns":1120},[1122,16473,16474,16573],{},[1125,16475,16476,16483,16485,16487,16489,16491,16493,16495,16497,16499,16501,16503,16505,16507,16509,16511,16513,16515,16517,16519,16525,16527,16529,16531,16533,16535,16537,16539,16541,16543,16545,16551,16553,16555,16557,16559,16561,16563,16565,16567],{},[1264,16477,16478,16480],{},[1128,16479,14523],{},[1128,16481,16482],{},"ρ",[1204,16484,1457],{"stretchy":1456},[1128,16486,1268],{},[1204,16488,1910],{"separator":1139},[1128,16490,16244],{},[1204,16492,1910],{"separator":1139},[1128,16494,5892],{},[1204,16496,1462],{"stretchy":1456},[1204,16498,1582],{},[1128,16500,1130],{},[1204,16502,1457],{"stretchy":1456},[1128,16504,1268],{},[1204,16506,1462],{"stretchy":1456},[1204,16508,2342],{},[1128,16510,1170],{},[1204,16512,1457],{"stretchy":1456},[1128,16514,16244],{},[1204,16516,1462],{"stretchy":1456},[1204,16518,2342],{},[6964,16520,16521,16523],{},[1128,16522,5892],{},[1128,16524,9471],{"mathvariant":1450},[1204,16526,1457],{"stretchy":1456},[1128,16528,16384],{},[1128,16530,1268],{},[1204,16532,2342],{},[1128,16534,16391],{},[1128,16536,16244],{},[1204,16538,5991],{},[1128,16540,16398],{},[1204,16542,1462],{"stretchy":1456},[1204,16544,2342],{},[4490,16546,16547,16549],{},[1128,16548,16482],{},[1208,16550,1917],{},[1128,16552,8631],{"mathvariant":1450},[1128,16554,16384],{},[1128,16556,1268],{},[1204,16558,2342],{},[1128,16560,16391],{},[1128,16562,16244],{},[1204,16564,5991],{},[1128,16566,16398],{},[6964,16568,16569,16571],{},[1128,16570,8631],{"mathvariant":1450},[1208,16572,1917],{},[1132,16574,16575],{"encoding":1134},"L_\\rho(x,z,\\lambda) = f(x) + g(z) + \\lambda^{\\top}(Ax + Bz - c) + \\frac{\\rho}{2}\\|Ax + Bz - c\\|^2",[1109,16577,16579,16661,16688,16715,16771,16792,16813,16907,16928],{"className":16578,"ariaHidden":1139},[1138],[1109,16580,16582,16585,16625,16628,16631,16634,16637,16640,16643,16646,16649,16652,16655,16658],{"className":16581},[1143],[1109,16583],{"className":16584,"style":16275},[1147],[1109,16586,16588,16591],{"className":16587},[1152],[1109,16589,14523],{"className":16590},[1152,1153],[1109,16592,16594],{"className":16593},[1293],[1109,16595,16597,16617],{"className":16596},[1297,1298],[1109,16598,16600,16614],{"className":16599},[1302],[1109,16601,16603],{"className":16602,"style":4547},[1306],[1109,16604,16605,16608],{"style":1310},[1109,16606],{"className":16607,"style":1315},[1314],[1109,16609,16611],{"className":16610},[1319,1320,1321,1322],[1109,16612,16482],{"className":16613},[1152,1153,1322],[1109,16615,1330],{"className":16616},[1329],[1109,16618,16620],{"className":16619},[1302],[1109,16621,16623],{"className":16622,"style":16324},[1306],[1109,16624],{},[1109,16626,1457],{"className":16627},[1508],[1109,16629,1268],{"className":16630},[1152,1153],[1109,16632,1910],{"className":16633},[1975],[1109,16635],{"className":16636,"style":1540},[1231],[1109,16638,16244],{"className":16639,"style":16314},[1152,1153],[1109,16641,1910],{"className":16642},[1975],[1109,16644],{"className":16645,"style":1540},[1231],[1109,16647,5892],{"className":16648},[1152,1153],[1109,16650,1462],{"className":16651},[1515],[1109,16653],{"className":16654,"style":1232},[1231],[1109,16656,1582],{"className":16657},[1236],[1109,16659],{"className":16660,"style":1232},[1231],[1109,16662,16664,16667,16670,16673,16676,16679,16682,16685],{"className":16663},[1143],[1109,16665],{"className":16666,"style":1498},[1147],[1109,16668,1130],{"className":16669,"style":1154},[1152,1153],[1109,16671,1457],{"className":16672},[1508],[1109,16674,1268],{"className":16675},[1152,1153],[1109,16677,1462],{"className":16678},[1515],[1109,16680],{"className":16681,"style":2497},[1231],[1109,16683,2342],{"className":16684},[2501],[1109,16686],{"className":16687,"style":2497},[1231],[1109,16689,16691,16694,16697,16700,16703,16706,16709,16712],{"className":16690},[1143],[1109,16692],{"className":16693,"style":1498},[1147],[1109,16695,1170],{"className":16696,"style":1186},[1152,1153],[1109,16698,1457],{"className":16699},[1508],[1109,16701,16244],{"className":16702,"style":16314},[1152,1153],[1109,16704,1462],{"className":16705},[1515],[1109,16707],{"className":16708,"style":2497},[1231],[1109,16710,2342],{"className":16711},[2501],[1109,16713],{"className":16714,"style":2497},[1231],[1109,16716,16718,16721,16753,16756,16759,16762,16765,16768],{"className":16717},[1143],[1109,16719],{"className":16720,"style":11150},[1147],[1109,16722,16724,16727],{"className":16723},[1152],[1109,16725,5892],{"className":16726},[1152,1153],[1109,16728,16730],{"className":16729},[1293],[1109,16731,16733],{"className":16732},[1297],[1109,16734,16736],{"className":16735},[1302],[1109,16737,16739],{"className":16738,"style":11218},[1306],[1109,16740,16741,16744],{"style":7044},[1109,16742],{"className":16743,"style":1315},[1314],[1109,16745,16747],{"className":16746},[1319,1320,1321,1322],[1109,16748,16750],{"className":16749},[1152,1322],[1109,16751,9471],{"className":16752},[1152,1322],[1109,16754,1457],{"className":16755},[1508],[1109,16757,16384],{"className":16758},[1152,1153],[1109,16760,1268],{"className":16761},[1152,1153],[1109,16763],{"className":16764,"style":2497},[1231],[1109,16766,2342],{"className":16767},[2501],[1109,16769],{"className":16770,"style":2497},[1231],[1109,16772,16774,16777,16780,16783,16786,16789],{"className":16773},[1143],[1109,16775],{"className":16776,"style":16411},[1147],[1109,16778,16391],{"className":16779,"style":16436},[1152,1153],[1109,16781,16244],{"className":16782,"style":16314},[1152,1153],[1109,16784],{"className":16785,"style":2497},[1231],[1109,16787,5991],{"className":16788},[2501],[1109,16790],{"className":16791,"style":2497},[1231],[1109,16793,16795,16798,16801,16804,16807,16810],{"className":16794},[1143],[1109,16796],{"className":16797,"style":1498},[1147],[1109,16799,16398],{"className":16800},[1152,1153],[1109,16802,1462],{"className":16803},[1515],[1109,16805],{"className":16806,"style":2497},[1231],[1109,16808,2342],{"className":16809},[2501],[1109,16811],{"className":16812,"style":2497},[1231],[1109,16814,16816,16820,16889,16892,16895,16898,16901,16904],{"className":16815},[1143],[1109,16817],{"className":16818,"style":16819},[1147],"height:1.095em;vertical-align:-0.345em;",[1109,16821,16823,16826,16886],{"className":16822},[1152],[1109,16824],{"className":16825},[1508,4589],[1109,16827,16829],{"className":16828},[4490],[1109,16830,16832,16878],{"className":16831},[1297,1298],[1109,16833,16835,16875],{"className":16834},[1302],[1109,16836,16839,16853,16861],{"className":16837,"style":16838},[1306],"height:0.7475em;",[1109,16840,16841,16844],{"style":4605},[1109,16842],{"className":16843,"style":4609},[1314],[1109,16845,16847],{"className":16846},[1319,1320,1321,1322],[1109,16848,16850],{"className":16849},[1152,1322],[1109,16851,1917],{"className":16852},[1152,1322],[1109,16854,16855,16858],{"style":4630},[1109,16856],{"className":16857,"style":4609},[1314],[1109,16859],{"className":16860,"style":4638},[4637],[1109,16862,16863,16866],{"style":9614},[1109,16864],{"className":16865,"style":4609},[1314],[1109,16867,16869],{"className":16868},[1319,1320,1321,1322],[1109,16870,16872],{"className":16871},[1152,1322],[1109,16873,16482],{"className":16874},[1152,1153,1322],[1109,16876,1330],{"className":16877},[1329],[1109,16879,16881],{"className":16880},[1302],[1109,16882,16884],{"className":16883,"style":8058},[1306],[1109,16885],{},[1109,16887],{"className":16888},[1515,4589],[1109,16890,8631],{"className":16891},[1152],[1109,16893,16384],{"className":16894},[1152,1153],[1109,16896,1268],{"className":16897},[1152,1153],[1109,16899],{"className":16900,"style":2497},[1231],[1109,16902,2342],{"className":16903},[2501],[1109,16905],{"className":16906,"style":2497},[1231],[1109,16908,16910,16913,16916,16919,16922,16925],{"className":16909},[1143],[1109,16911],{"className":16912,"style":16411},[1147],[1109,16914,16391],{"className":16915,"style":16436},[1152,1153],[1109,16917,16244],{"className":16918,"style":16314},[1152,1153],[1109,16920],{"className":16921,"style":2497},[1231],[1109,16923,5991],{"className":16924},[2501],[1109,16926],{"className":16927,"style":2497},[1231],[1109,16929,16931,16934,16937],{"className":16930},[1143],[1109,16932],{"className":16933,"style":8085},[1147],[1109,16935,16398],{"className":16936},[1152,1153],[1109,16938,16940,16943],{"className":16939},[1152],[1109,16941,8631],{"className":16942},[1152],[1109,16944,16946],{"className":16945},[1293],[1109,16947,16949],{"className":16948},[1297],[1109,16950,16952],{"className":16951},[1302],[1109,16953,16955],{"className":16954,"style":8144},[1306],[1109,16956,16957,16960],{"style":7044},[1109,16958],{"className":16959,"style":1315},[1314],[1109,16961,16963],{"className":16962},[1319,1320,1321,1322],[1109,16964,1917],{"className":16965},[1152,1322],[1101,16967,16968,16969,16972,16973,4024],{},"增广拉格朗日函数可以理解为在普通拉格朗日函数基础上加了一个",[1104,16970,16971],{},"二次惩罚项",",所以该方法是",[1104,16974,16975],{},"拉格朗日函数法与罚函数法的结合",[1101,16977,16978],{},"标准 ADMM 的优化迭代步骤(交替更新):",[1101,16980,16981,17320,17671],{},[1109,16982,16984,17053],{"className":16983},[1112],[1109,16985,16987],{"className":16986},[1116],[1118,16988,16989],{"xmlns":1120},[1122,16990,16991,17050],{},[1125,16992,16993,17005,17007,17010,17012,17022,17028,17030,17032,17034,17040,17042,17048],{},[1264,16994,16995,16997],{},[1128,16996,1268],{},[1125,16998,16999,17001,17003],{},[1128,17000,8620],{},[1204,17002,2342],{},[1208,17004,1685],{},[1204,17006,1582],{},[1128,17008,17009],{},"arg",[1204,17011,2412],{},[1264,17013,17014,17020],{},[1125,17015,17016,17018],{},[1128,17017,10241],{},[1204,17019,2412],{},[1128,17021,1268],{},[1264,17023,17024,17026],{},[1128,17025,14523],{},[1128,17027,16482],{},[1204,17029,1457],{"stretchy":1456},[1128,17031,1268],{},[1204,17033,1910],{"separator":1139},[1264,17035,17036,17038],{},[1128,17037,16244],{},[1128,17039,8620],{},[1204,17041,1910],{"separator":1139},[1264,17043,17044,17046],{},[1128,17045,5892],{},[1128,17047,8620],{},[1204,17049,1462],{"stretchy":1456},[1132,17051,17052],{"encoding":1134},"x_{k+1} = \\arg\\min_x L_\\rho(x, z_k, \\lambda_k)",[1109,17054,17056,17120],{"className":17055,"ariaHidden":1139},[1138],[1109,17057,17059,17062,17111,17114,17117],{"className":17058},[1143],[1109,17060],{"className":17061,"style":10615},[1147],[1109,17063,17065,17068],{"className":17064},[1152],[1109,17066,1268],{"className":17067},[1152,1153],[1109,17069,17071],{"className":17070},[1293],[1109,17072,17074,17103],{"className":17073},[1297,1298],[1109,17075,17077,17100],{"className":17076},[1302],[1109,17078,17080],{"className":17079,"style":8735},[1306],[1109,17081,17082,17085],{"style":1310},[1109,17083],{"className":17084,"style":1315},[1314],[1109,17086,17088],{"className":17087},[1319,1320,1321,1322],[1109,17089,17091,17094,17097],{"className":17090},[1152,1322],[1109,17092,8620],{"className":17093,"style":8750},[1152,1153,1322],[1109,17095,2342],{"className":17096},[2501,1322],[1109,17098,1685],{"className":17099},[1152,1322],[1109,17101,1330],{"className":17102},[1329],[1109,17104,17106],{"className":17105},[1302],[1109,17107,17109],{"className":17108,"style":10663},[1306],[1109,17110],{},[1109,17112],{"className":17113,"style":1232},[1231],[1109,17115,1582],{"className":17116},[1236],[1109,17118],{"className":17119,"style":1232},[1231],[1109,17121,17123,17126,17132,17135,17175,17178,17218,17221,17224,17227,17230,17271,17274,17277,17317],{"className":17122},[1143],[1109,17124],{"className":17125,"style":16275},[1147],[1109,17127,17129,17130],{"className":17128},[2878],"ar",[1109,17131,1170],{"style":15113},[1109,17133],{"className":17134,"style":1540},[1231],[1109,17136,17138,17141],{"className":17137},[2878],[1109,17139,10241],{"className":17140},[2878],[1109,17142,17144],{"className":17143},[1293],[1109,17145,17147,17167],{"className":17146},[1297,1298],[1109,17148,17150,17164],{"className":17149},[1302],[1109,17151,17153],{"className":17152,"style":4547},[1306],[1109,17154,17155,17158],{"style":4550},[1109,17156],{"className":17157,"style":1315},[1314],[1109,17159,17161],{"className":17160},[1319,1320,1321,1322],[1109,17162,1268],{"className":17163},[1152,1153,1322],[1109,17165,1330],{"className":17166},[1329],[1109,17168,17170],{"className":17169},[1302],[1109,17171,17173],{"className":17172,"style":1337},[1306],[1109,17174],{},[1109,17176],{"className":17177,"style":1540},[1231],[1109,17179,17181,17184],{"className":17180},[1152],[1109,17182,14523],{"className":17183},[1152,1153],[1109,17185,17187],{"className":17186},[1293],[1109,17188,17190,17210],{"className":17189},[1297,1298],[1109,17191,17193,17207],{"className":17192},[1302],[1109,17194,17196],{"className":17195,"style":4547},[1306],[1109,17197,17198,17201],{"style":1310},[1109,17199],{"className":17200,"style":1315},[1314],[1109,17202,17204],{"className":17203},[1319,1320,1321,1322],[1109,17205,16482],{"className":17206},[1152,1153,1322],[1109,17208,1330],{"className":17209},[1329],[1109,17211,17213],{"className":17212},[1302],[1109,17214,17216],{"className":17215,"style":16324},[1306],[1109,17217],{},[1109,17219,1457],{"className":17220},[1508],[1109,17222,1268],{"className":17223},[1152,1153],[1109,17225,1910],{"className":17226},[1975],[1109,17228],{"className":17229,"style":1540},[1231],[1109,17231,17233,17236],{"className":17232},[1152],[1109,17234,16244],{"className":17235,"style":16314},[1152,1153],[1109,17237,17239],{"className":17238},[1293],[1109,17240,17242,17263],{"className":17241},[1297,1298],[1109,17243,17245,17260],{"className":17244},[1302],[1109,17246,17248],{"className":17247,"style":8735},[1306],[1109,17249,17251,17254],{"style":17250},"top:-2.55em;margin-left:-0.044em;margin-right:0.05em;",[1109,17252],{"className":17253,"style":1315},[1314],[1109,17255,17257],{"className":17256},[1319,1320,1321,1322],[1109,17258,8620],{"className":17259,"style":8750},[1152,1153,1322],[1109,17261,1330],{"className":17262},[1329],[1109,17264,17266],{"className":17265},[1302],[1109,17267,17269],{"className":17268,"style":1337},[1306],[1109,17270],{},[1109,17272,1910],{"className":17273},[1975],[1109,17275],{"className":17276,"style":1540},[1231],[1109,17278,17280,17283],{"className":17279},[1152],[1109,17281,5892],{"className":17282},[1152,1153],[1109,17284,17286],{"className":17285},[1293],[1109,17287,17289,17309],{"className":17288},[1297,1298],[1109,17290,17292,17306],{"className":17291},[1302],[1109,17293,17295],{"className":17294,"style":8735},[1306],[1109,17296,17297,17300],{"style":1310},[1109,17298],{"className":17299,"style":1315},[1314],[1109,17301,17303],{"className":17302},[1319,1320,1321,1322],[1109,17304,8620],{"className":17305,"style":8750},[1152,1153,1322],[1109,17307,1330],{"className":17308},[1329],[1109,17310,17312],{"className":17311},[1302],[1109,17313,17315],{"className":17314,"style":1337},[1306],[1109,17316],{},[1109,17318,1462],{"className":17319},[1515],[1109,17321,17323,17397],{"className":17322},[1112],[1109,17324,17326],{"className":17325},[1116],[1118,17327,17328],{"xmlns":1120},[1122,17329,17330,17394],{},[1125,17331,17332,17344,17346,17348,17350,17360,17366,17368,17380,17382,17384,17386,17392],{},[1264,17333,17334,17336],{},[1128,17335,16244],{},[1125,17337,17338,17340,17342],{},[1128,17339,8620],{},[1204,17341,2342],{},[1208,17343,1685],{},[1204,17345,1582],{},[1128,17347,17009],{},[1204,17349,2412],{},[1264,17351,17352,17358],{},[1125,17353,17354,17356],{},[1128,17355,10241],{},[1204,17357,2412],{},[1128,17359,16244],{},[1264,17361,17362,17364],{},[1128,17363,14523],{},[1128,17365,16482],{},[1204,17367,1457],{"stretchy":1456},[1264,17369,17370,17372],{},[1128,17371,1268],{},[1125,17373,17374,17376,17378],{},[1128,17375,8620],{},[1204,17377,2342],{},[1208,17379,1685],{},[1204,17381,1910],{"separator":1139},[1128,17383,16244],{},[1204,17385,1910],{"separator":1139},[1264,17387,17388,17390],{},[1128,17389,5892],{},[1128,17391,8620],{},[1204,17393,1462],{"stretchy":1456},[1132,17395,17396],{"encoding":1134},"z_{k+1} = \\arg\\min_z L_\\rho(x_{k+1}, z, \\lambda_k)",[1109,17398,17400,17464],{"className":17399,"ariaHidden":1139},[1138],[1109,17401,17403,17406,17455,17458,17461],{"className":17402},[1143],[1109,17404],{"className":17405,"style":10615},[1147],[1109,17407,17409,17412],{"className":17408},[1152],[1109,17410,16244],{"className":17411,"style":16314},[1152,1153],[1109,17413,17415],{"className":17414},[1293],[1109,17416,17418,17447],{"className":17417},[1297,1298],[1109,17419,17421,17444],{"className":17420},[1302],[1109,17422,17424],{"className":17423,"style":8735},[1306],[1109,17425,17426,17429],{"style":17250},[1109,17427],{"className":17428,"style":1315},[1314],[1109,17430,17432],{"className":17431},[1319,1320,1321,1322],[1109,17433,17435,17438,17441],{"className":17434},[1152,1322],[1109,17436,8620],{"className":17437,"style":8750},[1152,1153,1322],[1109,17439,2342],{"className":17440},[2501,1322],[1109,17442,1685],{"className":17443},[1152,1322],[1109,17445,1330],{"className":17446},[1329],[1109,17448,17450],{"className":17449},[1302],[1109,17451,17453],{"className":17452,"style":10663},[1306],[1109,17454],{},[1109,17456],{"className":17457,"style":1232},[1231],[1109,17459,1582],{"className":17460},[1236],[1109,17462],{"className":17463,"style":1232},[1231],[1109,17465,17467,17470,17475,17478,17518,17521,17561,17564,17613,17616,17619,17622,17625,17628,17668],{"className":17466},[1143],[1109,17468],{"className":17469,"style":16275},[1147],[1109,17471,17129,17473],{"className":17472},[2878],[1109,17474,1170],{"style":15113},[1109,17476],{"className":17477,"style":1540},[1231],[1109,17479,17481,17484],{"className":17480},[2878],[1109,17482,10241],{"className":17483},[2878],[1109,17485,17487],{"className":17486},[1293],[1109,17488,17490,17510],{"className":17489},[1297,1298],[1109,17491,17493,17507],{"className":17492},[1302],[1109,17494,17496],{"className":17495,"style":4547},[1306],[1109,17497,17498,17501],{"style":4550},[1109,17499],{"className":17500,"style":1315},[1314],[1109,17502,17504],{"className":17503},[1319,1320,1321,1322],[1109,17505,16244],{"className":17506,"style":16314},[1152,1153,1322],[1109,17508,1330],{"className":17509},[1329],[1109,17511,17513],{"className":17512},[1302],[1109,17514,17516],{"className":17515,"style":1337},[1306],[1109,17517],{},[1109,17519],{"className":17520,"style":1540},[1231],[1109,17522,17524,17527],{"className":17523},[1152],[1109,17525,14523],{"className":17526},[1152,1153],[1109,17528,17530],{"className":17529},[1293],[1109,17531,17533,17553],{"className":17532},[1297,1298],[1109,17534,17536,17550],{"className":17535},[1302],[1109,17537,17539],{"className":17538,"style":4547},[1306],[1109,17540,17541,17544],{"style":1310},[1109,17542],{"className":17543,"style":1315},[1314],[1109,17545,17547],{"className":17546},[1319,1320,1321,1322],[1109,17548,16482],{"className":17549},[1152,1153,1322],[1109,17551,1330],{"className":17552},[1329],[1109,17554,17556],{"className":17555},[1302],[1109,17557,17559],{"className":17558,"style":16324},[1306],[1109,17560],{},[1109,17562,1457],{"className":17563},[1508],[1109,17565,17567,17570],{"className":17566},[1152],[1109,17568,1268],{"className":17569},[1152,1153],[1109,17571,17573],{"className":17572},[1293],[1109,17574,17576,17605],{"className":17575},[1297,1298],[1109,17577,17579,17602],{"className":17578},[1302],[1109,17580,17582],{"className":17581,"style":8735},[1306],[1109,17583,17584,17587],{"style":1310},[1109,17585],{"className":17586,"style":1315},[1314],[1109,17588,17590],{"className":17589},[1319,1320,1321,1322],[1109,17591,17593,17596,17599],{"className":17592},[1152,1322],[1109,17594,8620],{"className":17595,"style":8750},[1152,1153,1322],[1109,17597,2342],{"className":17598},[2501,1322],[1109,17600,1685],{"className":17601},[1152,1322],[1109,17603,1330],{"className":17604},[1329],[1109,17606,17608],{"className":17607},[1302],[1109,17609,17611],{"className":17610,"style":10663},[1306],[1109,17612],{},[1109,17614,1910],{"className":17615},[1975],[1109,17617],{"className":17618,"style":1540},[1231],[1109,17620,16244],{"className":17621,"style":16314},[1152,1153],[1109,17623,1910],{"className":17624},[1975],[1109,17626],{"className":17627,"style":1540},[1231],[1109,17629,17631,17634],{"className":17630},[1152],[1109,17632,5892],{"className":17633},[1152,1153],[1109,17635,17637],{"className":17636},[1293],[1109,17638,17640,17660],{"className":17639},[1297,1298],[1109,17641,17643,17657],{"className":17642},[1302],[1109,17644,17646],{"className":17645,"style":8735},[1306],[1109,17647,17648,17651],{"style":1310},[1109,17649],{"className":17650,"style":1315},[1314],[1109,17652,17654],{"className":17653},[1319,1320,1321,1322],[1109,17655,8620],{"className":17656,"style":8750},[1152,1153,1322],[1109,17658,1330],{"className":17659},[1329],[1109,17661,17663],{"className":17662},[1302],[1109,17664,17666],{"className":17665,"style":1337},[1306],[1109,17667],{},[1109,17669,1462],{"className":17670},[1515],[1109,17672,17674,17750],{"className":17673},[1112],[1109,17675,17677],{"className":17676},[1116],[1118,17678,17679],{"xmlns":1120},[1122,17680,17681,17747],{},[1125,17682,17683,17695,17697,17703,17705,17707,17709,17711,17713,17725,17727,17729,17741,17743,17745],{},[1264,17684,17685,17687],{},[1128,17686,5892],{},[1125,17688,17689,17691,17693],{},[1128,17690,8620],{},[1204,17692,2342],{},[1208,17694,1685],{},[1204,17696,1582],{},[1264,17698,17699,17701],{},[1128,17700,5892],{},[1128,17702,8620],{},[1204,17704,2342],{},[1128,17706,16482],{},[1471,17708,1473],{},[1204,17710,1457],{"stretchy":1456},[1128,17712,16384],{},[1264,17714,17715,17717],{},[1128,17716,1268],{},[1125,17718,17719,17721,17723],{},[1128,17720,8620],{},[1204,17722,2342],{},[1208,17724,1685],{},[1204,17726,2342],{},[1128,17728,16391],{},[1264,17730,17731,17733],{},[1128,17732,16244],{},[1125,17734,17735,17737,17739],{},[1128,17736,8620],{},[1204,17738,2342],{},[1208,17740,1685],{},[1204,17742,5991],{},[1128,17744,16398],{},[1204,17746,1462],{"stretchy":1456},[1132,17748,17749],{"encoding":1134},"\\lambda_{k+1} = \\lambda_k + \\rho\\,(Ax_{k+1} + Bz_{k+1} - c)",[1109,17751,17753,17817,17873,17949,18017],{"className":17752,"ariaHidden":1139},[1138],[1109,17754,17756,17759,17808,17811,17814],{"className":17755},[1143],[1109,17757],{"className":17758,"style":15948},[1147],[1109,17760,17762,17765],{"className":17761},[1152],[1109,17763,5892],{"className":17764},[1152,1153],[1109,17766,17768],{"className":17767},[1293],[1109,17769,17771,17800],{"className":17770},[1297,1298],[1109,17772,17774,17797],{"className":17773},[1302],[1109,17775,17777],{"className":17776,"style":8735},[1306],[1109,17778,17779,17782],{"style":1310},[1109,17780],{"className":17781,"style":1315},[1314],[1109,17783,17785],{"className":17784},[1319,1320,1321,1322],[1109,17786,17788,17791,17794],{"className":17787},[1152,1322],[1109,17789,8620],{"className":17790,"style":8750},[1152,1153,1322],[1109,17792,2342],{"className":17793},[2501,1322],[1109,17795,1685],{"className":17796},[1152,1322],[1109,17798,1330],{"className":17799},[1329],[1109,17801,17803],{"className":17802},[1302],[1109,17804,17806],{"className":17805,"style":10663},[1306],[1109,17807],{},[1109,17809],{"className":17810,"style":1232},[1231],[1109,17812,1582],{"className":17813},[1236],[1109,17815],{"className":17816,"style":1232},[1231],[1109,17818,17820,17824,17864,17867,17870],{"className":17819},[1143],[1109,17821],{"className":17822,"style":17823},[1147],"height:0.8444em;vertical-align:-0.15em;",[1109,17825,17827,17830],{"className":17826},[1152],[1109,17828,5892],{"className":17829},[1152,1153],[1109,17831,17833],{"className":17832},[1293],[1109,17834,17836,17856],{"className":17835},[1297,1298],[1109,17837,17839,17853],{"className":17838},[1302],[1109,17840,17842],{"className":17841,"style":8735},[1306],[1109,17843,17844,17847],{"style":1310},[1109,17845],{"className":17846,"style":1315},[1314],[1109,17848,17850],{"className":17849},[1319,1320,1321,1322],[1109,17851,8620],{"className":17852,"style":8750},[1152,1153,1322],[1109,17854,1330],{"className":17855},[1329],[1109,17857,17859],{"className":17858},[1302],[1109,17860,17862],{"className":17861,"style":1337},[1306],[1109,17863],{},[1109,17865],{"className":17866,"style":2497},[1231],[1109,17868,2342],{"className":17869},[2501],[1109,17871],{"className":17872,"style":2497},[1231],[1109,17874,17876,17879,17882,17885,17888,17891,17940,17943,17946],{"className":17875},[1143],[1109,17877],{"className":17878,"style":1498},[1147],[1109,17880,16482],{"className":17881},[1152,1153],[1109,17883],{"className":17884,"style":1540},[1231],[1109,17886,1457],{"className":17887},[1508],[1109,17889,16384],{"className":17890},[1152,1153],[1109,17892,17894,17897],{"className":17893},[1152],[1109,17895,1268],{"className":17896},[1152,1153],[1109,17898,17900],{"className":17899},[1293],[1109,17901,17903,17932],{"className":17902},[1297,1298],[1109,17904,17906,17929],{"className":17905},[1302],[1109,17907,17909],{"className":17908,"style":8735},[1306],[1109,17910,17911,17914],{"style":1310},[1109,17912],{"className":17913,"style":1315},[1314],[1109,17915,17917],{"className":17916},[1319,1320,1321,1322],[1109,17918,17920,17923,17926],{"className":17919},[1152,1322],[1109,17921,8620],{"className":17922,"style":8750},[1152,1153,1322],[1109,17924,2342],{"className":17925},[2501,1322],[1109,17927,1685],{"className":17928},[1152,1322],[1109,17930,1330],{"className":17931},[1329],[1109,17933,17935],{"className":17934},[1302],[1109,17936,17938],{"className":17937,"style":10663},[1306],[1109,17939],{},[1109,17941],{"className":17942,"style":2497},[1231],[1109,17944,2342],{"className":17945},[2501],[1109,17947],{"className":17948,"style":2497},[1231],[1109,17950,17952,17956,17959,18008,18011,18014],{"className":17951},[1143],[1109,17953],{"className":17954,"style":17955},[1147],"height:0.8917em;vertical-align:-0.2083em;",[1109,17957,16391],{"className":17958,"style":16436},[1152,1153],[1109,17960,17962,17965],{"className":17961},[1152],[1109,17963,16244],{"className":17964,"style":16314},[1152,1153],[1109,17966,17968],{"className":17967},[1293],[1109,17969,17971,18000],{"className":17970},[1297,1298],[1109,17972,17974,17997],{"className":17973},[1302],[1109,17975,17977],{"className":17976,"style":8735},[1306],[1109,17978,17979,17982],{"style":17250},[1109,17980],{"className":17981,"style":1315},[1314],[1109,17983,17985],{"className":17984},[1319,1320,1321,1322],[1109,17986,17988,17991,17994],{"className":17987},[1152,1322],[1109,17989,8620],{"className":17990,"style":8750},[1152,1153,1322],[1109,17992,2342],{"className":17993},[2501,1322],[1109,17995,1685],{"className":17996},[1152,1322],[1109,17998,1330],{"className":17999},[1329],[1109,18001,18003],{"className":18002},[1302],[1109,18004,18006],{"className":18005,"style":10663},[1306],[1109,18007],{},[1109,18009],{"className":18010,"style":2497},[1231],[1109,18012,5991],{"className":18013},[2501],[1109,18015],{"className":18016,"style":2497},[1231],[1109,18018,18020,18023,18026],{"className":18019},[1143],[1109,18021],{"className":18022,"style":1498},[1147],[1109,18024,16398],{"className":18025},[1152,1153],[1109,18027,1462],{"className":18028},[1515],[1096,18030,18032],{"id":18031},"例子lasso","例子:Lasso",[1101,18034,18035,18036,18106,18107,18408,18409,18437,18438,16369,18696,18769],{},"Lasso 是带 ",[1109,18037,18039,18057],{"className":18038},[1112],[1109,18040,18042],{"className":18041},[1116],[1118,18043,18044],{"xmlns":1120},[1122,18045,18046,18054],{},[1125,18047,18048],{},[1264,18049,18050,18052],{},[1128,18051,14523],{},[1208,18053,1685],{},[1132,18055,18056],{"encoding":1134},"L_1",[1109,18058,18060],{"className":18059,"ariaHidden":1139},[1138],[1109,18061,18063,18066],{"className":18062},[1143],[1109,18064],{"className":18065,"style":3045},[1147],[1109,18067,18069,18072],{"className":18068},[1152],[1109,18070,14523],{"className":18071},[1152,1153],[1109,18073,18075],{"className":18074},[1293],[1109,18076,18078,18098],{"className":18077},[1297,1298],[1109,18079,18081,18095],{"className":18080},[1302],[1109,18082,18084],{"className":18083,"style":1307},[1306],[1109,18085,18086,18089],{"style":1310},[1109,18087],{"className":18088,"style":1315},[1314],[1109,18090,18092],{"className":18091},[1319,1320,1321,1322],[1109,18093,1685],{"className":18094},[1152,1322],[1109,18096,1330],{"className":18097},[1329],[1109,18099,18101],{"className":18100},[1302],[1109,18102,18104],{"className":18103,"style":1337},[1306],[1109,18105],{}," 正则的回归:",[1109,18108,18110,18168],{"className":18109},[1112],[1109,18111,18113],{"className":18112},[1116],[1118,18114,18115],{"xmlns":1120},[1122,18116,18117,18165],{},[1125,18118,18119,18129,18135,18137,18139,18141,18143,18145,18151,18153,18155,18157,18159],{},[1264,18120,18121,18127],{},[1125,18122,18123,18125],{},[1128,18124,10241],{},[1204,18126,2412],{},[1128,18128,12502],{},[4490,18130,18131,18133],{},[1208,18132,1685],{},[1208,18134,1917],{},[1128,18136,8631],{"mathvariant":1450},[1128,18138,14113],{},[1204,18140,5991],{},[1128,18142,15377],{},[1128,18144,12502],{},[6964,18146,18147,18149],{},[1128,18148,8631],{"mathvariant":1450},[1208,18150,1917],{},[1204,18152,2342],{},[1128,18154,5892],{},[1128,18156,8631],{"mathvariant":1450},[1128,18158,12502],{},[1264,18160,18161,18163],{},[1128,18162,8631],{"mathvariant":1450},[1208,18164,1685],{},[1132,18166,18167],{"encoding":1134},"\\min_\\beta \\frac{1}{2}\\|y - X\\beta\\|^2 + \\lambda\\|\\beta\\|_1",[1109,18169,18171,18303,18353],{"className":18170,"ariaHidden":1139},[1138],[1109,18172,18174,18177,18217,18220,18288,18291,18294,18297,18300],{"className":18173},[1143],[1109,18175],{"className":18176,"style":11312},[1147],[1109,18178,18180,18183],{"className":18179},[2878],[1109,18181,10241],{"className":18182},[2878],[1109,18184,18186],{"className":18185},[1293],[1109,18187,18189,18209],{"className":18188},[1297,1298],[1109,18190,18192,18206],{"className":18191},[1302],[1109,18193,18195],{"className":18194,"style":8735},[1306],[1109,18196,18197,18200],{"style":4550},[1109,18198],{"className":18199,"style":1315},[1314],[1109,18201,18203],{"className":18202},[1319,1320,1321,1322],[1109,18204,12502],{"className":18205,"style":12530},[1152,1153,1322],[1109,18207,1330],{"className":18208},[1329],[1109,18210,18212],{"className":18211},[1302],[1109,18213,18215],{"className":18214,"style":16324},[1306],[1109,18216],{},[1109,18218],{"className":18219,"style":1540},[1231],[1109,18221,18223,18226,18285],{"className":18222},[1152],[1109,18224],{"className":18225},[1508,4589],[1109,18227,18229],{"className":18228},[4490],[1109,18230,18232,18277],{"className":18231},[1297,1298],[1109,18233,18235,18274],{"className":18234},[1302],[1109,18236,18238,18252,18260],{"className":18237,"style":11331},[1306],[1109,18239,18240,18243],{"style":4605},[1109,18241],{"className":18242,"style":4609},[1314],[1109,18244,18246],{"className":18245},[1319,1320,1321,1322],[1109,18247,18249],{"className":18248},[1152,1322],[1109,18250,1917],{"className":18251},[1152,1322],[1109,18253,18254,18257],{"style":4630},[1109,18255],{"className":18256,"style":4609},[1314],[1109,18258],{"className":18259,"style":4638},[4637],[1109,18261,18262,18265],{"style":11356},[1109,18263],{"className":18264,"style":4609},[1314],[1109,18266,18268],{"className":18267},[1319,1320,1321,1322],[1109,18269,18271],{"className":18270},[1152,1322],[1109,18272,1685],{"className":18273},[1152,1322],[1109,18275,1330],{"className":18276},[1329],[1109,18278,18280],{"className":18279},[1302],[1109,18281,18283],{"className":18282,"style":8058},[1306],[1109,18284],{},[1109,18286],{"className":18287},[1515,4589],[1109,18289,8631],{"className":18290},[1152],[1109,18292,14113],{"className":18293,"style":1186},[1152,1153],[1109,18295],{"className":18296,"style":2497},[1231],[1109,18298,5991],{"className":18299},[2501],[1109,18301],{"className":18302,"style":2497},[1231],[1109,18304,18306,18309,18312,18315,18344,18347,18350],{"className":18305},[1143],[1109,18307],{"className":18308,"style":8085},[1147],[1109,18310,15377],{"className":18311,"style":15471},[1152,1153],[1109,18313,12502],{"className":18314,"style":12530},[1152,1153],[1109,18316,18318,18321],{"className":18317},[1152],[1109,18319,8631],{"className":18320},[1152],[1109,18322,18324],{"className":18323},[1293],[1109,18325,18327],{"className":18326},[1297],[1109,18328,18330],{"className":18329},[1302],[1109,18331,18333],{"className":18332,"style":8144},[1306],[1109,18334,18335,18338],{"style":7044},[1109,18336],{"className":18337,"style":1315},[1314],[1109,18339,18341],{"className":18340},[1319,1320,1321,1322],[1109,18342,1917],{"className":18343},[1152,1322],[1109,18345],{"className":18346,"style":2497},[1231],[1109,18348,2342],{"className":18349},[2501],[1109,18351],{"className":18352,"style":2497},[1231],[1109,18354,18356,18359,18362,18365,18368],{"className":18355},[1143],[1109,18357],{"className":18358,"style":1498},[1147],[1109,18360,5892],{"className":18361},[1152,1153],[1109,18363,8631],{"className":18364},[1152],[1109,18366,12502],{"className":18367,"style":12530},[1152,1153],[1109,18369,18371,18374],{"className":18370},[1152],[1109,18372,8631],{"className":18373},[1152],[1109,18375,18377],{"className":18376},[1293],[1109,18378,18380,18400],{"className":18379},[1297,1298],[1109,18381,18383,18397],{"className":18382},[1302],[1109,18384,18386],{"className":18385,"style":1307},[1306],[1109,18387,18388,18391],{"style":1310},[1109,18389],{"className":18390,"style":1315},[1314],[1109,18392,18394],{"className":18393},[1319,1320,1321,1322],[1109,18395,1685],{"className":18396},[1152,1322],[1109,18398,1330],{"className":18399},[1329],[1109,18401,18403],{"className":18402},[1302],[1109,18404,18406],{"className":18405,"style":1337},[1306],[1109,18407],{},"。引入辅助变量 ",[1109,18410,18412,18425],{"className":18411},[1112],[1109,18413,18415],{"className":18414},[1116],[1118,18416,18417],{"xmlns":1120},[1122,18418,18419,18423],{},[1125,18420,18421],{},[1128,18422,16244],{},[1132,18424,16244],{"encoding":1134},[1109,18426,18428],{"className":18427,"ariaHidden":1139},[1138],[1109,18429,18431,18434],{"className":18430},[1143],[1109,18432],{"className":18433,"style":7312},[1147],[1109,18435,16244],{"className":18436,"style":16314},[1152,1153],",写成约束形式 ",[1109,18439,18441,18493],{"className":18440},[1112],[1109,18442,18444],{"className":18443},[1116],[1118,18445,18446],{"xmlns":1120},[1122,18447,18448,18490],{},[1125,18449,18450,18452,18454,18460,18462,18464,18466,18468,18470,18476,18478,18480,18482,18484],{},[1128,18451,10241],{},[1204,18453,2412],{},[4490,18455,18456,18458],{},[1208,18457,1685],{},[1208,18459,1917],{},[1128,18461,8631],{"mathvariant":1450},[1128,18463,14113],{},[1204,18465,5991],{},[1128,18467,15377],{},[1128,18469,12502],{},[6964,18471,18472,18474],{},[1128,18473,8631],{"mathvariant":1450},[1208,18475,1917],{},[1204,18477,2342],{},[1128,18479,5892],{},[1128,18481,8631],{"mathvariant":1450},[1128,18483,16244],{},[1264,18485,18486,18488],{},[1128,18487,8631],{"mathvariant":1450},[1208,18489,1685],{},[1132,18491,18492],{"encoding":1134},"\\min \\frac{1}{2}\\|y-X\\beta\\|^2 + \\lambda\\|z\\|_1",[1109,18494,18496,18591,18641],{"className":18495,"ariaHidden":1139},[1138],[1109,18497,18499,18502,18505,18508,18576,18579,18582,18585,18588],{"className":18498},[1143],[1109,18500],{"className":18501,"style":11312},[1147],[1109,18503,10241],{"className":18504},[2878],[1109,18506],{"className":18507,"style":1540},[1231],[1109,18509,18511,18514,18573],{"className":18510},[1152],[1109,18512],{"className":18513},[1508,4589],[1109,18515,18517],{"className":18516},[4490],[1109,18518,18520,18565],{"className":18519},[1297,1298],[1109,18521,18523,18562],{"className":18522},[1302],[1109,18524,18526,18540,18548],{"className":18525,"style":11331},[1306],[1109,18527,18528,18531],{"style":4605},[1109,18529],{"className":18530,"style":4609},[1314],[1109,18532,18534],{"className":18533},[1319,1320,1321,1322],[1109,18535,18537],{"className":18536},[1152,1322],[1109,18538,1917],{"className":18539},[1152,1322],[1109,18541,18542,18545],{"style":4630},[1109,18543],{"className":18544,"style":4609},[1314],[1109,18546],{"className":18547,"style":4638},[4637],[1109,18549,18550,18553],{"style":11356},[1109,18551],{"className":18552,"style":4609},[1314],[1109,18554,18556],{"className":18555},[1319,1320,1321,1322],[1109,18557,18559],{"className":18558},[1152,1322],[1109,18560,1685],{"className":18561},[1152,1322],[1109,18563,1330],{"className":18564},[1329],[1109,18566,18568],{"className":18567},[1302],[1109,18569,18571],{"className":18570,"style":8058},[1306],[1109,18572],{},[1109,18574],{"className":18575},[1515,4589],[1109,18577,8631],{"className":18578},[1152],[1109,18580,14113],{"className":18581,"style":1186},[1152,1153],[1109,18583],{"className":18584,"style":2497},[1231],[1109,18586,5991],{"className":18587},[2501],[1109,18589],{"className":18590,"style":2497},[1231],[1109,18592,18594,18597,18600,18603,18632,18635,18638],{"className":18593},[1143],[1109,18595],{"className":18596,"style":8085},[1147],[1109,18598,15377],{"className":18599,"style":15471},[1152,1153],[1109,18601,12502],{"className":18602,"style":12530},[1152,1153],[1109,18604,18606,18609],{"className":18605},[1152],[1109,18607,8631],{"className":18608},[1152],[1109,18610,18612],{"className":18611},[1293],[1109,18613,18615],{"className":18614},[1297],[1109,18616,18618],{"className":18617},[1302],[1109,18619,18621],{"className":18620,"style":8144},[1306],[1109,18622,18623,18626],{"style":7044},[1109,18624],{"className":18625,"style":1315},[1314],[1109,18627,18629],{"className":18628},[1319,1320,1321,1322],[1109,18630,1917],{"className":18631},[1152,1322],[1109,18633],{"className":18634,"style":2497},[1231],[1109,18636,2342],{"className":18637},[2501],[1109,18639],{"className":18640,"style":2497},[1231],[1109,18642,18644,18647,18650,18653,18656],{"className":18643},[1143],[1109,18645],{"className":18646,"style":1498},[1147],[1109,18648,5892],{"className":18649},[1152,1153],[1109,18651,8631],{"className":18652},[1152],[1109,18654,16244],{"className":18655,"style":16314},[1152,1153],[1109,18657,18659,18662],{"className":18658},[1152],[1109,18660,8631],{"className":18661},[1152],[1109,18663,18665],{"className":18664},[1293],[1109,18666,18668,18688],{"className":18667},[1297,1298],[1109,18669,18671,18685],{"className":18670},[1302],[1109,18672,18674],{"className":18673,"style":1307},[1306],[1109,18675,18676,18679],{"style":1310},[1109,18677],{"className":18678,"style":1315},[1314],[1109,18680,18682],{"className":18681},[1319,1320,1321,1322],[1109,18683,1685],{"className":18684},[1152,1322],[1109,18686,1330],{"className":18687},[1329],[1109,18689,18691],{"className":18690},[1302],[1109,18692,18694],{"className":18693,"style":1337},[1306],[1109,18695],{},[1109,18697,18699,18721],{"className":18698},[1112],[1109,18700,18702],{"className":18701},[1116],[1118,18703,18704],{"xmlns":1120},[1122,18705,18706,18718],{},[1125,18707,18708,18710,18712,18714,18716],{},[1128,18709,12502],{},[1204,18711,5991],{},[1128,18713,16244],{},[1204,18715,1582],{},[1208,18717,1210],{},[1132,18719,18720],{"encoding":1134},"\\beta - z = 0",[1109,18722,18724,18742,18760],{"className":18723,"ariaHidden":1139},[1138],[1109,18725,18727,18730,18733,18736,18739],{"className":18726},[1143],[1109,18728],{"className":18729,"style":1148},[1147],[1109,18731,12502],{"className":18732,"style":12530},[1152,1153],[1109,18734],{"className":18735,"style":2497},[1231],[1109,18737,5991],{"className":18738},[2501],[1109,18740],{"className":18741,"style":2497},[1231],[1109,18743,18745,18748,18751,18754,18757],{"className":18744},[1143],[1109,18746],{"className":18747,"style":7312},[1147],[1109,18749,16244],{"className":18750,"style":16314},[1152,1153],[1109,18752],{"className":18753,"style":1232},[1231],[1109,18755,1582],{"className":18756},[1236],[1109,18758],{"className":18759,"style":1232},[1231],[1109,18761,18763,18766],{"className":18762},[1143],[1109,18764],{"className":18765,"style":1246},[1147],[1109,18767,1210],{"className":18768},[1152],"。增广拉格朗日函数为",[1101,18771,18772],{},[1109,18773,18775,18872],{"className":18774},[1112],[1109,18776,18778],{"className":18777},[1116],[1118,18779,18780],{"xmlns":1120},[1122,18781,18782,18869],{},[1125,18783,18784,18790,18792,18798,18800,18802,18804,18806,18808,18814,18816,18818,18820,18822,18828,18830,18837,18839,18841,18843,18845,18847,18849,18855,18857,18859,18861,18863],{},[1264,18785,18786,18788],{},[1128,18787,14523],{},[1128,18789,16482],{},[1204,18791,1582],{},[4490,18793,18794,18796],{},[1208,18795,1685],{},[1208,18797,1917],{},[1128,18799,8631],{"mathvariant":1450},[1128,18801,14113],{},[1204,18803,5991],{},[1128,18805,15377],{},[1128,18807,12502],{},[6964,18809,18810,18812],{},[1128,18811,8631],{"mathvariant":1450},[1208,18813,1917],{},[1204,18815,2342],{},[1128,18817,5892],{},[1128,18819,8631],{"mathvariant":1450},[1128,18821,16244],{},[1264,18823,18824,18826],{},[1128,18825,8631],{"mathvariant":1450},[1208,18827,1685],{},[1204,18829,2342],{},[6964,18831,18832,18835],{},[1128,18833,18834],{},"u",[1128,18836,9471],{"mathvariant":1450},[1204,18838,1457],{"stretchy":1456},[1128,18840,12502],{},[1204,18842,5991],{},[1128,18844,16244],{},[1204,18846,1462],{"stretchy":1456},[1204,18848,2342],{},[4490,18850,18851,18853],{},[1128,18852,16482],{},[1208,18854,1917],{},[1128,18856,8631],{"mathvariant":1450},[1128,18858,12502],{},[1204,18860,5991],{},[1128,18862,16244],{},[6964,18864,18865,18867],{},[1128,18866,8631],{"mathvariant":1450},[1208,18868,1917],{},[1132,18870,18871],{"encoding":1134},"L_\\rho = \\frac{1}{2}\\|y - X\\beta\\|^2 + \\lambda\\|z\\|_1 + u^{\\top}(\\beta - z) + \\frac{\\rho}{2}\\|\\beta - z\\|^2",[1109,18873,18875,18931,19020,19070,19134,19187,19208,19297],{"className":18874,"ariaHidden":1139},[1138],[1109,18876,18878,18882,18922,18925,18928],{"className":18877},[1143],[1109,18879],{"className":18880,"style":18881},[1147],"height:0.9694em;vertical-align:-0.2861em;",[1109,18883,18885,18888],{"className":18884},[1152],[1109,18886,14523],{"className":18887},[1152,1153],[1109,18889,18891],{"className":18890},[1293],[1109,18892,18894,18914],{"className":18893},[1297,1298],[1109,18895,18897,18911],{"className":18896},[1302],[1109,18898,18900],{"className":18899,"style":4547},[1306],[1109,18901,18902,18905],{"style":1310},[1109,18903],{"className":18904,"style":1315},[1314],[1109,18906,18908],{"className":18907},[1319,1320,1321,1322],[1109,18909,16482],{"className":18910},[1152,1153,1322],[1109,18912,1330],{"className":18913},[1329],[1109,18915,18917],{"className":18916},[1302],[1109,18918,18920],{"className":18919,"style":16324},[1306],[1109,18921],{},[1109,18923],{"className":18924,"style":1232},[1231],[1109,18926,1582],{"className":18927},[1236],[1109,18929],{"className":18930,"style":1232},[1231],[1109,18932,18934,18937,19005,19008,19011,19014,19017],{"className":18933},[1143],[1109,18935],{"className":18936,"style":11312},[1147],[1109,18938,18940,18943,19002],{"className":18939},[1152],[1109,18941],{"className":18942},[1508,4589],[1109,18944,18946],{"className":18945},[4490],[1109,18947,18949,18994],{"className":18948},[1297,1298],[1109,18950,18952,18991],{"className":18951},[1302],[1109,18953,18955,18969,18977],{"className":18954,"style":11331},[1306],[1109,18956,18957,18960],{"style":4605},[1109,18958],{"className":18959,"style":4609},[1314],[1109,18961,18963],{"className":18962},[1319,1320,1321,1322],[1109,18964,18966],{"className":18965},[1152,1322],[1109,18967,1917],{"className":18968},[1152,1322],[1109,18970,18971,18974],{"style":4630},[1109,18972],{"className":18973,"style":4609},[1314],[1109,18975],{"className":18976,"style":4638},[4637],[1109,18978,18979,18982],{"style":11356},[1109,18980],{"className":18981,"style":4609},[1314],[1109,18983,18985],{"className":18984},[1319,1320,1321,1322],[1109,18986,18988],{"className":18987},[1152,1322],[1109,18989,1685],{"className":18990},[1152,1322],[1109,18992,1330],{"className":18993},[1329],[1109,18995,18997],{"className":18996},[1302],[1109,18998,19000],{"className":18999,"style":8058},[1306],[1109,19001],{},[1109,19003],{"className":19004},[1515,4589],[1109,19006,8631],{"className":19007},[1152],[1109,19009,14113],{"className":19010,"style":1186},[1152,1153],[1109,19012],{"className":19013,"style":2497},[1231],[1109,19015,5991],{"className":19016},[2501],[1109,19018],{"className":19019,"style":2497},[1231],[1109,19021,19023,19026,19029,19032,19061,19064,19067],{"className":19022},[1143],[1109,19024],{"className":19025,"style":8085},[1147],[1109,19027,15377],{"className":19028,"style":15471},[1152,1153],[1109,19030,12502],{"className":19031,"style":12530},[1152,1153],[1109,19033,19035,19038],{"className":19034},[1152],[1109,19036,8631],{"className":19037},[1152],[1109,19039,19041],{"className":19040},[1293],[1109,19042,19044],{"className":19043},[1297],[1109,19045,19047],{"className":19046},[1302],[1109,19048,19050],{"className":19049,"style":8144},[1306],[1109,19051,19052,19055],{"style":7044},[1109,19053],{"className":19054,"style":1315},[1314],[1109,19056,19058],{"className":19057},[1319,1320,1321,1322],[1109,19059,1917],{"className":19060},[1152,1322],[1109,19062],{"className":19063,"style":2497},[1231],[1109,19065,2342],{"className":19066},[2501],[1109,19068],{"className":19069,"style":2497},[1231],[1109,19071,19073,19076,19079,19082,19085,19125,19128,19131],{"className":19072},[1143],[1109,19074],{"className":19075,"style":1498},[1147],[1109,19077,5892],{"className":19078},[1152,1153],[1109,19080,8631],{"className":19081},[1152],[1109,19083,16244],{"className":19084,"style":16314},[1152,1153],[1109,19086,19088,19091],{"className":19087},[1152],[1109,19089,8631],{"className":19090},[1152],[1109,19092,19094],{"className":19093},[1293],[1109,19095,19097,19117],{"className":19096},[1297,1298],[1109,19098,19100,19114],{"className":19099},[1302],[1109,19101,19103],{"className":19102,"style":1307},[1306],[1109,19104,19105,19108],{"style":1310},[1109,19106],{"className":19107,"style":1315},[1314],[1109,19109,19111],{"className":19110},[1319,1320,1321,1322],[1109,19112,1685],{"className":19113},[1152,1322],[1109,19115,1330],{"className":19116},[1329],[1109,19118,19120],{"className":19119},[1302],[1109,19121,19123],{"className":19122,"style":1337},[1306],[1109,19124],{},[1109,19126],{"className":19127,"style":2497},[1231],[1109,19129,2342],{"className":19130},[2501],[1109,19132],{"className":19133,"style":2497},[1231],[1109,19135,19137,19140,19172,19175,19178,19181,19184],{"className":19136},[1143],[1109,19138],{"className":19139,"style":11150},[1147],[1109,19141,19143,19146],{"className":19142},[1152],[1109,19144,18834],{"className":19145},[1152,1153],[1109,19147,19149],{"className":19148},[1293],[1109,19150,19152],{"className":19151},[1297],[1109,19153,19155],{"className":19154},[1302],[1109,19156,19158],{"className":19157,"style":11218},[1306],[1109,19159,19160,19163],{"style":7044},[1109,19161],{"className":19162,"style":1315},[1314],[1109,19164,19166],{"className":19165},[1319,1320,1321,1322],[1109,19167,19169],{"className":19168},[1152,1322],[1109,19170,9471],{"className":19171},[1152,1322],[1109,19173,1457],{"className":19174},[1508],[1109,19176,12502],{"className":19177,"style":12530},[1152,1153],[1109,19179],{"className":19180,"style":2497},[1231],[1109,19182,5991],{"className":19183},[2501],[1109,19185],{"className":19186,"style":2497},[1231],[1109,19188,19190,19193,19196,19199,19202,19205],{"className":19189},[1143],[1109,19191],{"className":19192,"style":1498},[1147],[1109,19194,16244],{"className":19195,"style":16314},[1152,1153],[1109,19197,1462],{"className":19198},[1515],[1109,19200],{"className":19201,"style":2497},[1231],[1109,19203,2342],{"className":19204},[2501],[1109,19206],{"className":19207,"style":2497},[1231],[1109,19209,19211,19214,19282,19285,19288,19291,19294],{"className":19210},[1143],[1109,19212],{"className":19213,"style":16819},[1147],[1109,19215,19217,19220,19279],{"className":19216},[1152],[1109,19218],{"className":19219},[1508,4589],[1109,19221,19223],{"className":19222},[4490],[1109,19224,19226,19271],{"className":19225},[1297,1298],[1109,19227,19229,19268],{"className":19228},[1302],[1109,19230,19232,19246,19254],{"className":19231,"style":16838},[1306],[1109,19233,19234,19237],{"style":4605},[1109,19235],{"className":19236,"style":4609},[1314],[1109,19238,19240],{"className":19239},[1319,1320,1321,1322],[1109,19241,19243],{"className":19242},[1152,1322],[1109,19244,1917],{"className":19245},[1152,1322],[1109,19247,19248,19251],{"style":4630},[1109,19249],{"className":19250,"style":4609},[1314],[1109,19252],{"className":19253,"style":4638},[4637],[1109,19255,19256,19259],{"style":9614},[1109,19257],{"className":19258,"style":4609},[1314],[1109,19260,19262],{"className":19261},[1319,1320,1321,1322],[1109,19263,19265],{"className":19264},[1152,1322],[1109,19266,16482],{"className":19267},[1152,1153,1322],[1109,19269,1330],{"className":19270},[1329],[1109,19272,19274],{"className":19273},[1302],[1109,19275,19277],{"className":19276,"style":8058},[1306],[1109,19278],{},[1109,19280],{"className":19281},[1515,4589],[1109,19283,8631],{"className":19284},[1152],[1109,19286,12502],{"className":19287,"style":12530},[1152,1153],[1109,19289],{"className":19290,"style":2497},[1231],[1109,19292,5991],{"className":19293},[2501],[1109,19295],{"className":19296,"style":2497},[1231],[1109,19298,19300,19303,19306],{"className":19299},[1143],[1109,19301],{"className":19302,"style":8085},[1147],[1109,19304,16244],{"className":19305,"style":16314},[1152,1153],[1109,19307,19309,19312],{"className":19308},[1152],[1109,19310,8631],{"className":19311},[1152],[1109,19313,19315],{"className":19314},[1293],[1109,19316,19318],{"className":19317},[1297],[1109,19319,19321],{"className":19320},[1302],[1109,19322,19324],{"className":19323,"style":8144},[1306],[1109,19325,19326,19329],{"style":7044},[1109,19327],{"className":19328,"style":1315},[1314],[1109,19330,19332],{"className":19331},[1319,1320,1321,1322],[1109,19333,1917],{"className":19334},[1152,1322],[1101,19336,15316,19337,19365,19366,19394],{},[1109,19338,19340,19353],{"className":19339},[1112],[1109,19341,19343],{"className":19342},[1116],[1118,19344,19345],{"xmlns":1120},[1122,19346,19347,19351],{},[1125,19348,19349],{},[1128,19350,12502],{},[1132,19352,15333],{"encoding":1134},[1109,19354,19356],{"className":19355,"ariaHidden":1139},[1138],[1109,19357,19359,19362],{"className":19358},[1143],[1109,19360],{"className":19361,"style":1148},[1147],[1109,19363,12502],{"className":19364,"style":12530},[1152,1153]," 求导并令其为 0,得到 ",[1109,19367,19369,19382],{"className":19368},[1112],[1109,19370,19372],{"className":19371},[1116],[1118,19373,19374],{"xmlns":1120},[1122,19375,19376,19380],{},[1125,19377,19378],{},[1128,19379,12502],{},[1132,19381,15333],{"encoding":1134},[1109,19383,19385],{"className":19384,"ariaHidden":1139},[1138],[1109,19386,19388,19391],{"className":19387},[1143],[1109,19389],{"className":19390,"style":1148},[1147],[1109,19392,12502],{"className":19393,"style":12530},[1152,1153]," 更新是一个岭回归式解:",[1101,19396,19397],{},[1109,19398,19400,19483],{"className":19399},[1112],[1109,19401,19403],{"className":19402},[1116],[1118,19404,19405],{"xmlns":1120},[1122,19406,19407,19480],{},[1125,19408,19409,19421,19423,19425,19431,19433,19435,19437,19440,19450,19452,19458,19460,19462,19464,19470,19472,19478],{},[1264,19410,19411,19413],{},[1128,19412,12502],{},[1125,19414,19415,19417,19419],{},[1128,19416,8620],{},[1204,19418,2342],{},[1208,19420,1685],{},[1204,19422,1582],{},[1204,19424,1457],{"stretchy":1456},[6964,19426,19427,19429],{},[1128,19428,15377],{},[1128,19430,9471],{"mathvariant":1450},[1128,19432,15377],{},[1204,19434,2342],{},[1128,19436,16482],{},[1128,19438,19439],{},"I",[6964,19441,19442,19444],{},[1204,19443,1462],{"stretchy":1456},[1125,19445,19446,19448],{},[1204,19447,5991],{},[1208,19449,1685],{},[1204,19451,1457],{"fence":1456,"stretchy":1139,"minsize":5974,"maxsize":5974},[6964,19453,19454,19456],{},[1128,19455,15377],{},[1128,19457,9471],{"mathvariant":1450},[1128,19459,14113],{},[1204,19461,2342],{},[1128,19463,16482],{},[1264,19465,19466,19468],{},[1128,19467,16244],{},[1128,19469,8620],{},[1204,19471,5991],{},[1264,19473,19474,19476],{},[1128,19475,18834],{},[1128,19477,8620],{},[1204,19479,1462],{"fence":1456,"stretchy":1139,"minsize":5974,"maxsize":5974},[1132,19481,19482],{"encoding":1134},"\\beta_{k+1} = (X^{\\top}X + \\rho I)^{-1}\\big(X^{\\top}y + \\rho z_k - u_k\\big)",[1109,19484,19486,19550,19603,19700,19759],{"className":19485,"ariaHidden":1139},[1138],[1109,19487,19489,19492,19541,19544,19547],{"className":19488},[1143],[1109,19490],{"className":19491,"style":15948},[1147],[1109,19493,19495,19498],{"className":19494},[1152],[1109,19496,12502],{"className":19497,"style":12530},[1152,1153],[1109,19499,19501],{"className":19500},[1293],[1109,19502,19504,19533],{"className":19503},[1297,1298],[1109,19505,19507,19530],{"className":19506},[1302],[1109,19508,19510],{"className":19509,"style":8735},[1306],[1109,19511,19512,19515],{"style":15969},[1109,19513],{"className":19514,"style":1315},[1314],[1109,19516,19518],{"className":19517},[1319,1320,1321,1322],[1109,19519,19521,19524,19527],{"className":19520},[1152,1322],[1109,19522,8620],{"className":19523,"style":8750},[1152,1153,1322],[1109,19525,2342],{"className":19526},[2501,1322],[1109,19528,1685],{"className":19529},[1152,1322],[1109,19531,1330],{"className":19532},[1329],[1109,19534,19536],{"className":19535},[1302],[1109,19537,19539],{"className":19538,"style":10663},[1306],[1109,19540],{},[1109,19542],{"className":19543,"style":1232},[1231],[1109,19545,1582],{"className":19546},[1236],[1109,19548],{"className":19549,"style":1232},[1231],[1109,19551,19553,19556,19559,19591,19594,19597,19600],{"className":19552},[1143],[1109,19554],{"className":19555,"style":11150},[1147],[1109,19557,1457],{"className":19558},[1508],[1109,19560,19562,19565],{"className":19561},[1152],[1109,19563,15377],{"className":19564,"style":15471},[1152,1153],[1109,19566,19568],{"className":19567},[1293],[1109,19569,19571],{"className":19570},[1297],[1109,19572,19574],{"className":19573},[1302],[1109,19575,19577],{"className":19576,"style":11218},[1306],[1109,19578,19579,19582],{"style":7044},[1109,19580],{"className":19581,"style":1315},[1314],[1109,19583,19585],{"className":19584},[1319,1320,1321,1322],[1109,19586,19588],{"className":19587},[1152,1322],[1109,19589,9471],{"className":19590},[1152,1322],[1109,19592,15377],{"className":19593,"style":15471},[1152,1153],[1109,19595],{"className":19596,"style":2497},[1231],[1109,19598,2342],{"className":19599},[2501],[1109,19601],{"className":19602,"style":2497},[1231],[1109,19604,19606,19609,19612,19615,19650,19656,19688,19691,19694,19697],{"className":19605},[1143],[1109,19607],{"className":19608,"style":6056},[1147],[1109,19610,16482],{"className":19611},[1152,1153],[1109,19613,19439],{"className":19614,"style":15471},[1152,1153],[1109,19616,19618,19621],{"className":19617},[1515],[1109,19619,1462],{"className":19620},[1515],[1109,19622,19624],{"className":19623},[1293],[1109,19625,19627],{"className":19626},[1297],[1109,19628,19630],{"className":19629},[1302],[1109,19631,19633],{"className":19632,"style":8144},[1306],[1109,19634,19635,19638],{"style":7044},[1109,19636],{"className":19637,"style":1315},[1314],[1109,19639,19641],{"className":19640},[1319,1320,1321,1322],[1109,19642,19644,19647],{"className":19643},[1152,1322],[1109,19645,5991],{"className":19646},[1152,1322],[1109,19648,1685],{"className":19649},[1152,1322],[1109,19651,19653],{"className":19652},[1152],[1109,19654,1457],{"className":19655},[6066,6067],[1109,19657,19659,19662],{"className":19658},[1152],[1109,19660,15377],{"className":19661,"style":15471},[1152,1153],[1109,19663,19665],{"className":19664},[1293],[1109,19666,19668],{"className":19667},[1297],[1109,19669,19671],{"className":19670},[1302],[1109,19672,19674],{"className":19673,"style":11218},[1306],[1109,19675,19676,19679],{"style":7044},[1109,19677],{"className":19678,"style":1315},[1314],[1109,19680,19682],{"className":19681},[1319,1320,1321,1322],[1109,19683,19685],{"className":19684},[1152,1322],[1109,19686,9471],{"className":19687},[1152,1322],[1109,19689,14113],{"className":19690,"style":1186},[1152,1153],[1109,19692],{"className":19693,"style":2497},[1231],[1109,19695,2342],{"className":19696},[2501],[1109,19698],{"className":19699,"style":2497},[1231],[1109,19701,19703,19707,19710,19750,19753,19756],{"className":19702},[1143],[1109,19704],{"className":19705,"style":19706},[1147],"height:0.7778em;vertical-align:-0.1944em;",[1109,19708,16482],{"className":19709},[1152,1153],[1109,19711,19713,19716],{"className":19712},[1152],[1109,19714,16244],{"className":19715,"style":16314},[1152,1153],[1109,19717,19719],{"className":19718},[1293],[1109,19720,19722,19742],{"className":19721},[1297,1298],[1109,19723,19725,19739],{"className":19724},[1302],[1109,19726,19728],{"className":19727,"style":8735},[1306],[1109,19729,19730,19733],{"style":17250},[1109,19731],{"className":19732,"style":1315},[1314],[1109,19734,19736],{"className":19735},[1319,1320,1321,1322],[1109,19737,8620],{"className":19738,"style":8750},[1152,1153,1322],[1109,19740,1330],{"className":19741},[1329],[1109,19743,19745],{"className":19744},[1302],[1109,19746,19748],{"className":19747,"style":1337},[1306],[1109,19749],{},[1109,19751],{"className":19752,"style":2497},[1231],[1109,19754,5991],{"className":19755},[2501],[1109,19757],{"className":19758,"style":2497},[1231],[1109,19760,19762,19765,19805],{"className":19761},[1143],[1109,19763],{"className":19764,"style":6056},[1147],[1109,19766,19768,19771],{"className":19767},[1152],[1109,19769,18834],{"className":19770},[1152,1153],[1109,19772,19774],{"className":19773},[1293],[1109,19775,19777,19797],{"className":19776},[1297,1298],[1109,19778,19780,19794],{"className":19779},[1302],[1109,19781,19783],{"className":19782,"style":8735},[1306],[1109,19784,19785,19788],{"style":1310},[1109,19786],{"className":19787,"style":1315},[1314],[1109,19789,19791],{"className":19790},[1319,1320,1321,1322],[1109,19792,8620],{"className":19793,"style":8750},[1152,1153,1322],[1109,19795,1330],{"className":19796},[1329],[1109,19798,19800],{"className":19799},[1302],[1109,19801,19803],{"className":19802,"style":1337},[1306],[1109,19804],{},[1109,19806,19808],{"className":19807},[1152],[1109,19809,1462],{"className":19810},[6066,6067],[1101,19812,19813,19814,19842,19843,1662],{},"而 ",[1109,19815,19817,19830],{"className":19816},[1112],[1109,19818,19820],{"className":19819},[1116],[1118,19821,19822],{"xmlns":1120},[1122,19823,19824,19828],{},[1125,19825,19826],{},[1128,19827,16244],{},[1132,19829,16244],{"encoding":1134},[1109,19831,19833],{"className":19832,"ariaHidden":1139},[1138],[1109,19834,19836,19839],{"className":19835},[1143],[1109,19837],{"className":19838,"style":7312},[1147],[1109,19840,16244],{"className":19841,"style":16314},[1152,1153]," 的更新公式是",[1104,19844,19845],{},"软阈值(soft-thresholding)算子",[1101,19847,19848],{},[1109,19849,19851,19972],{"className":19850},[1112],[1109,19852,19854],{"className":19853},[1116],[1118,19855,19856],{"xmlns":1120},[1122,19857,19858,19969],{},[1125,19859,19860,19872,19874,19888,19890,19902,19904,19910,19912,19914,19916,19918,19920,19927,19929,19931,19933,19935,19938,19940,19942,19944,19946,19948,19950,19952,19954,19956,19958,19960,19962,19965,19967],{},[1264,19861,19862,19864],{},[1128,19863,16244],{},[1125,19865,19866,19868,19870],{},[1128,19867,8620],{},[1204,19869,2342],{},[1208,19871,1685],{},[1204,19873,1582],{},[1264,19875,19876,19879],{},[1128,19877,19878],{},"S",[1125,19880,19881,19883,19886],{},[1128,19882,5892],{},[1128,19884,19885],{"mathvariant":1450},"\u002F",[1128,19887,16482],{},[1204,19889,1457],{"fence":1456,"stretchy":1139,"minsize":5974,"maxsize":5974},[1264,19891,19892,19894],{},[1128,19893,12502],{},[1125,19895,19896,19898,19900],{},[1128,19897,8620],{},[1204,19899,2342],{},[1208,19901,1685],{},[1204,19903,2342],{},[1264,19905,19906,19908],{},[1128,19907,18834],{},[1128,19909,8620],{},[1128,19911,19885],{"mathvariant":1450},[1128,19913,16482],{},[1204,19915,1462],{"fence":1456,"stretchy":1139,"minsize":5974,"maxsize":5974},[1204,19917,1910],{"separator":1139},[1231,19919],{"width":14583},[1264,19921,19922,19924],{},[1128,19923,19878],{},[1128,19925,19926],{},"κ",[1204,19928,1457],{"stretchy":1456},[1128,19930,5522],{},[1204,19932,1462],{"stretchy":1456},[1204,19934,1582],{},[1471,19936,19937],{},"sign",[1204,19939,1457],{"stretchy":1456},[1128,19941,5522],{},[1204,19943,1462],{"stretchy":1456},[1128,19945,2409],{},[1204,19947,2412],{},[1204,19949,1457],{"stretchy":1456},[1128,19951,1451],{"mathvariant":1450},[1128,19953,5522],{},[1128,19955,1451],{"mathvariant":1450},[1204,19957,5991],{},[1128,19959,19926],{},[1204,19961,1910],{"separator":1139},[1471,19963,19964],{}," ",[1208,19966,1210],{},[1204,19968,1462],{"stretchy":1456},[1132,19970,19971],{"encoding":1134},"z_{k+1} = S_{\\lambda\u002F\\rho}\\big(\\beta_{k+1} + u_k\u002F\\rho\\big), \\qquad S_\\kappa(a) = \\text{sign}(a)\\max(|a| - \\kappa,\\ 0)",[1109,19973,19975,20039,20162,20288,20336],{"className":19974,"ariaHidden":1139},[1138],[1109,19976,19978,19981,20030,20033,20036],{"className":19977},[1143],[1109,19979],{"className":19980,"style":10615},[1147],[1109,19982,19984,19987],{"className":19983},[1152],[1109,19985,16244],{"className":19986,"style":16314},[1152,1153],[1109,19988,19990],{"className":19989},[1293],[1109,19991,19993,20022],{"className":19992},[1297,1298],[1109,19994,19996,20019],{"className":19995},[1302],[1109,19997,19999],{"className":19998,"style":8735},[1306],[1109,20000,20001,20004],{"style":17250},[1109,20002],{"className":20003,"style":1315},[1314],[1109,20005,20007],{"className":20006},[1319,1320,1321,1322],[1109,20008,20010,20013,20016],{"className":20009},[1152,1322],[1109,20011,8620],{"className":20012,"style":8750},[1152,1153,1322],[1109,20014,2342],{"className":20015},[2501,1322],[1109,20017,1685],{"className":20018},[1152,1322],[1109,20020,1330],{"className":20021},[1329],[1109,20023,20025],{"className":20024},[1302],[1109,20026,20028],{"className":20027,"style":10663},[1306],[1109,20029],{},[1109,20031],{"className":20032,"style":1232},[1231],[1109,20034,1582],{"className":20035},[1236],[1109,20037],{"className":20038,"style":1232},[1231],[1109,20040,20042,20046,20098,20104,20153,20156,20159],{"className":20041},[1143],[1109,20043],{"className":20044,"style":20045},[1147],"height:1.2052em;vertical-align:-0.3552em;",[1109,20047,20049,20053],{"className":20048},[1152],[1109,20050,19878],{"className":20051,"style":20052},[1152,1153],"margin-right:0.0576em;",[1109,20054,20056],{"className":20055},[1293],[1109,20057,20059,20089],{"className":20058},[1297,1298],[1109,20060,20062,20086],{"className":20061},[1302],[1109,20063,20065],{"className":20064,"style":8822},[1306],[1109,20066,20068,20071],{"style":20067},"top:-2.5198em;margin-left:-0.0576em;margin-right:0.05em;",[1109,20069],{"className":20070,"style":1315},[1314],[1109,20072,20074],{"className":20073},[1319,1320,1321,1322],[1109,20075,20077,20080,20083],{"className":20076},[1152,1322],[1109,20078,5892],{"className":20079},[1152,1153,1322],[1109,20081,19885],{"className":20082},[1152,1322],[1109,20084,16482],{"className":20085},[1152,1153,1322],[1109,20087,1330],{"className":20088},[1329],[1109,20090,20092],{"className":20091},[1302],[1109,20093,20096],{"className":20094,"style":20095},[1306],"height:0.3552em;",[1109,20097],{},[1109,20099,20101],{"className":20100},[1152],[1109,20102,1457],{"className":20103},[6066,6067],[1109,20105,20107,20110],{"className":20106},[1152],[1109,20108,12502],{"className":20109,"style":12530},[1152,1153],[1109,20111,20113],{"className":20112},[1293],[1109,20114,20116,20145],{"className":20115},[1297,1298],[1109,20117,20119,20142],{"className":20118},[1302],[1109,20120,20122],{"className":20121,"style":8735},[1306],[1109,20123,20124,20127],{"style":15969},[1109,20125],{"className":20126,"style":1315},[1314],[1109,20128,20130],{"className":20129},[1319,1320,1321,1322],[1109,20131,20133,20136,20139],{"className":20132},[1152,1322],[1109,20134,8620],{"className":20135,"style":8750},[1152,1153,1322],[1109,20137,2342],{"className":20138},[2501,1322],[1109,20140,1685],{"className":20141},[1152,1322],[1109,20143,1330],{"className":20144},[1329],[1109,20146,20148],{"className":20147},[1302],[1109,20149,20151],{"className":20150,"style":10663},[1306],[1109,20152],{},[1109,20154],{"className":20155,"style":2497},[1231],[1109,20157,2342],{"className":20158},[2501],[1109,20160],{"className":20161,"style":2497},[1231],[1109,20163,20165,20168,20208,20211,20214,20220,20223,20226,20229,20270,20273,20276,20279,20282,20285],{"className":20164},[1143],[1109,20166],{"className":20167,"style":6056},[1147],[1109,20169,20171,20174],{"className":20170},[1152],[1109,20172,18834],{"className":20173},[1152,1153],[1109,20175,20177],{"className":20176},[1293],[1109,20178,20180,20200],{"className":20179},[1297,1298],[1109,20181,20183,20197],{"className":20182},[1302],[1109,20184,20186],{"className":20185,"style":8735},[1306],[1109,20187,20188,20191],{"style":1310},[1109,20189],{"className":20190,"style":1315},[1314],[1109,20192,20194],{"className":20193},[1319,1320,1321,1322],[1109,20195,8620],{"className":20196,"style":8750},[1152,1153,1322],[1109,20198,1330],{"className":20199},[1329],[1109,20201,20203],{"className":20202},[1302],[1109,20204,20206],{"className":20205,"style":1337},[1306],[1109,20207],{},[1109,20209,19885],{"className":20210},[1152],[1109,20212,16482],{"className":20213},[1152,1153],[1109,20215,20217],{"className":20216},[1152],[1109,20218,1462],{"className":20219},[6066,6067],[1109,20221,1910],{"className":20222},[1975],[1109,20224],{"className":20225,"style":14984},[1231],[1109,20227],{"className":20228,"style":1540},[1231],[1109,20230,20232,20235],{"className":20231},[1152],[1109,20233,19878],{"className":20234,"style":20052},[1152,1153],[1109,20236,20238],{"className":20237},[1293],[1109,20239,20241,20262],{"className":20240},[1297,1298],[1109,20242,20244,20259],{"className":20243},[1302],[1109,20245,20247],{"className":20246,"style":4547},[1306],[1109,20248,20250,20253],{"style":20249},"top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;",[1109,20251],{"className":20252,"style":1315},[1314],[1109,20254,20256],{"className":20255},[1319,1320,1321,1322],[1109,20257,19926],{"className":20258},[1152,1153,1322],[1109,20260,1330],{"className":20261},[1329],[1109,20263,20265],{"className":20264},[1302],[1109,20266,20268],{"className":20267,"style":1337},[1306],[1109,20269],{},[1109,20271,1457],{"className":20272},[1508],[1109,20274,5522],{"className":20275},[1152,1153],[1109,20277,1462],{"className":20278},[1515],[1109,20280],{"className":20281,"style":1232},[1231],[1109,20283,1582],{"className":20284},[1236],[1109,20286],{"className":20287,"style":1232},[1231],[1109,20289,20291,20294,20300,20303,20306,20309,20312,20315,20318,20321,20324,20327,20330,20333],{"className":20290},[1143],[1109,20292],{"className":20293,"style":1498},[1147],[1109,20295,20297],{"className":20296},[1152,15712],[1109,20298,19937],{"className":20299},[1152],[1109,20301,1457],{"className":20302},[1508],[1109,20304,5522],{"className":20305},[1152,1153],[1109,20307,1462],{"className":20308},[1515],[1109,20310],{"className":20311,"style":1540},[1231],[1109,20313,2409],{"className":20314},[2878],[1109,20316,1457],{"className":20317},[1508],[1109,20319,1451],{"className":20320},[1152],[1109,20322,5522],{"className":20323},[1152,1153],[1109,20325,1451],{"className":20326},[1152],[1109,20328],{"className":20329,"style":2497},[1231],[1109,20331,5991],{"className":20332},[2501],[1109,20334],{"className":20335,"style":2497},[1231],[1109,20337,20339,20342,20345,20348,20351,20354,20357],{"className":20338},[1143],[1109,20340],{"className":20341,"style":1498},[1147],[1109,20343,19926],{"className":20344},[1152,1153],[1109,20346,1910],{"className":20347},[1975],[1109,20349,19964],{"className":20350},[1231],[1109,20352],{"className":20353,"style":1540},[1231],[1109,20355,1210],{"className":20356},[1152],[1109,20358,1462],{"className":20359},[1515],[1096,20361,20363],{"id":20362},"例子分位数回归","例子:分位数回归",[1101,20365,20366,20369,20370,20401,20402,1662],{},[1104,20367,20368],{},"分位数的定义与损失",":分位数回归希望找一条直线拟合样本的 ",[1109,20371,20373,20388],{"className":20372},[1112],[1109,20374,20376],{"className":20375},[1116],[1118,20377,20378],{"xmlns":1120},[1122,20379,20380,20385],{},[1125,20381,20382],{},[1128,20383,20384],{},"τ",[1132,20386,20387],{"encoding":1134},"\\tau",[1109,20389,20391],{"className":20390,"ariaHidden":1139},[1138],[1109,20392,20394,20397],{"className":20393},[1143],[1109,20395],{"className":20396,"style":7312},[1147],[1109,20398,20384],{"className":20399,"style":20400},[1152,1153],"margin-right:0.1132em;"," 分位数。它用的是",[1104,20403,20404],{},"检验损失(pinball loss)",[1101,20406,20407],{},[1109,20408,20410,20537],{"className":20409},[1112],[1109,20411,20413],{"className":20412},[1116],[1118,20414,20415],{"xmlns":1120},[1122,20416,20417,20534],{},[1125,20418,20419,20425,20427,20429,20431,20433,20435,20437,20439,20441,20444,20446,20448,20450,20452,20454,20456,20458],{},[1264,20420,20421,20423],{},[1128,20422,16482],{},[1128,20424,20384],{},[1204,20426,1457],{"stretchy":1456},[1128,20428,18834],{},[1204,20430,1462],{"stretchy":1456},[1204,20432,1582],{},[1128,20434,18834],{},[1204,20436,1457],{"fence":1456,"stretchy":1139,"minsize":5974,"maxsize":5974},[1128,20438,20384],{},[1204,20440,5991],{},[1208,20442,1685],{"mathvariant":20443},"double-struck",[1204,20445,5519],{"stretchy":1456},[1128,20447,18834],{},[1204,20449,6687],{},[1208,20451,1210],{},[1204,20453,5530],{"stretchy":1456},[1204,20455,1462],{"fence":1456,"stretchy":1139,"minsize":5974,"maxsize":5974},[1204,20457,1582],{},[1125,20459,20460,20463],{},[1204,20461,20462],{"fence":1139},"{",[20464,20465,20468,20498],"mtable",{"rowspacing":20466,"columnalign":20467,"columnspacing":8689},"0.36em","left left",[20469,20470,20471,20486],"mtr",{},[20472,20473,20474],"mtd",{},[13396,20475,20476],{"scriptlevel":1210,"displaystyle":1456},[1125,20477,20478,20480,20482,20484],{},[1128,20479,20384],{},[1471,20481,1473],{},[1128,20483,18834],{},[1204,20485,1910],{"separator":1139},[20472,20487,20488],{},[13396,20489,20490],{"scriptlevel":1210,"displaystyle":1456},[1125,20491,20492,20494,20496],{},[1128,20493,18834],{},[1204,20495,1357],{},[1208,20497,1210],{},[20469,20499,20500,20522],{},[20472,20501,20502],{},[13396,20503,20504],{"scriptlevel":1210,"displaystyle":1456},[1125,20505,20506,20508,20510,20512,20514,20516,20518,20520],{},[1204,20507,1457],{"stretchy":1456},[1128,20509,20384],{},[1204,20511,5991],{},[1208,20513,1685],{},[1204,20515,1462],{"stretchy":1456},[1471,20517,1473],{},[1128,20519,18834],{},[1204,20521,1910],{"separator":1139},[20472,20523,20524],{},[13396,20525,20526],{"scriptlevel":1210,"displaystyle":1456},[1125,20527,20528,20530,20532],{},[1128,20529,18834],{},[1204,20531,6687],{},[1208,20533,1210],{},[1132,20535,20536],{"encoding":1134},"\\rho_\\tau(u) = u\\big(\\tau - \\mathbb{1}[u \u003C 0]\\big) = \\begin{cases} \\tau\\,u, & u \\ge 0 \\\\ (\\tau-1)\\,u, & u \u003C 0 \\end{cases}",[1109,20538,20540,20604,20631,20655,20682],{"className":20539,"ariaHidden":1139},[1138],[1109,20541,20543,20546,20586,20589,20592,20595,20598,20601],{"className":20542},[1143],[1109,20544],{"className":20545,"style":1498},[1147],[1109,20547,20549,20552],{"className":20548},[1152],[1109,20550,16482],{"className":20551},[1152,1153],[1109,20553,20555],{"className":20554},[1293],[1109,20556,20558,20578],{"className":20557},[1297,1298],[1109,20559,20561,20575],{"className":20560},[1302],[1109,20562,20564],{"className":20563,"style":4547},[1306],[1109,20565,20566,20569],{"style":1310},[1109,20567],{"className":20568,"style":1315},[1314],[1109,20570,20572],{"className":20571},[1319,1320,1321,1322],[1109,20573,20384],{"className":20574,"style":20400},[1152,1153,1322],[1109,20576,1330],{"className":20577},[1329],[1109,20579,20581],{"className":20580},[1302],[1109,20582,20584],{"className":20583,"style":1337},[1306],[1109,20585],{},[1109,20587,1457],{"className":20588},[1508],[1109,20590,18834],{"className":20591},[1152,1153],[1109,20593,1462],{"className":20594},[1515],[1109,20596],{"className":20597,"style":1232},[1231],[1109,20599,1582],{"className":20600},[1236],[1109,20602],{"className":20603,"style":1232},[1231],[1109,20605,20607,20610,20613,20619,20622,20625,20628],{"className":20606},[1143],[1109,20608],{"className":20609,"style":6056},[1147],[1109,20611,18834],{"className":20612},[1152,1153],[1109,20614,20616],{"className":20615},[1152],[1109,20617,1457],{"className":20618},[6066,6067],[1109,20620,20384],{"className":20621,"style":20400},[1152,1153],[1109,20623],{"className":20624,"style":2497},[1231],[1109,20626,5991],{"className":20627},[2501],[1109,20629],{"className":20630,"style":2497},[1231],[1109,20632,20634,20637,20640,20643,20646,20649,20652],{"className":20633},[1143],[1109,20635],{"className":20636,"style":1498},[1147],[1109,20638,1685],{"className":20639},[1152],[1109,20641,5519],{"className":20642},[1508],[1109,20644,18834],{"className":20645},[1152,1153],[1109,20647],{"className":20648,"style":1232},[1231],[1109,20650,6687],{"className":20651},[1236],[1109,20653],{"className":20654,"style":1232},[1231],[1109,20656,20658,20661,20664,20667,20673,20676,20679],{"className":20657},[1143],[1109,20659],{"className":20660,"style":6056},[1147],[1109,20662,1210],{"className":20663},[1152],[1109,20665,5530],{"className":20666},[1515],[1109,20668,20670],{"className":20669},[1152],[1109,20671,1462],{"className":20672},[6066,6067],[1109,20674],{"className":20675,"style":1232},[1231],[1109,20677,1582],{"className":20678},[1236],[1109,20680],{"className":20681,"style":1232},[1231],[1109,20683,20685,20689],{"className":20684},[1143],[1109,20686],{"className":20687,"style":20688},[1147],"height:3em;vertical-align:-1.25em;",[1109,20690,20692,20699,20866],{"className":20691},[8175],[1109,20693,20695],{"className":20694,"style":9525},[1508,9524],[1109,20696,20462],{"className":20697},[6066,20698],"size4",[1109,20700,20702],{"className":20701},[1152],[1109,20703,20705,20792,20797],{"className":20704},[20464],[1109,20706,20709],{"className":20707},[20708],"col-align-l",[1109,20710,20712,20783],{"className":20711},[1297,1298],[1109,20713,20715,20780],{"className":20714},[1302],[1109,20716,20719,20741],{"className":20717,"style":20718},[1306],"height:1.69em;",[1109,20720,20722,20726],{"style":20721},"top:-3.69em;",[1109,20723],{"className":20724,"style":20725},[1314],"height:3.008em;",[1109,20727,20729,20732,20735,20738],{"className":20728},[1152],[1109,20730,20384],{"className":20731,"style":20400},[1152,1153],[1109,20733],{"className":20734,"style":1540},[1231],[1109,20736,18834],{"className":20737},[1152,1153],[1109,20739,1910],{"className":20740},[1975],[1109,20742,20744,20747],{"style":20743},"top:-2.25em;",[1109,20745],{"className":20746,"style":20725},[1314],[1109,20748,20750,20753,20756,20759,20762,20765,20768,20771,20774,20777],{"className":20749},[1152],[1109,20751,1457],{"className":20752},[1508],[1109,20754,20384],{"className":20755,"style":20400},[1152,1153],[1109,20757],{"className":20758,"style":2497},[1231],[1109,20760,5991],{"className":20761},[2501],[1109,20763],{"className":20764,"style":2497},[1231],[1109,20766,1685],{"className":20767},[1152],[1109,20769,1462],{"className":20770},[1515],[1109,20772],{"className":20773,"style":1540},[1231],[1109,20775,18834],{"className":20776},[1152,1153],[1109,20778,1910],{"className":20779},[1975],[1109,20781,1330],{"className":20782},[1329],[1109,20784,20786],{"className":20785},[1302],[1109,20787,20790],{"className":20788,"style":20789},[1306],"height:1.19em;",[1109,20791],{},[1109,20793],{"className":20794,"style":20796},[20795],"arraycolsep","width:1em;",[1109,20798,20800],{"className":20799},[20708],[1109,20801,20803,20858],{"className":20802},[1297,1298],[1109,20804,20806,20855],{"className":20805},[1302],[1109,20807,20809,20832],{"className":20808,"style":20718},[1306],[1109,20810,20811,20814],{"style":20721},[1109,20812],{"className":20813,"style":20725},[1314],[1109,20815,20817,20820,20823,20826,20829],{"className":20816},[1152],[1109,20818,18834],{"className":20819},[1152,1153],[1109,20821],{"className":20822,"style":1232},[1231],[1109,20824,1357],{"className":20825},[1236],[1109,20827],{"className":20828,"style":1232},[1231],[1109,20830,1210],{"className":20831},[1152],[1109,20833,20834,20837],{"style":20743},[1109,20835],{"className":20836,"style":20725},[1314],[1109,20838,20840,20843,20846,20849,20852],{"className":20839},[1152],[1109,20841,18834],{"className":20842},[1152,1153],[1109,20844],{"className":20845,"style":1232},[1231],[1109,20847,6687],{"className":20848},[1236],[1109,20850],{"className":20851,"style":1232},[1231],[1109,20853,1210],{"className":20854},[1152],[1109,20856,1330],{"className":20857},[1329],[1109,20859,20861],{"className":20860},[1302],[1109,20862,20864],{"className":20863,"style":20789},[1306],[1109,20865],{},[1109,20867],{"className":20868},[1515,4589],[1101,20870,20871],{},"可以理解为:给小于分位数的点(残差为负)一个权重,给大于分位数的点另一个权重(给多数点小权重、少数点大权重)。",[1101,20873,20874,20877,20878,20906],{},[1104,20875,20876],{},"几何意义",":希望回归曲线之下能够包含 ",[1109,20879,20881,20894],{"className":20880},[1112],[1109,20882,20884],{"className":20883},[1116],[1118,20885,20886],{"xmlns":1120},[1122,20887,20888,20892],{},[1125,20889,20890],{},[1128,20891,20384],{},[1132,20893,20387],{"encoding":1134},[1109,20895,20897],{"className":20896,"ariaHidden":1139},[1138],[1109,20898,20900,20903],{"className":20899},[1143],[1109,20901],{"className":20902,"style":7312},[1147],[1109,20904,20384],{"className":20905,"style":20400},[1152,1153]," 比例的数据点。",[1101,20908,20909,20912,20913,21046,21047,21363,21364,21392,21393,21421,21422,21450,21451,21643],{},[1104,20910,20911],{},"优化",":对样本 ",[1109,20914,20916,20945],{"className":20915},[1112],[1109,20917,20919],{"className":20918},[1116],[1118,20920,20921],{"xmlns":1120},[1122,20922,20923,20943],{},[1125,20924,20925,20927,20933,20935,20941],{},[1204,20926,1457],{"stretchy":1456},[1264,20928,20929,20931],{},[1128,20930,1268],{},[1128,20932,9875],{},[1204,20934,1910],{"separator":1139},[1264,20936,20937,20939],{},[1128,20938,14113],{},[1128,20940,9875],{},[1204,20942,1462],{"stretchy":1456},[1132,20944,14120],{"encoding":1134},[1109,20946,20948],{"className":20947,"ariaHidden":1139},[1138],[1109,20949,20951,20954,20957,20997,21000,21003,21043],{"className":20950},[1143],[1109,20952],{"className":20953,"style":1498},[1147],[1109,20955,1457],{"className":20956},[1508],[1109,20958,20960,20963],{"className":20959},[1152],[1109,20961,1268],{"className":20962},[1152,1153],[1109,20964,20966],{"className":20965},[1293],[1109,20967,20969,20989],{"className":20968},[1297,1298],[1109,20970,20972,20986],{"className":20971},[1302],[1109,20973,20975],{"className":20974,"style":14151},[1306],[1109,20976,20977,20980],{"style":1310},[1109,20978],{"className":20979,"style":1315},[1314],[1109,20981,20983],{"className":20982},[1319,1320,1321,1322],[1109,20984,9875],{"className":20985},[1152,1153,1322],[1109,20987,1330],{"className":20988},[1329],[1109,20990,20992],{"className":20991},[1302],[1109,20993,20995],{"className":20994,"style":1337},[1306],[1109,20996],{},[1109,20998,1910],{"className":20999},[1975],[1109,21001],{"className":21002,"style":1540},[1231],[1109,21004,21006,21009],{"className":21005},[1152],[1109,21007,14113],{"className":21008,"style":1186},[1152,1153],[1109,21010,21012],{"className":21011},[1293],[1109,21013,21015,21035],{"className":21014},[1297,1298],[1109,21016,21018,21032],{"className":21017},[1302],[1109,21019,21021],{"className":21020,"style":14151},[1306],[1109,21022,21023,21026],{"style":2146},[1109,21024],{"className":21025,"style":1315},[1314],[1109,21027,21029],{"className":21028},[1319,1320,1321,1322],[1109,21030,9875],{"className":21031},[1152,1153,1322],[1109,21033,1330],{"className":21034},[1329],[1109,21036,21038],{"className":21037},[1302],[1109,21039,21041],{"className":21040,"style":1337},[1306],[1109,21042],{},[1109,21044,1462],{"className":21045},[1515],",考虑 ",[1109,21048,21050,21106],{"className":21049},[1112],[1109,21051,21053],{"className":21052},[1116],[1118,21054,21055],{"xmlns":1120},[1122,21056,21057,21103],{},[1125,21058,21059,21069,21075,21081,21083,21089,21091,21099,21101],{},[1264,21060,21061,21067],{},[1125,21062,21063,21065],{},[1128,21064,10241],{},[1204,21066,2412],{},[1128,21068,12502],{},[1264,21070,21071,21073],{},[1204,21072,14599],{},[1128,21074,9875],{},[1264,21076,21077,21079],{},[1128,21078,16482],{},[1128,21080,20384],{},[1204,21082,1457],{"stretchy":1456},[1264,21084,21085,21087],{},[1128,21086,14113],{},[1128,21088,9875],{},[1204,21090,5991],{},[14264,21092,21093,21095,21097],{},[1128,21094,1268],{},[1128,21096,9875],{},[1128,21098,9471],{"mathvariant":1450},[1128,21100,12502],{},[1204,21102,1462],{"stretchy":1456},[1132,21104,21105],{"encoding":1134},"\\min_\\beta \\sum_i \\rho_\\tau(y_i - x_i^{\\top}\\beta)",[1109,21107,21109,21294],{"className":21108,"ariaHidden":1139},[1138],[1109,21110,21112,21116,21156,21159,21199,21202,21242,21245,21285,21288,21291],{"className":21111},[1143],[1109,21113],{"className":21114,"style":21115},[1147],"height:1.0497em;vertical-align:-0.2997em;",[1109,21117,21119,21122],{"className":21118},[2878],[1109,21120,10241],{"className":21121},[2878],[1109,21123,21125],{"className":21124},[1293],[1109,21126,21128,21148],{"className":21127},[1297,1298],[1109,21129,21131,21145],{"className":21130},[1302],[1109,21132,21134],{"className":21133,"style":8735},[1306],[1109,21135,21136,21139],{"style":4550},[1109,21137],{"className":21138,"style":1315},[1314],[1109,21140,21142],{"className":21141},[1319,1320,1321,1322],[1109,21143,12502],{"className":21144,"style":12530},[1152,1153,1322],[1109,21146,1330],{"className":21147},[1329],[1109,21149,21151],{"className":21150},[1302],[1109,21152,21154],{"className":21153,"style":16324},[1306],[1109,21155],{},[1109,21157],{"className":21158,"style":1540},[1231],[1109,21160,21162,21165],{"className":21161},[2878],[1109,21163,14599],{"className":21164,"style":14705},[2878,14703,14704],[1109,21166,21168],{"className":21167},[1293],[1109,21169,21171,21191],{"className":21170},[1297,1298],[1109,21172,21174,21188],{"className":21173},[1302],[1109,21175,21177],{"className":21176,"style":14718},[1306],[1109,21178,21179,21182],{"style":14721},[1109,21180],{"className":21181,"style":1315},[1314],[1109,21183,21185],{"className":21184},[1319,1320,1321,1322],[1109,21186,9875],{"className":21187},[1152,1153,1322],[1109,21189,1330],{"className":21190},[1329],[1109,21192,21194],{"className":21193},[1302],[1109,21195,21197],{"className":21196,"style":14740},[1306],[1109,21198],{},[1109,21200],{"className":21201,"style":1540},[1231],[1109,21203,21205,21208],{"className":21204},[1152],[1109,21206,16482],{"className":21207},[1152,1153],[1109,21209,21211],{"className":21210},[1293],[1109,21212,21214,21234],{"className":21213},[1297,1298],[1109,21215,21217,21231],{"className":21216},[1302],[1109,21218,21220],{"className":21219,"style":4547},[1306],[1109,21221,21222,21225],{"style":1310},[1109,21223],{"className":21224,"style":1315},[1314],[1109,21226,21228],{"className":21227},[1319,1320,1321,1322],[1109,21229,20384],{"className":21230,"style":20400},[1152,1153,1322],[1109,21232,1330],{"className":21233},[1329],[1109,21235,21237],{"className":21236},[1302],[1109,21238,21240],{"className":21239,"style":1337},[1306],[1109,21241],{},[1109,21243,1457],{"className":21244},[1508],[1109,21246,21248,21251],{"className":21247},[1152],[1109,21249,14113],{"className":21250,"style":1186},[1152,1153],[1109,21252,21254],{"className":21253},[1293],[1109,21255,21257,21277],{"className":21256},[1297,1298],[1109,21258,21260,21274],{"className":21259},[1302],[1109,21261,21263],{"className":21262,"style":14151},[1306],[1109,21264,21265,21268],{"style":2146},[1109,21266],{"className":21267,"style":1315},[1314],[1109,21269,21271],{"className":21270},[1319,1320,1321,1322],[1109,21272,9875],{"className":21273},[1152,1153,1322],[1109,21275,1330],{"className":21276},[1329],[1109,21278,21280],{"className":21279},[1302],[1109,21281,21283],{"className":21282,"style":1337},[1306],[1109,21284],{},[1109,21286],{"className":21287,"style":2497},[1231],[1109,21289,5991],{"className":21290},[2501],[1109,21292],{"className":21293,"style":2497},[1231],[1109,21295,21297,21301,21357,21360],{"className":21296},[1143],[1109,21298],{"className":21299,"style":21300},[1147],"height:1.1078em;vertical-align:-0.2587em;",[1109,21302,21304,21307],{"className":21303},[1152],[1109,21305,1268],{"className":21306},[1152,1153],[1109,21308,21310],{"className":21309},[1293],[1109,21311,21313,21348],{"className":21312},[1297,1298],[1109,21314,21316,21345],{"className":21315},[1302],[1109,21317,21319,21331],{"className":21318,"style":11218},[1306],[1109,21320,21322,21325],{"style":21321},"top:-2.4413em;margin-left:0em;margin-right:0.05em;",[1109,21323],{"className":21324,"style":1315},[1314],[1109,21326,21328],{"className":21327},[1319,1320,1321,1322],[1109,21329,9875],{"className":21330},[1152,1153,1322],[1109,21332,21333,21336],{"style":7044},[1109,21334],{"className":21335,"style":1315},[1314],[1109,21337,21339],{"className":21338},[1319,1320,1321,1322],[1109,21340,21342],{"className":21341},[1152,1322],[1109,21343,9471],{"className":21344},[1152,1322],[1109,21346,1330],{"className":21347},[1329],[1109,21349,21351],{"className":21350},[1302],[1109,21352,21355],{"className":21353,"style":21354},[1306],"height:0.2587em;",[1109,21356],{},[1109,21358,12502],{"className":21359,"style":12530},[1152,1153],[1109,21361,1462],{"className":21362},[1515],"。同样引入辅助变量写成约束形式,构造增广拉格朗日函数,按 ADMM 交替更新 ",[1109,21365,21367,21380],{"className":21366},[1112],[1109,21368,21370],{"className":21369},[1116],[1118,21371,21372],{"xmlns":1120},[1122,21373,21374,21378],{},[1125,21375,21376],{},[1128,21377,12502],{},[1132,21379,15333],{"encoding":1134},[1109,21381,21383],{"className":21382,"ariaHidden":1139},[1138],[1109,21384,21386,21389],{"className":21385},[1143],[1109,21387],{"className":21388,"style":1148},[1147],[1109,21390,12502],{"className":21391,"style":12530},[1152,1153],"、辅助变量、对偶变量;对 ",[1109,21394,21396,21409],{"className":21395},[1112],[1109,21397,21399],{"className":21398},[1116],[1118,21400,21401],{"xmlns":1120},[1122,21402,21403,21407],{},[1125,21404,21405],{},[1128,21406,12502],{},[1132,21408,15333],{"encoding":1134},[1109,21410,21412],{"className":21411,"ariaHidden":1139},[1138],[1109,21413,21415,21418],{"className":21414},[1143],[1109,21416],{"className":21417,"style":1148},[1147],[1109,21419,12502],{"className":21420,"style":12530},[1152,1153]," 子问题求偏导并令其为 0,得到 ",[1109,21423,21425,21438],{"className":21424},[1112],[1109,21426,21428],{"className":21427},[1116],[1118,21429,21430],{"xmlns":1120},[1122,21431,21432,21436],{},[1125,21433,21434],{},[1128,21435,12502],{},[1132,21437,15333],{"encoding":1134},[1109,21439,21441],{"className":21440,"ariaHidden":1139},[1138],[1109,21442,21444,21447],{"className":21443},[1143],[1109,21445],{"className":21446,"style":1148},[1147],[1109,21448,12502],{"className":21449,"style":12530},[1152,1153]," 的闭式更新(形如 ",[1109,21452,21454,21502],{"className":21453},[1112],[1109,21455,21457],{"className":21456},[1116],[1118,21458,21459],{"xmlns":1120},[1122,21460,21461,21499],{},[1125,21462,21463,21465,21467,21469,21475,21477,21487,21493,21495,21497],{},[1128,21464,12502],{},[1204,21466,1582],{},[1204,21468,1457],{"stretchy":1456},[6964,21470,21471,21473],{},[1128,21472,15377],{},[1128,21474,9471],{"mathvariant":1450},[1128,21476,15377],{},[6964,21478,21479,21481],{},[1204,21480,1462],{"stretchy":1456},[1125,21482,21483,21485],{},[1204,21484,5991],{},[1208,21486,1685],{},[6964,21488,21489,21491],{},[1128,21490,15377],{},[1128,21492,9471],{"mathvariant":1450},[1204,21494,1457],{"stretchy":1456},[1204,21496,3762],{},[1204,21498,1462],{"stretchy":1456},[1132,21500,21501],{"encoding":1134},"\\beta = (X^{\\top}X)^{-1}X^{\\top}(\\cdot)",[1109,21503,21505,21523],{"className":21504,"ariaHidden":1139},[1138],[1109,21506,21508,21511,21514,21517,21520],{"className":21507},[1143],[1109,21509],{"className":21510,"style":1148},[1147],[1109,21512,12502],{"className":21513,"style":12530},[1152,1153],[1109,21515],{"className":21516,"style":1232},[1231],[1109,21518,1582],{"className":21519},[1236],[1109,21521],{"className":21522,"style":1232},[1231],[1109,21524,21526,21529,21532,21564,21567,21602,21634,21637,21640],{"className":21525},[1143],[1109,21527],{"className":21528,"style":11150},[1147],[1109,21530,1457],{"className":21531},[1508],[1109,21533,21535,21538],{"className":21534},[1152],[1109,21536,15377],{"className":21537,"style":15471},[1152,1153],[1109,21539,21541],{"className":21540},[1293],[1109,21542,21544],{"className":21543},[1297],[1109,21545,21547],{"className":21546},[1302],[1109,21548,21550],{"className":21549,"style":11218},[1306],[1109,21551,21552,21555],{"style":7044},[1109,21553],{"className":21554,"style":1315},[1314],[1109,21556,21558],{"className":21557},[1319,1320,1321,1322],[1109,21559,21561],{"className":21560},[1152,1322],[1109,21562,9471],{"className":21563},[1152,1322],[1109,21565,15377],{"className":21566,"style":15471},[1152,1153],[1109,21568,21570,21573],{"className":21569},[1515],[1109,21571,1462],{"className":21572},[1515],[1109,21574,21576],{"className":21575},[1293],[1109,21577,21579],{"className":21578},[1297],[1109,21580,21582],{"className":21581},[1302],[1109,21583,21585],{"className":21584,"style":8144},[1306],[1109,21586,21587,21590],{"style":7044},[1109,21588],{"className":21589,"style":1315},[1314],[1109,21591,21593],{"className":21592},[1319,1320,1321,1322],[1109,21594,21596,21599],{"className":21595},[1152,1322],[1109,21597,5991],{"className":21598},[1152,1322],[1109,21600,1685],{"className":21601},[1152,1322],[1109,21603,21605,21608],{"className":21604},[1152],[1109,21606,15377],{"className":21607,"style":15471},[1152,1153],[1109,21609,21611],{"className":21610},[1293],[1109,21612,21614],{"className":21613},[1297],[1109,21615,21617],{"className":21616},[1302],[1109,21618,21620],{"className":21619,"style":11218},[1306],[1109,21621,21622,21625],{"style":7044},[1109,21623],{"className":21624,"style":1315},[1314],[1109,21626,21628],{"className":21627},[1319,1320,1321,1322],[1109,21629,21631],{"className":21630},[1152,1322],[1109,21632,9471],{"className":21633},[1152,1322],[1109,21635,1457],{"className":21636},[1508],[1109,21638,3762],{"className":21639},[1152],[1109,21641,1462],{"className":21642},[1515],"),辅助变量更新则用与 pinball 损失对应的阈值算子。",[10214,21645],{},[1091,21647,21649],{"id":21648},"四梯度下降家族","四、梯度下降家族",[1096,21651,21653],{"id":21652},"梯度下降算法gd","梯度下降算法(GD)",[1101,21655,21656,21657,21046,21790,22121,22122,22327,22328,22398],{},"给定一组样本 ",[1109,21658,21660,21689],{"className":21659},[1112],[1109,21661,21663],{"className":21662},[1116],[1118,21664,21665],{"xmlns":1120},[1122,21666,21667,21687],{},[1125,21668,21669,21671,21677,21679,21685],{},[1204,21670,1457],{"stretchy":1456},[1264,21672,21673,21675],{},[1128,21674,1268],{},[1128,21676,9875],{},[1204,21678,1910],{"separator":1139},[1264,21680,21681,21683],{},[1128,21682,14113],{},[1128,21684,9875],{},[1204,21686,1462],{"stretchy":1456},[1132,21688,14120],{"encoding":1134},[1109,21690,21692],{"className":21691,"ariaHidden":1139},[1138],[1109,21693,21695,21698,21701,21741,21744,21747,21787],{"className":21694},[1143],[1109,21696],{"className":21697,"style":1498},[1147],[1109,21699,1457],{"className":21700},[1508],[1109,21702,21704,21707],{"className":21703},[1152],[1109,21705,1268],{"className":21706},[1152,1153],[1109,21708,21710],{"className":21709},[1293],[1109,21711,21713,21733],{"className":21712},[1297,1298],[1109,21714,21716,21730],{"className":21715},[1302],[1109,21717,21719],{"className":21718,"style":14151},[1306],[1109,21720,21721,21724],{"style":1310},[1109,21722],{"className":21723,"style":1315},[1314],[1109,21725,21727],{"className":21726},[1319,1320,1321,1322],[1109,21728,9875],{"className":21729},[1152,1153,1322],[1109,21731,1330],{"className":21732},[1329],[1109,21734,21736],{"className":21735},[1302],[1109,21737,21739],{"className":21738,"style":1337},[1306],[1109,21740],{},[1109,21742,1910],{"className":21743},[1975],[1109,21745],{"className":21746,"style":1540},[1231],[1109,21748,21750,21753],{"className":21749},[1152],[1109,21751,14113],{"className":21752,"style":1186},[1152,1153],[1109,21754,21756],{"className":21755},[1293],[1109,21757,21759,21779],{"className":21758},[1297,1298],[1109,21760,21762,21776],{"className":21761},[1302],[1109,21763,21765],{"className":21764,"style":14151},[1306],[1109,21766,21767,21770],{"style":2146},[1109,21768],{"className":21769,"style":1315},[1314],[1109,21771,21773],{"className":21772},[1319,1320,1321,1322],[1109,21774,9875],{"className":21775},[1152,1153,1322],[1109,21777,1330],{"className":21778},[1329],[1109,21780,21782],{"className":21781},[1302],[1109,21783,21785],{"className":21784,"style":1337},[1306],[1109,21786],{},[1109,21788,1462],{"className":21789},[1515],[1109,21791,21793,21851],{"className":21792},[1112],[1109,21794,21796],{"className":21795},[1116],[1118,21797,21798],{"xmlns":1120},[1122,21799,21800,21848],{},[1125,21801,21802,21812,21818,21824,21826,21828,21834,21836,21842,21844,21846],{},[1264,21803,21804,21810],{},[1125,21805,21806,21808],{},[1128,21807,10241],{},[1204,21809,2412],{},[1128,21811,12502],{},[4490,21813,21814,21816],{},[1208,21815,1685],{},[1128,21817,7300],{},[1264,21819,21820,21822],{},[1204,21821,14599],{},[1128,21823,9875],{},[1128,21825,14586],{"mathvariant":1450},[1204,21827,1457],{"stretchy":1456},[1264,21829,21830,21832],{},[1128,21831,1268],{},[1128,21833,9875],{},[1204,21835,1910],{"separator":1139},[1264,21837,21838,21840],{},[1128,21839,14113],{},[1128,21841,9875],{},[1204,21843,12261],{"separator":1139},[1128,21845,12502],{},[1204,21847,1462],{"stretchy":1456},[1132,21849,21850],{"encoding":1134},"\\min_\\beta \\frac{1}{n}\\sum_i \\ell(x_i, y_i; \\beta)",[1109,21852,21854],{"className":21853,"ariaHidden":1139},[1138],[1109,21855,21857,21860,21900,21903,21971,21974,22014,22017,22020,22023,22063,22066,22069,22109,22112,22115,22118],{"className":21856},[1143],[1109,21858],{"className":21859,"style":11312},[1147],[1109,21861,21863,21866],{"className":21862},[2878],[1109,21864,10241],{"className":21865},[2878],[1109,21867,21869],{"className":21868},[1293],[1109,21870,21872,21892],{"className":21871},[1297,1298],[1109,21873,21875,21889],{"className":21874},[1302],[1109,21876,21878],{"className":21877,"style":8735},[1306],[1109,21879,21880,21883],{"style":4550},[1109,21881],{"className":21882,"style":1315},[1314],[1109,21884,21886],{"className":21885},[1319,1320,1321,1322],[1109,21887,12502],{"className":21888,"style":12530},[1152,1153,1322],[1109,21890,1330],{"className":21891},[1329],[1109,21893,21895],{"className":21894},[1302],[1109,21896,21898],{"className":21897,"style":16324},[1306],[1109,21899],{},[1109,21901],{"className":21902,"style":1540},[1231],[1109,21904,21906,21909,21968],{"className":21905},[1152],[1109,21907],{"className":21908},[1508,4589],[1109,21910,21912],{"className":21911},[4490],[1109,21913,21915,21960],{"className":21914},[1297,1298],[1109,21916,21918,21957],{"className":21917},[1302],[1109,21919,21921,21935,21943],{"className":21920,"style":11331},[1306],[1109,21922,21923,21926],{"style":4605},[1109,21924],{"className":21925,"style":4609},[1314],[1109,21927,21929],{"className":21928},[1319,1320,1321,1322],[1109,21930,21932],{"className":21931},[1152,1322],[1109,21933,7300],{"className":21934},[1152,1153,1322],[1109,21936,21937,21940],{"style":4630},[1109,21938],{"className":21939,"style":4609},[1314],[1109,21941],{"className":21942,"style":4638},[4637],[1109,21944,21945,21948],{"style":11356},[1109,21946],{"className":21947,"style":4609},[1314],[1109,21949,21951],{"className":21950},[1319,1320,1321,1322],[1109,21952,21954],{"className":21953},[1152,1322],[1109,21955,1685],{"className":21956},[1152,1322],[1109,21958,1330],{"className":21959},[1329],[1109,21961,21963],{"className":21962},[1302],[1109,21964,21966],{"className":21965,"style":8058},[1306],[1109,21967],{},[1109,21969],{"className":21970},[1515,4589],[1109,21972],{"className":21973,"style":1540},[1231],[1109,21975,21977,21980],{"className":21976},[2878],[1109,21978,14599],{"className":21979,"style":14705},[2878,14703,14704],[1109,21981,21983],{"className":21982},[1293],[1109,21984,21986,22006],{"className":21985},[1297,1298],[1109,21987,21989,22003],{"className":21988},[1302],[1109,21990,21992],{"className":21991,"style":14718},[1306],[1109,21993,21994,21997],{"style":14721},[1109,21995],{"className":21996,"style":1315},[1314],[1109,21998,22000],{"className":21999},[1319,1320,1321,1322],[1109,22001,9875],{"className":22002},[1152,1153,1322],[1109,22004,1330],{"className":22005},[1329],[1109,22007,22009],{"className":22008},[1302],[1109,22010,22012],{"className":22011,"style":14740},[1306],[1109,22013],{},[1109,22015],{"className":22016,"style":1540},[1231],[1109,22018,14586],{"className":22019},[1152],[1109,22021,1457],{"className":22022},[1508],[1109,22024,22026,22029],{"className":22025},[1152],[1109,22027,1268],{"className":22028},[1152,1153],[1109,22030,22032],{"className":22031},[1293],[1109,22033,22035,22055],{"className":22034},[1297,1298],[1109,22036,22038,22052],{"className":22037},[1302],[1109,22039,22041],{"className":22040,"style":14151},[1306],[1109,22042,22043,22046],{"style":1310},[1109,22044],{"className":22045,"style":1315},[1314],[1109,22047,22049],{"className":22048},[1319,1320,1321,1322],[1109,22050,9875],{"className":22051},[1152,1153,1322],[1109,22053,1330],{"className":22054},[1329],[1109,22056,22058],{"className":22057},[1302],[1109,22059,22061],{"className":22060,"style":1337},[1306],[1109,22062],{},[1109,22064,1910],{"className":22065},[1975],[1109,22067],{"className":22068,"style":1540},[1231],[1109,22070,22072,22075],{"className":22071},[1152],[1109,22073,14113],{"className":22074,"style":1186},[1152,1153],[1109,22076,22078],{"className":22077},[1293],[1109,22079,22081,22101],{"className":22080},[1297,1298],[1109,22082,22084,22098],{"className":22083},[1302],[1109,22085,22087],{"className":22086,"style":14151},[1306],[1109,22088,22089,22092],{"style":2146},[1109,22090],{"className":22091,"style":1315},[1314],[1109,22093,22095],{"className":22094},[1319,1320,1321,1322],[1109,22096,9875],{"className":22097},[1152,1153,1322],[1109,22099,1330],{"className":22100},[1329],[1109,22102,22104],{"className":22103},[1302],[1109,22105,22107],{"className":22106,"style":1337},[1306],[1109,22108],{},[1109,22110,12261],{"className":22111},[1975],[1109,22113],{"className":22114,"style":1540},[1231],[1109,22116,12502],{"className":22117,"style":12530},[1152,1153],[1109,22119,1462],{"className":22120},[1515],"。若只考虑线性模型 ",[1109,22123,22125,22161],{"className":22124},[1112],[1109,22126,22128],{"className":22127},[1116],[1118,22129,22130],{"xmlns":1120},[1122,22131,22132,22158],{},[1125,22133,22134,22146,22148,22156],{},[1264,22135,22136,22144],{},[22137,22138,22139,22141],"mover",{"accent":1139},[1128,22140,14113],{},[1204,22142,22143],{},"^",[1128,22145,9875],{},[1204,22147,1582],{},[14264,22149,22150,22152,22154],{},[1128,22151,1268],{},[1128,22153,9875],{},[1128,22155,9471],{"mathvariant":1450},[1128,22157,12502],{},[1132,22159,22160],{"encoding":1134},"\\hat y_i = x_i^{\\top}\\beta",[1109,22162,22164,22264],{"className":22163,"ariaHidden":1139},[1138],[1109,22165,22167,22170,22255,22258,22261],{"className":22166},[1143],[1109,22168],{"className":22169,"style":1148},[1147],[1109,22171,22173,22221],{"className":22172},[1152],[1109,22174,22177],{"className":22175},[1152,22176],"accent",[1109,22178,22180,22212],{"className":22179},[1297,1298],[1109,22181,22183,22209],{"className":22182},[1302],[1109,22184,22187,22196],{"className":22185,"style":22186},[1306],"height:0.6944em;",[1109,22188,22190,22193],{"style":22189},"top:-3em;",[1109,22191],{"className":22192,"style":4609},[1314],[1109,22194,14113],{"className":22195,"style":1186},[1152,1153],[1109,22197,22198,22201],{"style":22189},[1109,22199],{"className":22200,"style":4609},[1314],[1109,22202,22206],{"className":22203,"style":22205},[22204],"accent-body","left:-0.1944em;",[1109,22207,22143],{"className":22208},[1152],[1109,22210,1330],{"className":22211},[1329],[1109,22213,22215],{"className":22214},[1302],[1109,22216,22219],{"className":22217,"style":22218},[1306],"height:0.1944em;",[1109,22220],{},[1109,22222,22224],{"className":22223},[1293],[1109,22225,22227,22247],{"className":22226},[1297,1298],[1109,22228,22230,22244],{"className":22229},[1302],[1109,22231,22233],{"className":22232,"style":14151},[1306],[1109,22234,22235,22238],{"style":2146},[1109,22236],{"className":22237,"style":1315},[1314],[1109,22239,22241],{"className":22240},[1319,1320,1321,1322],[1109,22242,9875],{"className":22243},[1152,1153,1322],[1109,22245,1330],{"className":22246},[1329],[1109,22248,22250],{"className":22249},[1302],[1109,22251,22253],{"className":22252,"style":1337},[1306],[1109,22254],{},[1109,22256],{"className":22257,"style":1232},[1231],[1109,22259,1582],{"className":22260},[1236],[1109,22262],{"className":22263,"style":1232},[1231],[1109,22265,22267,22270,22324],{"className":22266},[1143],[1109,22268],{"className":22269,"style":21300},[1147],[1109,22271,22273,22276],{"className":22272},[1152],[1109,22274,1268],{"className":22275},[1152,1153],[1109,22277,22279],{"className":22278},[1293],[1109,22280,22282,22316],{"className":22281},[1297,1298],[1109,22283,22285,22313],{"className":22284},[1302],[1109,22286,22288,22299],{"className":22287,"style":11218},[1306],[1109,22289,22290,22293],{"style":21321},[1109,22291],{"className":22292,"style":1315},[1314],[1109,22294,22296],{"className":22295},[1319,1320,1321,1322],[1109,22297,9875],{"className":22298},[1152,1153,1322],[1109,22300,22301,22304],{"style":7044},[1109,22302],{"className":22303,"style":1315},[1314],[1109,22305,22307],{"className":22306},[1319,1320,1321,1322],[1109,22308,22310],{"className":22309},[1152,1322],[1109,22311,9471],{"className":22312},[1152,1322],[1109,22314,1330],{"className":22315},[1329],[1109,22317,22319],{"className":22318},[1302],[1109,22320,22322],{"className":22321,"style":21354},[1306],[1109,22323],{},[1109,22325,12502],{"className":22326,"style":12530},[1152,1153],",目标就是均方误差。对当前估计 ",[1109,22329,22331,22349],{"className":22330},[1112],[1109,22332,22334],{"className":22333},[1116],[1118,22335,22336],{"xmlns":1120},[1122,22337,22338,22346],{},[1125,22339,22340],{},[1264,22341,22342,22344],{},[1128,22343,12502],{},[1128,22345,8620],{},[1132,22347,22348],{"encoding":1134},"\\beta_k",[1109,22350,22352],{"className":22351,"ariaHidden":1139},[1138],[1109,22353,22355,22358],{"className":22354},[1143],[1109,22356],{"className":22357,"style":1148},[1147],[1109,22359,22361,22364],{"className":22360},[1152],[1109,22362,12502],{"className":22363,"style":12530},[1152,1153],[1109,22365,22367],{"className":22366},[1293],[1109,22368,22370,22390],{"className":22369},[1297,1298],[1109,22371,22373,22387],{"className":22372},[1302],[1109,22374,22376],{"className":22375,"style":8735},[1306],[1109,22377,22378,22381],{"style":15969},[1109,22379],{"className":22380,"style":1315},[1314],[1109,22382,22384],{"className":22383},[1319,1320,1321,1322],[1109,22385,8620],{"className":22386,"style":8750},[1152,1153,1322],[1109,22388,1330],{"className":22389},[1329],[1109,22391,22393],{"className":22392},[1302],[1109,22394,22396],{"className":22395,"style":1337},[1306],[1109,22397],{},",梯度下降的迭代步骤为",[1101,22400,22401],{},[1109,22402,22404,22526],{"className":22403},[1112],[1109,22405,22407],{"className":22406},[1116],[1118,22408,22409],{"xmlns":1120},[1122,22410,22411,22523],{},[1125,22412,22413,22425,22427,22433,22435,22437,22439,22441,22443,22445,22451,22453,22455,22457,22459,22461,22463,22469,22471,22473,22479,22493,22495,22497,22499,22505,22507,22513,22515,22521],{},[1264,22414,22415,22417],{},[1128,22416,12502],{},[1125,22418,22419,22421,22423],{},[1128,22420,8620],{},[1204,22422,2342],{},[1208,22424,1685],{},[1204,22426,1582],{},[1264,22428,22429,22431],{},[1128,22430,12502],{},[1128,22432,8620],{},[1204,22434,5991],{},[1128,22436,12579],{},[1471,22438,1473],{},[1128,22440,5611],{"mathvariant":1450},[1128,22442,1130],{},[1204,22444,1457],{"stretchy":1456},[1264,22446,22447,22449],{},[1128,22448,12502],{},[1128,22450,8620],{},[1204,22452,1462],{"stretchy":1456},[1204,22454,1910],{"separator":1139},[1231,22456],{"width":14583},[1128,22458,5611],{"mathvariant":1450},[1128,22460,1130],{},[1204,22462,1457],{"stretchy":1456},[1264,22464,22465,22467],{},[1128,22466,12502],{},[1128,22468,8620],{},[1204,22470,1462],{"stretchy":1456},[1204,22472,1582],{},[4490,22474,22475,22477],{},[1208,22476,1685],{},[1128,22478,7300],{},[14264,22480,22481,22483,22491],{},[1204,22482,14599],{},[1125,22484,22485,22487,22489],{},[1128,22486,9875],{},[1204,22488,1582],{},[1208,22490,1685],{},[1128,22492,7300],{},[1128,22494,5611],{"mathvariant":1450},[1128,22496,14586],{"mathvariant":1450},[1204,22498,1457],{"stretchy":1456},[1264,22500,22501,22503],{},[1128,22502,1268],{},[1128,22504,9875],{},[1204,22506,1910],{"separator":1139},[1264,22508,22509,22511],{},[1128,22510,14113],{},[1128,22512,9875],{},[1204,22514,12261],{"separator":1139},[1264,22516,22517,22519],{},[1128,22518,12502],{},[1128,22520,8620],{},[1204,22522,1462],{"stretchy":1456},[1132,22524,22525],{"encoding":1134},"\\beta_{k+1} = \\beta_k - t\\,\\nabla f(\\beta_k), \\qquad \\nabla f(\\beta_k) = \\frac{1}{n}\\sum_{i=1}^{n}\\nabla\\ell(x_i, y_i; \\beta_k)",[1109,22527,22529,22593,22648,22782],{"className":22528,"ariaHidden":1139},[1138],[1109,22530,22532,22535,22584,22587,22590],{"className":22531},[1143],[1109,22533],{"className":22534,"style":15948},[1147],[1109,22536,22538,22541],{"className":22537},[1152],[1109,22539,12502],{"className":22540,"style":12530},[1152,1153],[1109,22542,22544],{"className":22543},[1293],[1109,22545,22547,22576],{"className":22546},[1297,1298],[1109,22548,22550,22573],{"className":22549},[1302],[1109,22551,22553],{"className":22552,"style":8735},[1306],[1109,22554,22555,22558],{"style":15969},[1109,22556],{"className":22557,"style":1315},[1314],[1109,22559,22561],{"className":22560},[1319,1320,1321,1322],[1109,22562,22564,22567,22570],{"className":22563},[1152,1322],[1109,22565,8620],{"className":22566,"style":8750},[1152,1153,1322],[1109,22568,2342],{"className":22569},[2501,1322],[1109,22571,1685],{"className":22572},[1152,1322],[1109,22574,1330],{"className":22575},[1329],[1109,22577,22579],{"className":22578},[1302],[1109,22580,22582],{"className":22581,"style":10663},[1306],[1109,22583],{},[1109,22585],{"className":22586,"style":1232},[1231],[1109,22588,1582],{"className":22589},[1236],[1109,22591],{"className":22592,"style":1232},[1231],[1109,22594,22596,22599,22639,22642,22645],{"className":22595},[1143],[1109,22597],{"className":22598,"style":1148},[1147],[1109,22600,22602,22605],{"className":22601},[1152],[1109,22603,12502],{"className":22604,"style":12530},[1152,1153],[1109,22606,22608],{"className":22607},[1293],[1109,22609,22611,22631],{"className":22610},[1297,1298],[1109,22612,22614,22628],{"className":22613},[1302],[1109,22615,22617],{"className":22616,"style":8735},[1306],[1109,22618,22619,22622],{"style":15969},[1109,22620],{"className":22621,"style":1315},[1314],[1109,22623,22625],{"className":22624},[1319,1320,1321,1322],[1109,22626,8620],{"className":22627,"style":8750},[1152,1153,1322],[1109,22629,1330],{"className":22630},[1329],[1109,22632,22634],{"className":22633},[1302],[1109,22635,22637],{"className":22636,"style":1337},[1306],[1109,22638],{},[1109,22640],{"className":22641,"style":2497},[1231],[1109,22643,5991],{"className":22644},[2501],[1109,22646],{"className":22647,"style":2497},[1231],[1109,22649,22651,22654,22657,22660,22663,22666,22669,22709,22712,22715,22718,22721,22724,22727,22730,22770,22773,22776,22779],{"className":22650},[1143],[1109,22652],{"className":22653,"style":1498},[1147],[1109,22655,12579],{"className":22656},[1152,1153],[1109,22658],{"className":22659,"style":1540},[1231],[1109,22661,5611],{"className":22662},[1152],[1109,22664,1130],{"className":22665,"style":1154},[1152,1153],[1109,22667,1457],{"className":22668},[1508],[1109,22670,22672,22675],{"className":22671},[1152],[1109,22673,12502],{"className":22674,"style":12530},[1152,1153],[1109,22676,22678],{"className":22677},[1293],[1109,22679,22681,22701],{"className":22680},[1297,1298],[1109,22682,22684,22698],{"className":22683},[1302],[1109,22685,22687],{"className":22686,"style":8735},[1306],[1109,22688,22689,22692],{"style":15969},[1109,22690],{"className":22691,"style":1315},[1314],[1109,22693,22695],{"className":22694},[1319,1320,1321,1322],[1109,22696,8620],{"className":22697,"style":8750},[1152,1153,1322],[1109,22699,1330],{"className":22700},[1329],[1109,22702,22704],{"className":22703},[1302],[1109,22705,22707],{"className":22706,"style":1337},[1306],[1109,22708],{},[1109,22710,1462],{"className":22711},[1515],[1109,22713,1910],{"className":22714},[1975],[1109,22716],{"className":22717,"style":14984},[1231],[1109,22719],{"className":22720,"style":1540},[1231],[1109,22722,5611],{"className":22723},[1152],[1109,22725,1130],{"className":22726,"style":1154},[1152,1153],[1109,22728,1457],{"className":22729},[1508],[1109,22731,22733,22736],{"className":22732},[1152],[1109,22734,12502],{"className":22735,"style":12530},[1152,1153],[1109,22737,22739],{"className":22738},[1293],[1109,22740,22742,22762],{"className":22741},[1297,1298],[1109,22743,22745,22759],{"className":22744},[1302],[1109,22746,22748],{"className":22747,"style":8735},[1306],[1109,22749,22750,22753],{"style":15969},[1109,22751],{"className":22752,"style":1315},[1314],[1109,22754,22756],{"className":22755},[1319,1320,1321,1322],[1109,22757,8620],{"className":22758,"style":8750},[1152,1153,1322],[1109,22760,1330],{"className":22761},[1329],[1109,22763,22765],{"className":22764},[1302],[1109,22766,22768],{"className":22767,"style":1337},[1306],[1109,22769],{},[1109,22771,1462],{"className":22772},[1515],[1109,22774],{"className":22775,"style":1232},[1231],[1109,22777,1582],{"className":22778},[1236],[1109,22780],{"className":22781,"style":1232},[1231],[1109,22783,22785,22788,22856,22859,22924,22927,22930,22933,22936,22976,22979,22982,23022,23025,23028,23068],{"className":22784},[1143],[1109,22786],{"className":22787,"style":11312},[1147],[1109,22789,22791,22794,22853],{"className":22790},[1152],[1109,22792],{"className":22793},[1508,4589],[1109,22795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className":23078},[1116],[1118,23080,23081],{"xmlns":1120},[1122,23082,23083,23087],{},[1125,23084,23085],{},[1128,23086,12579],{},[1132,23088,12579],{"encoding":1134},[1109,23090,23092],{"className":23091,"ariaHidden":1139},[1138],[1109,23093,23095,23098],{"className":23094},[1143],[1109,23096],{"className":23097,"style":12596},[1147],[1109,23099,12579],{"className":23100},[1152,1153]," 为步长。",[1101,23103,23104,23136,23137,23325,23326,23381],{},[1104,23105,23106,23107,23135],{},"步长 ",[1109,23108,23110,23123],{"className":23109},[1112],[1109,23111,23113],{"className":23112},[1116],[1118,23114,23115],{"xmlns":1120},[1122,23116,23117,23121],{},[1125,23118,23119],{},[1128,23120,12579],{},[1132,23122,12579],{"encoding":1134},[1109,23124,23126],{"className":23125,"ariaHidden":1139},[1138],[1109,23127,23129,23132],{"className":23128},[1143],[1109,23130],{"className":23131,"style":12596},[1147],[1109,23133,12579],{"className":23134},[1152,1153]," 的选择",":通常用 Backtracking line search 筛选出足够长、又能保证目标函数递减的步长。给定当前估计,若充分下降条件 ",[1109,23138,23140,23196],{"className":23139},[1112],[1109,23141,23143],{"className":23142},[1116],[1118,23144,23145],{"xmlns":1120},[1122,23146,23147,23193],{},[1125,23148,23149,23151,23153,23155,23157,23159,23161,23163,23165,23167,23169,23171,23173,23175,23177,23179,23181,23183,23185,23187],{},[1128,23150,1130],{},[1204,23152,1457],{"stretchy":1456},[1128,23154,12502],{},[1204,23156,5991],{},[1128,23158,12579],{},[1128,23160,5611],{"mathvariant":1450},[1128,23162,1130],{},[1204,23164,1462],{"stretchy":1456},[1204,23166,1467],{},[1128,23168,1130],{},[1204,23170,1457],{"stretchy":1456},[1128,23172,12502],{},[1204,23174,1462],{"stretchy":1456},[1204,23176,5991],{},[1128,23178,12425],{},[1128,23180,12579],{},[1128,23182,8631],{"mathvariant":1450},[1128,23184,5611],{"mathvariant":1450},[1128,23186,1130],{},[6964,23188,23189,23191],{},[1128,23190,8631],{"mathvariant":1450},[1208,23192,1917],{},[1132,23194,23195],{"encoding":1134},"f(\\beta - t\\nabla f) \\le f(\\beta) - \\alpha t\\|\\nabla f\\|^2",[1109,23197,23199,23223,23250,23277],{"className":23198,"ariaHidden":1139},[1138],[1109,23200,23202,23205,23208,23211,23214,23217,23220],{"className":23201},[1143],[1109,23203],{"className":23204,"style":1498},[1147],[1109,23206,1130],{"className":23207,"style":1154},[1152,1153],[1109,23209,1457],{"className":23210},[1508],[1109,23212,12502],{"className":23213,"style":12530},[1152,1153],[1109,23215],{"className":23216,"style":2497},[1231],[1109,23218,5991],{"className":23219},[2501],[1109,23221],{"className":23222,"style":2497},[1231],[1109,23224,23226,23229,23232,23235,23238,23241,23244,23247],{"className":23225},[1143],[1109,23227],{"className":23228,"style":1498},[1147],[1109,23230,12579],{"className":23231},[1152,1153],[1109,23233,5611],{"className":23234},[1152],[1109,23236,1130],{"className":23237,"style":1154},[1152,1153],[1109,23239,1462],{"className":23240},[1515],[1109,23242],{"className":23243,"style":1232},[1231],[1109,23245,1467],{"className":23246},[1236],[1109,23248],{"className":23249,"style":1232},[1231],[1109,23251,23253,23256,23259,23262,23265,23268,23271,23274],{"className":23252},[1143],[1109,23254],{"className":23255,"style":1498},[1147],[1109,23257,1130],{"className":23258,"style":1154},[1152,1153],[1109,23260,1457],{"className":23261},[1508],[1109,23263,12502],{"className":23264,"style":12530},[1152,1153],[1109,23266,1462],{"className":23267},[1515],[1109,23269],{"className":23270,"style":2497},[1231],[1109,23272,5991],{"className":23273},[2501],[1109,23275],{"className":23276,"style":2497},[1231],[1109,23278,23280,23283,23286,23289,23293,23296],{"className":23279},[1143],[1109,23281],{"className":23282,"style":8085},[1147],[1109,23284,12425],{"className":23285,"style":12454},[1152,1153],[1109,23287,12579],{"className":23288},[1152,1153],[1109,23290,23292],{"className":23291},[1152],"∥∇",[1109,23294,1130],{"className":23295,"style":1154},[1152,1153],[1109,23297,23299,23302],{"className":23298},[1152],[1109,23300,8631],{"className":23301},[1152],[1109,23303,23305],{"className":23304},[1293],[1109,23306,23308],{"className":23307},[1297],[1109,23309,23311],{"className":23310},[1302],[1109,23312,23314],{"className":23313,"style":8144},[1306],[1109,23315,23316,23319],{"style":7044},[1109,23317],{"className":23318,"style":1315},[1314],[1109,23320,23322],{"className":23321},[1319,1320,1321,1322],[1109,23323,1917],{"className":23324},[1152,1322]," 不成立,就令 ",[1109,23327,23329,23348],{"className":23328},[1112],[1109,23330,23332],{"className":23331},[1116],[1118,23333,23334],{"xmlns":1120},[1122,23335,23336,23346],{},[1125,23337,23338,23340,23342,23344],{},[1128,23339,12579],{},[1204,23341,12905],{},[1128,23343,12502],{},[1128,23345,12579],{},[1132,23347,12998],{"encoding":1134},[1109,23349,23351,23369],{"className":23350,"ariaHidden":1139},[1138],[1109,23352,23354,23357,23360,23363,23366],{"className":23353},[1143],[1109,23355],{"className":23356,"style":12596},[1147],[1109,23358,12579],{"className":23359},[1152,1153],[1109,23361],{"className":23362,"style":1232},[1231],[1109,23364,12905],{"className":23365},[1236],[1109,23367],{"className":23368,"style":1232},[1231],[1109,23370,23372,23375,23378],{"className":23371},[1143],[1109,23373],{"className":23374,"style":1148},[1147],[1109,23376,12502],{"className":23377,"style":12530},[1152,1153],[1109,23379,12579],{"className":23380},[1152,1153]," 缩小步长再试。",[1096,23383,23385],{"id":23384},"随机梯度下降法sgd","随机梯度下降法(SGD)",[1101,23387,23388,23389,23458,23459,4729,23462,23595],{},"给定当前估计值 ",[1109,23390,23392,23409],{"className":23391},[1112],[1109,23393,23395],{"className":23394},[1116],[1118,23396,23397],{"xmlns":1120},[1122,23398,23399,23407],{},[1125,23400,23401],{},[1264,23402,23403,23405],{},[1128,23404,12502],{},[1128,23406,8620],{},[1132,23408,22348],{"encoding":1134},[1109,23410,23412],{"className":23411,"ariaHidden":1139},[1138],[1109,23413,23415,23418],{"className":23414},[1143],[1109,23416],{"className":23417,"style":1148},[1147],[1109,23419,23421,23424],{"className":23420},[1152],[1109,23422,12502],{"className":23423,"style":12530},[1152,1153],[1109,23425,23427],{"className":23426},[1293],[1109,23428,23430,23450],{"className":23429},[1297,1298],[1109,23431,23433,23447],{"className":23432},[1302],[1109,23434,23436],{"className":23435,"style":8735},[1306],[1109,23437,23438,23441],{"style":15969},[1109,23439],{"className":23440,"style":1315},[1314],[1109,23442,23444],{"className":23443},[1319,1320,1321,1322],[1109,23445,8620],{"className":23446,"style":8750},[1152,1153,1322],[1109,23448,1330],{"className":23449},[1329],[1109,23451,23453],{"className":23452},[1302],[1109,23454,23456],{"className":23455,"style":1337},[1306],[1109,23457],{},",随机梯度下降每步",[1104,23460,23461],{},"随机抽取一个样本",[1109,23463,23465,23494],{"className":23464},[1112],[1109,23466,23468],{"className":23467},[1116],[1118,23469,23470],{"xmlns":1120},[1122,23471,23472,23492],{},[1125,23473,23474,23476,23482,23484,23490],{},[1204,23475,1457],{"stretchy":1456},[1264,23477,23478,23480],{},[1128,23479,1268],{},[1128,23481,9875],{},[1204,23483,1910],{"separator":1139},[1264,23485,23486,23488],{},[1128,23487,14113],{},[1128,23489,9875],{},[1204,23491,1462],{"stretchy":1456},[1132,23493,14120],{"encoding":1134},[1109,23495,23497],{"className":23496,"ariaHidden":1139},[1138],[1109,23498,23500,23503,23506,23546,23549,23552,23592],{"className":23499},[1143],[1109,23501],{"className":23502,"style":1498},[1147],[1109,23504,1457],{"className":23505},[1508],[1109,23507,23509,23512],{"className":23508},[1152],[1109,23510,1268],{"className":23511},[1152,1153],[1109,23513,23515],{"className":23514},[1293],[1109,23516,23518,23538],{"className":23517},[1297,1298],[1109,23519,23521,23535],{"className":23520},[1302],[1109,23522,23524],{"className":23523,"style":14151},[1306],[1109,23525,23526,23529],{"style":1310},[1109,23527],{"className":23528,"style":1315},[1314],[1109,23530,23532],{"className":23531},[1319,1320,1321,1322],[1109,23533,9875],{"className":23534},[1152,1153,1322],[1109,23536,1330],{"className":23537},[1329],[1109,23539,23541],{"className":23540},[1302],[1109,23542,23544],{"className":23543,"style":1337},[1306],[1109,23545],{},[1109,23547,1910],{"className":23548},[1975],[1109,23550],{"className":23551,"style":1540},[1231],[1109,23553,23555,23558],{"className":23554},[1152],[1109,23556,14113],{"className":23557,"style":1186},[1152,1153],[1109,23559,23561],{"className":23560},[1293],[1109,23562,23564,23584],{"className":23563},[1297,1298],[1109,23565,23567,23581],{"className":23566},[1302],[1109,23568,23570],{"className":23569,"style":14151},[1306],[1109,23571,23572,23575],{"style":2146},[1109,23573],{"className":23574,"style":1315},[1314],[1109,23576,23578],{"className":23577},[1319,1320,1321,1322],[1109,23579,9875],{"className":23580},[1152,1153,1322],[1109,23582,1330],{"className":23583},[1329],[1109,23585,23587],{"className":23586},[1302],[1109,23588,23590],{"className":23589,"style":1337},[1306],[1109,23591],{},[1109,23593,1462],{"className":23594},[1515],",只用它算梯度来更新:",[1101,23597,23598],{},[1109,23599,23601,23669],{"className":23600},[1112],[1109,23602,23604],{"className":23603},[1116],[1118,23605,23606],{"xmlns":1120},[1122,23607,23608,23666],{},[1125,23609,23610,23622,23624,23630,23632,23634,23636,23638,23640,23642,23648,23650,23656,23658,23664],{},[1264,23611,23612,23614],{},[1128,23613,12502],{},[1125,23615,23616,23618,23620],{},[1128,23617,8620],{},[1204,23619,2342],{},[1208,23621,1685],{},[1204,23623,1582],{},[1264,23625,23626,23628],{},[1128,23627,12502],{},[1128,23629,8620],{},[1204,23631,5991],{},[1128,23633,12579],{},[1471,23635,1473],{},[1128,23637,5611],{"mathvariant":1450},[1128,23639,14586],{"mathvariant":1450},[1204,23641,1457],{"stretchy":1456},[1264,23643,23644,23646],{},[1128,23645,1268],{},[1128,23647,9875],{},[1204,23649,1910],{"separator":1139},[1264,23651,23652,23654],{},[1128,23653,14113],{},[1128,23655,9875],{},[1204,23657,12261],{"separator":1139},[1264,23659,23660,23662],{},[1128,23661,12502],{},[1128,23663,8620],{},[1204,23665,1462],{"stretchy":1456},[1132,23667,23668],{"encoding":1134},"\\beta_{k+1} = \\beta_k - t\\,\\nabla\\ell(x_i, y_i; \\beta_k)",[1109,23670,23672,23736,23791],{"className":23671,"ariaHidden":1139},[1138],[1109,23673,23675,23678,23727,23730,23733],{"className":23674},[1143],[1109,23676],{"className":23677,"style":15948},[1147],[1109,23679,23681,23684],{"className":23680},[1152],[1109,23682,12502],{"className":23683,"style":12530},[1152,1153],[1109,23685,23687],{"className":23686},[1293],[1109,23688,23690,23719],{"className":23689},[1297,1298],[1109,23691,23693,23716],{"className":23692},[1302],[1109,23694,23696],{"className":23695,"style":8735},[1306],[1109,23697,23698,23701],{"style":15969},[1109,23699],{"className":23700,"style":1315},[1314],[1109,23702,23704],{"className":23703},[1319,1320,1321,1322],[1109,23705,23707,23710,23713],{"className":23706},[1152,1322],[1109,23708,8620],{"className":23709,"style":8750},[1152,1153,1322],[1109,23711,2342],{"className":23712},[2501,1322],[1109,23714,1685],{"className":23715},[1152,1322],[1109,23717,1330],{"className":23718},[1329],[1109,23720,23722],{"className":23721},[1302],[1109,23723,23725],{"className":23724,"style":10663},[1306],[1109,23726],{},[1109,23728],{"className":23729,"style":1232},[1231],[1109,23731,1582],{"className":23732},[1236],[1109,23734],{"className":23735,"style":1232},[1231],[1109,23737,23739,23742,23782,23785,23788],{"className":23738},[1143],[1109,23740],{"className":23741,"style":1148},[1147],[1109,23743,23745,23748],{"className":23744},[1152],[1109,23746,12502],{"className":23747,"style":12530},[1152,1153],[1109,23749,23751],{"className":23750},[1293],[1109,23752,23754,23774],{"className":23753},[1297,1298],[1109,23755,23757,23771],{"className":23756},[1302],[1109,23758,23760],{"className":23759,"style":8735},[1306],[1109,23761,23762,23765],{"style":15969},[1109,23763],{"className":23764,"style":1315},[1314],[1109,23766,23768],{"className":23767},[1319,1320,1321,1322],[1109,23769,8620],{"className":23770,"style":8750},[1152,1153,1322],[1109,23772,1330],{"className":23773},[1329],[1109,23775,23777],{"className":23776},[1302],[1109,23778,23780],{"className":23779,"style":1337},[1306],[1109,23781],{},[1109,23783],{"className":23784,"style":2497},[1231],[1109,23786,5991],{"className":23787},[2501],[1109,23789],{"className":23790,"style":2497},[1231],[1109,23792,23794,23797,23800,23803,23806,23809,23812,23852,23855,23858,23898,23901,23904,23944],{"className":23793},[1143],[1109,23795],{"className":23796,"style":1498},[1147],[1109,23798,12579],{"className":23799},[1152,1153],[1109,23801],{"className":23802,"style":1540},[1231],[1109,23804,5611],{"className":23805},[1152],[1109,23807,14586],{"className":23808},[1152],[1109,23810,1457],{"className":23811},[1508],[1109,23813,23815,23818],{"className":23814},[1152],[1109,23816,1268],{"className":23817},[1152,1153],[1109,23819,23821],{"className":23820},[1293],[1109,23822,23824,23844],{"className":23823},[1297,1298],[1109,23825,23827,23841],{"className":23826},[1302],[1109,23828,23830],{"className":23829,"style":14151},[1306],[1109,23831,23832,23835],{"style":1310},[1109,23833],{"className":23834,"style":1315},[1314],[1109,23836,23838],{"className":23837},[1319,1320,1321,1322],[1109,23839,9875],{"className":23840},[1152,1153,1322],[1109,23842,1330],{"className":23843},[1329],[1109,23845,23847],{"className":23846},[1302],[1109,23848,23850],{"className":23849,"style":1337},[1306],[1109,23851],{},[1109,23853,1910],{"className":23854},[1975],[1109,23856],{"className":23857,"style":1540},[1231],[1109,23859,23861,23864],{"className":23860},[1152],[1109,23862,14113],{"className":23863,"style":1186},[1152,1153],[1109,23865,23867],{"className":23866},[1293],[1109,23868,23870,23890],{"className":23869},[1297,1298],[1109,23871,23873,23887],{"className":23872},[1302],[1109,23874,23876],{"className":23875,"style":14151},[1306],[1109,23877,23878,23881],{"style":2146},[1109,23879],{"className":23880,"style":1315},[1314],[1109,23882,23884],{"className":23883},[1319,1320,1321,1322],[1109,23885,9875],{"className":23886},[1152,1153,1322],[1109,23888,1330],{"className":23889},[1329],[1109,23891,23893],{"className":23892},[1302],[1109,23894,23896],{"className":23895,"style":1337},[1306],[1109,23897],{},[1109,23899,12261],{"className":23900},[1975],[1109,23902],{"className":23903,"style":1540},[1231],[1109,23905,23907,23910],{"className":23906},[1152],[1109,23908,12502],{"className":23909,"style":12530},[1152,1153],[1109,23911,23913],{"className":23912},[1293],[1109,23914,23916,23936],{"className":23915},[1297,1298],[1109,23917,23919,23933],{"className":23918},[1302],[1109,23920,23922],{"className":23921,"style":8735},[1306],[1109,23923,23924,23927],{"style":15969},[1109,23925],{"className":23926,"style":1315},[1314],[1109,23928,23930],{"className":23929},[1319,1320,1321,1322],[1109,23931,8620],{"className":23932,"style":8750},[1152,1153,1322],[1109,23934,1330],{"className":23935},[1329],[1109,23937,23939],{"className":23938},[1302],[1109,23940,23942],{"className":23941,"style":1337},[1306],[1109,23943],{},[1109,23945,1462],{"className":23946},[1515],[1101,23948,23949],{},"直到事先给定的收敛准则被满足后停止。",[1096,23951,23953],{"id":23952},"gd-与-sgd-对比","GD 与 SGD 对比",[12249,23955,23956,23966],{},[12252,23957,23958,23961,23962,23965],{},[1104,23959,23960],{},"梯度下降法",":每步用",[1104,23963,23964],{},"全体样本","求平均梯度;",[12252,23967,23968,23971,23972,23975],{},[1104,23969,23970],{},"随机梯度下降法",":每步随机抽取",[1104,23973,23974],{},"一个样本","估计梯度。",[1101,23977,23978,23979,23982,23983,23986,23987,23990,23991,23994,23995,23998],{},"当样本数量巨大时,GD 每步都要算全体样本的平均,",[1104,23980,23981],{},"计算量极大","。SGD ",[1104,23984,23985],{},"不能保证每一步都让目标函数下降",",相比 GD 往往需要",[1104,23988,23989],{},"更多迭代次数",",但",[1104,23992,23993],{},"每次迭代的计算量小得多","。折中办法是",[1104,23996,23997],{},"小批量(mini-batch)",":每次随机抽取一小组样本估计梯度,兼顾稳定与速度。",[1096,24000,24002],{"id":24001},"bfgs-与-l-bfgs","BFGS 与 L-BFGS",[1101,24004,24005,24008,24009,24012,24013,24084,24085,24255,24256,24325],{},[1104,24006,24007],{},"BFGS 算法适用条件",":当 Hessian 矩阵维度超大时,求逆非常困难、计算复杂度极高。BFGS 通过",[1104,24010,24011],{},"迭代逼近 Hessian 的逆",",即在牛顿迭代过程中,用一个矩阵 ",[1109,24014,24016,24034],{"className":24015},[1112],[1109,24017,24019],{"className":24018},[1116],[1118,24020,24021],{"xmlns":1120},[1122,24022,24023,24031],{},[1125,24024,24025],{},[1264,24026,24027,24029],{},[1128,24028,16391],{},[1128,24030,8620],{},[1132,24032,24033],{"encoding":1134},"B_k",[1109,24035,24037],{"className":24036,"ariaHidden":1139},[1138],[1109,24038,24040,24043],{"className":24039},[1143],[1109,24041],{"className":24042,"style":3045},[1147],[1109,24044,24046,24049],{"className":24045},[1152],[1109,24047,16391],{"className":24048,"style":16436},[1152,1153],[1109,24050,24052],{"className":24051},[1293],[1109,24053,24055,24076],{"className":24054},[1297,1298],[1109,24056,24058,24073],{"className":24057},[1302],[1109,24059,24061],{"className":24060,"style":8735},[1306],[1109,24062,24064,24067],{"style":24063},"top:-2.55em;margin-left:-0.0502em;margin-right:0.05em;",[1109,24065],{"className":24066,"style":1315},[1314],[1109,24068,24070],{"className":24069},[1319,1320,1321,1322],[1109,24071,8620],{"className":24072,"style":8750},[1152,1153,1322],[1109,24074,1330],{"className":24075},[1329],[1109,24077,24079],{"className":24078},[1302],[1109,24080,24082],{"className":24081,"style":1337},[1306],[1109,24083],{}," 来代替 ",[1109,24086,24088,24130],{"className":24087},[1112],[1109,24089,24091],{"className":24090},[1116],[1118,24092,24093],{"xmlns":1120},[1122,24094,24095,24127],{},[1125,24096,24097,24099,24105,24107,24109,24115,24117],{},[1204,24098,5519],{"stretchy":1456},[6964,24100,24101,24103],{},[1128,24102,5611],{"mathvariant":1450},[1208,24104,1917],{},[1128,24106,1130],{},[1204,24108,1457],{"stretchy":1456},[1264,24110,24111,24113],{},[1128,24112,1268],{},[1128,24114,8620],{},[1204,24116,1462],{"stretchy":1456},[6964,24118,24119,24121],{},[1204,24120,5530],{"stretchy":1456},[1125,24122,24123,24125],{},[1204,24124,5991],{},[1208,24126,1685],{},[1132,24128,24129],{"encoding":1134},"[\\nabla^2 f(x_k)]^{-1}",[1109,24131,24133],{"className":24132,"ariaHidden":1139},[1138],[1109,24134,24136,24139,24142,24171,24174,24177,24217,24220],{"className":24135},[1143],[1109,24137],{"className":24138,"style":8085},[1147],[1109,24140,5519],{"className":24141},[1508],[1109,24143,24145,24148],{"className":24144},[1152],[1109,24146,5611],{"className":24147},[1152],[1109,24149,24151],{"className":24150},[1293],[1109,24152,24154],{"className":24153},[1297],[1109,24155,24157],{"className":24156},[1302],[1109,24158,24160],{"className":24159,"style":8144},[1306],[1109,24161,24162,24165],{"style":7044},[1109,24163],{"className":24164,"style":1315},[1314],[1109,24166,24168],{"className":24167},[1319,1320,1321,1322],[1109,24169,1917],{"className":24170},[1152,1322],[1109,24172,1130],{"className":24173,"style":1154},[1152,1153],[1109,24175,1457],{"className":24176},[1508],[1109,24178,24180,24183],{"className":24179},[1152],[1109,24181,1268],{"className":24182},[1152,1153],[1109,24184,24186],{"className":24185},[1293],[1109,24187,24189,24209],{"className":24188},[1297,1298],[1109,24190,24192,24206],{"className":24191},[1302],[1109,24193,24195],{"className":24194,"style":8735},[1306],[1109,24196,24197,24200],{"style":1310},[1109,24198],{"className":24199,"style":1315},[1314],[1109,24201,24203],{"className":24202},[1319,1320,1321,1322],[1109,24204,8620],{"className":24205,"style":8750},[1152,1153,1322],[1109,24207,1330],{"className":24208},[1329],[1109,24210,24212],{"className":24211},[1302],[1109,24213,24215],{"className":24214,"style":1337},[1306],[1109,24216],{},[1109,24218,1462],{"className":24219},[1515],[1109,24221,24223,24226],{"className":24222},[1515],[1109,24224,5530],{"className":24225},[1515],[1109,24227,24229],{"className":24228},[1293],[1109,24230,24232],{"className":24231},[1297],[1109,24233,24235],{"className":24234},[1302],[1109,24236,24238],{"className":24237,"style":8144},[1306],[1109,24239,24240,24243],{"style":7044},[1109,24241],{"className":24242,"style":1315},[1314],[1109,24244,24246],{"className":24245},[1319,1320,1321,1322],[1109,24247,24249,24252],{"className":24248},[1152,1322],[1109,24250,5991],{"className":24251},[1152,1322],[1109,24253,1685],{"className":24254},[1152,1322],",每步用梯度差和位置差对 ",[1109,24257,24259,24276],{"className":24258},[1112],[1109,24260,24262],{"className":24261},[1116],[1118,24263,24264],{"xmlns":1120},[1122,24265,24266,24274],{},[1125,24267,24268],{},[1264,24269,24270,24272],{},[1128,24271,16391],{},[1128,24273,8620],{},[1132,24275,24033],{"encoding":1134},[1109,24277,24279],{"className":24278,"ariaHidden":1139},[1138],[1109,24280,24282,24285],{"className":24281},[1143],[1109,24283],{"className":24284,"style":3045},[1147],[1109,24286,24288,24291],{"className":24287},[1152],[1109,24289,16391],{"className":24290,"style":16436},[1152,1153],[1109,24292,24294],{"className":24293},[1293],[1109,24295,24297,24317],{"className":24296},[1297,1298],[1109,24298,24300,24314],{"className":24299},[1302],[1109,24301,24303],{"className":24302,"style":8735},[1306],[1109,24304,24305,24308],{"style":24063},[1109,24306],{"className":24307,"style":1315},[1314],[1109,24309,24311],{"className":24310},[1319,1320,1321,1322],[1109,24312,8620],{"className":24313,"style":8750},[1152,1153,1322],[1109,24315,1330],{"className":24316},[1329],[1109,24318,24320],{"className":24319},[1302],[1109,24321,24323],{"className":24322,"style":1337},[1306],[1109,24324],{}," 做秩更新,避免直接求逆。",[1101,24327,24328,24331,24332,24383,24384,24417],{},[1104,24329,24330],{},"L-BFGS 算法",":由于 BFGS 每次都要存储 ",[1109,24333,24335,24353],{"className":24334},[1112],[1109,24336,24338],{"className":24337},[1116],[1118,24339,24340],{"xmlns":1120},[1122,24341,24342,24350],{},[1125,24343,24344,24346,24348],{},[1128,24345,1101],{},[1204,24347,9893],{},[1128,24349,1101],{},[1132,24351,24352],{"encoding":1134},"p\\times p",[1109,24354,24356,24374],{"className":24355,"ariaHidden":1139},[1138],[1109,24357,24359,24362,24365,24368,24371],{"className":24358},[1143],[1109,24360],{"className":24361,"style":19706},[1147],[1109,24363,1101],{"className":24364},[1152,1153],[1109,24366],{"className":24367,"style":2497},[1231],[1109,24369,9893],{"className":24370},[2501],[1109,24372],{"className":24373,"style":2497},[1231],[1109,24375,24377,24380],{"className":24376},[1143],[1109,24378],{"className":24379,"style":1182},[1147],[1109,24381,1101],{"className":24382},[1152,1153]," 的近似矩阵,内存开销很大,数据维度高时机器可能吃不消。L-BFGS 只保存",[1104,24385,24386,24387,24416],{},"最近 ",[1109,24388,24390,24404],{"className":24389},[1112],[1109,24391,24393],{"className":24392},[1116],[1118,24394,24395],{"xmlns":1120},[1122,24396,24397,24402],{},[1125,24398,24399],{},[1128,24400,24401],{},"m",[1132,24403,24401],{"encoding":1134},[1109,24405,24407],{"className":24406,"ariaHidden":1139},[1138],[1109,24408,24410,24413],{"className":24409},[1143],[1109,24411],{"className":24412,"style":7312},[1147],[1109,24414,24401],{"className":24415},[1152,1153]," 次迭代的信息","(向量对)用于隐式重构更新方向,大幅节省内存,适合超高维问题。",[10214,24419],{},[1091,24421,24423],{"id":24422},"五总结","五、总结",[12249,24425,24426,24679,24741,24848,25118,25424,25532,25538,25547],{},[12252,24427,24428,1662,24431,24459,24460,24536,24537,24565,24566,24536,24627,4024],{},[1104,24429,24430],{},"大 O \u002F 小 o",[1109,24432,24434,24447],{"className":24433},[1112],[1109,24435,24437],{"className":24436},[1116],[1118,24438,24439],{"xmlns":1120},[1122,24440,24441,24445],{},[1125,24442,24443],{},[1128,24444,1585],{},[1132,24446,1585],{"encoding":1134},[1109,24448,24450],{"className":24449,"ariaHidden":1139},[1138],[1109,24451,24453,24456],{"className":24452},[1143],[1109,24454],{"className":24455,"style":3026},[1147],[1109,24457,1585],{"className":24458,"style":1640},[1152,1153]," 是\"不超过量级\"(",[1109,24461,24463,24491],{"className":24462},[1112],[1109,24464,24466],{"className":24465},[1116],[1118,24467,24468],{"xmlns":1120},[1122,24469,24470,24488],{},[1125,24471,24472,24474,24476,24478,24480,24482,24484,24486],{},[1128,24473,1451],{"mathvariant":1450},[1128,24475,1130],{},[1128,24477,1451],{"mathvariant":1450},[1204,24479,1467],{},[1128,24481,1202],{},[1128,24483,1451],{"mathvariant":1450},[1128,24485,1170],{},[1128,24487,1451],{"mathvariant":1450},[1132,24489,24490],{"encoding":1134},"|f|\\le C|g|",[1109,24492,24494,24518],{"className":24493,"ariaHidden":1139},[1138],[1109,24495,24497,24500,24503,24506,24509,24512,24515],{"className":24496},[1143],[1109,24498],{"className":24499,"style":1498},[1147],[1109,24501,1451],{"className":24502},[1152],[1109,24504,1130],{"className":24505,"style":1154},[1152,1153],[1109,24507,1451],{"className":24508},[1152],[1109,24510],{"className":24511,"style":1232},[1231],[1109,24513,1467],{"className":24514},[1236],[1109,24516],{"className":24517,"style":1232},[1231],[1109,24519,24521,24524,24527,24530,24533],{"className":24520},[1143],[1109,24522],{"className":24523,"style":1498},[1147],[1109,24525,1202],{"className":24526,"style":1227},[1152,1153],[1109,24528,1451],{"className":24529},[1152],[1109,24531,1170],{"className":24532,"style":1186},[1152,1153],[1109,24534,1451],{"className":24535},[1152],");",[1109,24538,24540,24553],{"className":24539},[1112],[1109,24541,24543],{"className":24542},[1116],[1118,24544,24545],{"xmlns":1120},[1122,24546,24547,24551],{},[1125,24548,24549],{},[1128,24550,4385],{},[1132,24552,4385],{"encoding":1134},[1109,24554,24556],{"className":24555,"ariaHidden":1139},[1138],[1109,24557,24559,24562],{"className":24558},[1143],[1109,24560],{"className":24561,"style":7312},[1147],[1109,24563,4385],{"className":24564},[1152,1153]," 是\"严格更低量级\"(",[1109,24567,24569,24591],{"className":24568},[1112],[1109,24570,24572],{"className":24571},[1116],[1118,24573,24574],{"xmlns":1120},[1122,24575,24576,24588],{},[1125,24577,24578,24580,24582,24584,24586],{},[1128,24579,1130],{},[1128,24581,19885],{"mathvariant":1450},[1128,24583,1170],{},[1204,24585,4485],{},[1208,24587,1210],{},[1132,24589,24590],{"encoding":1134},"f\u002Fg\\to0",[1109,24592,24594,24618],{"className":24593,"ariaHidden":1139},[1138],[1109,24595,24597,24600,24603,24606,24609,24612,24615],{"className":24596},[1143],[1109,24598],{"className":24599,"style":1498},[1147],[1109,24601,1130],{"className":24602,"style":1154},[1152,1153],[1109,24604,19885],{"className":24605},[1152],[1109,24607,1170],{"className":24608,"style":1186},[1152,1153],[1109,24610],{"className":24611,"style":1232},[1231],[1109,24613,4485],{"className":24614},[1236],[1109,24616],{"className":24617,"style":1232},[1231],[1109,24619,24621,24624],{"className":24620},[1143],[1109,24622],{"className":24623,"style":1246},[1147],[1109,24625,1210],{"className":24626},[1152],[1109,24628,24630,24649],{"className":24629},[1112],[1109,24631,24633],{"className":24632},[1116],[1118,24634,24635],{"xmlns":1120},[1122,24636,24637,24646],{},[1125,24638,24639,24641,24644],{},[1128,24640,4385],{},[1204,24642,24643],{},"⇒",[1128,24645,1585],{},[1132,24647,24648],{"encoding":1134},"o \\Rightarrow O",[1109,24650,24652,24670],{"className":24651,"ariaHidden":1139},[1138],[1109,24653,24655,24658,24661,24664,24667],{"className":24654},[1143],[1109,24656],{"className":24657,"style":7312},[1147],[1109,24659,4385],{"className":24660},[1152,1153],[1109,24662],{"className":24663,"style":1232},[1231],[1109,24665,24643],{"className":24666},[1236],[1109,24668],{"className":24669,"style":1232},[1231],[1109,24671,24673,24676],{"className":24672},[1143],[1109,24674],{"className":24675,"style":3026},[1147],[1109,24677,1585],{"className":24678,"style":1640},[1152,1153],[12252,24680,24681,1662,24684,24740],{},[1104,24682,24683],{},"驻点 ≠ 最优",[1109,24685,24687,24707],{"className":24686},[1112],[1109,24688,24690],{"className":24689},[1116],[1118,24691,24692],{"xmlns":1120},[1122,24693,24694,24704],{},[1125,24695,24696,24698,24700,24702],{},[1128,24697,5611],{"mathvariant":1450},[1128,24699,1130],{},[1204,24701,1582],{},[1208,24703,1210],{},[1132,24705,24706],{"encoding":1134},"\\nabla f=0",[1109,24708,24710,24731],{"className":24709,"ariaHidden":1139},[1138],[1109,24711,24713,24716,24719,24722,24725,24728],{"className":24712},[1143],[1109,24714],{"className":24715,"style":1148},[1147],[1109,24717,5611],{"className":24718},[1152],[1109,24720,1130],{"className":24721,"style":1154},[1152,1153],[1109,24723],{"className":24724,"style":1232},[1231],[1109,24726,1582],{"className":24727},[1236],[1109,24729],{"className":24730,"style":1232},[1231],[1109,24732,24734,24737],{"className":24733},[1143],[1109,24735],{"className":24736,"style":1246},[1147],[1109,24738,1210],{"className":24739},[1152]," 可能是局部最优或鞍点;凸函数才能保证局部即全局。",[12252,24742,24743,1662,24746,24796,24797,24847],{},[1104,24744,24745],{},"收敛速率",[1109,24747,24749,24766],{"className":24748},[1112],[1109,24750,24752],{"className":24751},[1116],[1118,24753,24754],{"xmlns":1120},[1122,24755,24756,24764],{},[1125,24757,24758,24760,24762],{},[1128,24759,8579],{},[1204,24761,1582],{},[1208,24763,1685],{},[1132,24765,9291],{"encoding":1134},[1109,24767,24769,24787],{"className":24768,"ariaHidden":1139},[1138],[1109,24770,24772,24775,24778,24781,24784],{"className":24771},[1143],[1109,24773],{"className":24774,"style":1182},[1147],[1109,24776,8579],{"className":24777,"style":1186},[1152,1153],[1109,24779],{"className":24780,"style":1232},[1231],[1109,24782,1582],{"className":24783},[1236],[1109,24785],{"className":24786,"style":1232},[1231],[1109,24788,24790,24793],{"className":24789},[1143],[1109,24791],{"className":24792,"style":1246},[1147],[1109,24794,1685],{"className":24795},[1152]," 线性、",[1109,24798,24800,24817],{"className":24799},[1112],[1109,24801,24803],{"className":24802},[1116],[1118,24804,24805],{"xmlns":1120},[1122,24806,24807,24815],{},[1125,24808,24809,24811,24813],{},[1128,24810,8579],{},[1204,24812,1582],{},[1208,24814,1917],{},[1132,24816,9346],{"encoding":1134},[1109,24818,24820,24838],{"className":24819,"ariaHidden":1139},[1138],[1109,24821,24823,24826,24829,24832,24835],{"className":24822},[1143],[1109,24824],{"className":24825,"style":1182},[1147],[1109,24827,8579],{"className":24828,"style":1186},[1152,1153],[1109,24830],{"className":24831,"style":1232},[1231],[1109,24833,1582],{"className":24834},[1236],[1109,24836],{"className":24837,"style":1232},[1231],[1109,24839,24841,24844],{"className":24840},[1143],[1109,24842],{"className":24843,"style":1246},[1147],[1109,24845,1917],{"className":24846},[1152]," 二次;牛顿法二次收敛。",[12252,24849,24850,1662,24852,25117],{},[1104,24851,335],{},[1109,24853,24855,24913],{"className":24854},[1112],[1109,24856,24858],{"className":24857},[1116],[1118,24859,24860],{"xmlns":1120},[1122,24861,24862,24910],{},[1125,24863,24864,24876,24878,24884,24886,24888,24894,24896,24906,24908],{},[1264,24865,24866,24868],{},[1128,24867,1268],{},[1125,24869,24870,24872,24874],{},[1128,24871,8620],{},[1204,24873,2342],{},[1208,24875,1685],{},[1204,24877,1582],{},[1264,24879,24880,24882],{},[1128,24881,1268],{},[1128,24883,8620],{},[1204,24885,5991],{},[1204,24887,5519],{"stretchy":1456},[6964,24889,24890,24892],{},[1128,24891,5611],{"mathvariant":1450},[1208,24893,1917],{},[1128,24895,1130],{},[6964,24897,24898,24900],{},[1204,24899,5530],{"stretchy":1456},[1125,24901,24902,24904],{},[1204,24903,5991],{},[1208,24905,1685],{},[1128,24907,5611],{"mathvariant":1450},[1128,24909,1130],{},[1132,24911,24912],{"encoding":1134},"x_{k+1}=x_k-[\\nabla^2 f]^{-1}\\nabla f",[1109,24914,24916,24980,25035],{"className":24915,"ariaHidden":1139},[1138],[1109,24917,24919,24922,24971,24974,24977],{"className":24918},[1143],[1109,24920],{"className":24921,"style":10615},[1147],[1109,24923,24925,24928],{"className":24924},[1152],[1109,24926,1268],{"className":24927},[1152,1153],[1109,24929,24931],{"className":24930},[1293],[1109,24932,24934,24963],{"className":24933},[1297,1298],[1109,24935,24937,24960],{"className":24936},[1302],[1109,24938,24940],{"className":24939,"style":8735},[1306],[1109,24941,24942,24945],{"style":1310},[1109,24943],{"className":24944,"style":1315},[1314],[1109,24946,24948],{"className":24947},[1319,1320,1321,1322],[1109,24949,24951,24954,24957],{"className":24950},[1152,1322],[1109,24952,8620],{"className":24953,"style":8750},[1152,1153,1322],[1109,24955,2342],{"className":24956},[2501,1322],[1109,24958,1685],{"className":24959},[1152,1322],[1109,24961,1330],{"className":24962},[1329],[1109,24964,24966],{"className":24965},[1302],[1109,24967,24969],{"className":24968,"style":10663},[1306],[1109,24970],{},[1109,24972],{"className":24973,"style":1232},[1231],[1109,24975,1582],{"className":24976},[1236],[1109,24978],{"className":24979,"style":1232},[1231],[1109,24981,24983,24986,25026,25029,25032],{"className":24982},[1143],[1109,24984],{"className":24985,"style":10681},[1147],[1109,24987,24989,24992],{"className":24988},[1152],[1109,24990,1268],{"className":24991},[1152,1153],[1109,24993,24995],{"className":24994},[1293],[1109,24996,24998,25018],{"className":24997},[1297,1298],[1109,24999,25001,25015],{"className":25000},[1302],[1109,25002,25004],{"className":25003,"style":8735},[1306],[1109,25005,25006,25009],{"style":1310},[1109,25007],{"className":25008,"style":1315},[1314],[1109,25010,25012],{"className":25011},[1319,1320,1321,1322],[1109,25013,8620],{"className":25014,"style":8750},[1152,1153,1322],[1109,25016,1330],{"className":25017},[1329],[1109,25019,25021],{"className":25020},[1302],[1109,25022,25024],{"className":25023,"style":1337},[1306],[1109,25025],{},[1109,25027],{"className":25028,"style":2497},[1231],[1109,25030,5991],{"className":25031},[2501],[1109,25033],{"className":25034,"style":2497},[1231],[1109,25036,25038,25041,25044,25073,25076,25111,25114],{"className":25037},[1143],[1109,25039],{"className":25040,"style":8085},[1147],[1109,25042,5519],{"className":25043},[1508],[1109,25045,25047,25050],{"className":25046},[1152],[1109,25048,5611],{"className":25049},[1152],[1109,25051,25053],{"className":25052},[1293],[1109,25054,25056],{"className":25055},[1297],[1109,25057,25059],{"className":25058},[1302],[1109,25060,25062],{"className":25061,"style":8144},[1306],[1109,25063,25064,25067],{"style":7044},[1109,25065],{"className":25066,"style":1315},[1314],[1109,25068,25070],{"className":25069},[1319,1320,1321,1322],[1109,25071,1917],{"className":25072},[1152,1322],[1109,25074,1130],{"className":25075,"style":1154},[1152,1153],[1109,25077,25079,25082],{"className":25078},[1515],[1109,25080,5530],{"className":25081},[1515],[1109,25083,25085],{"className":25084},[1293],[1109,25086,25088],{"className":25087},[1297],[1109,25089,25091],{"className":25090},[1302],[1109,25092,25094],{"className":25093,"style":8144},[1306],[1109,25095,25096,25099],{"style":7044},[1109,25097],{"className":25098,"style":1315},[1314],[1109,25100,25102],{"className":25101},[1319,1320,1321,1322],[1109,25103,25105,25108],{"className":25104},[1152,1322],[1109,25106,5991],{"className":25107},[1152,1322],[1109,25109,1685],{"className":25110},[1152,1322],[1109,25112,5611],{"className":25113},[1152],[1109,25115,1130],{"className":25116,"style":1154},[1152,1153],";问题是 Hessian 求逆贵\u002F可能不可逆、值不一定递减、依赖初值。",[12252,25119,25120,1662,25123,25174,25175,25366,25367,25423],{},[1104,25121,25122],{},"回溯线搜索",[1109,25124,25126,25144],{"className":25125},[1112],[1109,25127,25129],{"className":25128},[1116],[1118,25130,25131],{"xmlns":1120},[1122,25132,25133,25141],{},[1125,25134,25135,25137,25139],{},[1128,25136,12579],{},[1204,25138,1582],{},[1208,25140,1685],{},[1132,25142,25143],{"encoding":1134},"t=1",[1109,25145,25147,25165],{"className":25146,"ariaHidden":1139},[1138],[1109,25148,25150,25153,25156,25159,25162],{"className":25149},[1143],[1109,25151],{"className":25152,"style":12596},[1147],[1109,25154,12579],{"className":25155},[1152,1153],[1109,25157],{"className":25158,"style":1232},[1231],[1109,25160,1582],{"className":25161},[1236],[1109,25163],{"className":25164,"style":1232},[1231],[1109,25166,25168,25171],{"className":25167},[1143],[1109,25169],{"className":25170,"style":1246},[1147],[1109,25172,1685],{"className":25173},[1152]," 起,不满足充分下降条件 ",[1109,25176,25178,25234],{"className":25177},[1112],[1109,25179,25181],{"className":25180},[1116],[1118,25182,25183],{"xmlns":1120},[1122,25184,25185,25231],{},[1125,25186,25187,25189,25191,25193,25195,25197,25199,25201,25203,25205,25207,25209,25211,25213,25215,25217,25219,25221,25227,25229],{},[1128,25188,1130],{},[1204,25190,1457],{"stretchy":1456},[1128,25192,1268],{},[1204,25194,2342],{},[1128,25196,12579],{},[1128,25198,12653],{"mathvariant":1450},[1128,25200,1268],{},[1204,25202,1462],{"stretchy":1456},[1204,25204,1467],{},[1128,25206,1130],{},[1204,25208,1457],{"stretchy":1456},[1128,25210,1268],{},[1204,25212,1462],{"stretchy":1456},[1204,25214,2342],{},[1128,25216,12425],{},[1128,25218,12579],{},[1128,25220,5611],{"mathvariant":1450},[6964,25222,25223,25225],{},[1128,25224,1130],{},[1128,25226,9471],{"mathvariant":1450},[1128,25228,12653],{"mathvariant":1450},[1128,25230,1268],{},[1132,25232,25233],{"encoding":1134},"f(x+t\\Delta x)\\le f(x)+\\alpha t\\nabla f^\\top\\Delta x",[1109,25235,25237,25261,25288,25315],{"className":25236,"ariaHidden":1139},[1138],[1109,25238,25240,25243,25246,25249,25252,25255,25258],{"className":25239},[1143],[1109,25241],{"className":25242,"style":1498},[1147],[1109,25244,1130],{"className":25245,"style":1154},[1152,1153],[1109,25247,1457],{"className":25248},[1508],[1109,25250,1268],{"className":25251},[1152,1153],[1109,25253],{"className":25254,"style":2497},[1231],[1109,25256,2342],{"className":25257},[2501],[1109,25259],{"className":25260,"style":2497},[1231],[1109,25262,25264,25267,25270,25273,25276,25279,25282,25285],{"className":25263},[1143],[1109,25265],{"className":25266,"style":1498},[1147],[1109,25268,12579],{"className":25269},[1152,1153],[1109,25271,12653],{"className":25272},[1152],[1109,25274,1268],{"className":25275},[1152,1153],[1109,25277,1462],{"className":25278},[1515],[1109,25280],{"className":25281,"style":1232},[1231],[1109,25283,1467],{"className":25284},[1236],[1109,25286],{"className":25287,"style":1232},[1231],[1109,25289,25291,25294,25297,25300,25303,25306,25309,25312],{"className":25290},[1143],[1109,25292],{"className":25293,"style":1498},[1147],[1109,25295,1130],{"className":25296,"style":1154},[1152,1153],[1109,25298,1457],{"className":25299},[1508],[1109,25301,1268],{"className":25302},[1152,1153],[1109,25304,1462],{"className":25305},[1515],[1109,25307],{"className":25308,"style":2497},[1231],[1109,25310,2342],{"className":25311},[2501],[1109,25313],{"className":25314,"style":2497},[1231],[1109,25316,25318,25322,25325,25328,25331,25360,25363],{"className":25317},[1143],[1109,25319],{"className":25320,"style":25321},[1147],"height:1.0435em;vertical-align:-0.1944em;",[1109,25323,12425],{"className":25324,"style":12454},[1152,1153],[1109,25326,12579],{"className":25327},[1152,1153],[1109,25329,5611],{"className":25330},[1152],[1109,25332,25334,25337],{"className":25333},[1152],[1109,25335,1130],{"className":25336,"style":1154},[1152,1153],[1109,25338,25340],{"className":25339},[1293],[1109,25341,25343],{"className":25342},[1297],[1109,25344,25346],{"className":25345},[1302],[1109,25347,25349],{"className":25348,"style":11218},[1306],[1109,25350,25351,25354],{"style":7044},[1109,25352],{"className":25353,"style":1315},[1314],[1109,25355,25357],{"className":25356},[1319,1320,1321,1322],[1109,25358,9471],{"className":25359},[1152,1322],[1109,25361,12653],{"className":25362},[1152],[1109,25364,1268],{"className":25365},[1152,1153]," 就 ",[1109,25368,25370,25390],{"className":25369},[1112],[1109,25371,25373],{"className":25372},[1116],[1118,25374,25375],{"xmlns":1120},[1122,25376,25377,25387],{},[1125,25378,25379,25381,25383,25385],{},[1128,25380,12579],{},[1204,25382,12905],{},[1128,25384,12502],{},[1128,25386,12579],{},[1132,25388,25389],{"encoding":1134},"t\\leftarrow\\beta t",[1109,25391,25393,25411],{"className":25392,"ariaHidden":1139},[1138],[1109,25394,25396,25399,25402,25405,25408],{"className":25395},[1143],[1109,25397],{"className":25398,"style":12596},[1147],[1109,25400,12579],{"className":25401},[1152,1153],[1109,25403],{"className":25404,"style":1232},[1231],[1109,25406,12905],{"className":25407},[1236],[1109,25409],{"className":25410,"style":1232},[1231],[1109,25412,25414,25417,25420],{"className":25413},[1143],[1109,25415],{"className":25416,"style":1148},[1147],[1109,25418,12502],{"className":25419,"style":12530},[1152,1153],[1109,25421,12579],{"className":25422},[1152,1153]," 缩小。",[12252,25425,25426,25428,25429,25531],{},[1104,25427,13190],{},":绝对 \u002F 相对 \u002F 调整相对(分母 ",[1109,25430,25432,25458],{"className":25431},[1112],[1109,25433,25435],{"className":25434},[1116],[1118,25436,25437],{"xmlns":1120},[1122,25438,25439,25455],{},[1125,25440,25441,25443,25449,25451,25453],{},[1128,25442,8631],{"mathvariant":1450},[1264,25444,25445,25447],{},[1128,25446,1268],{},[1128,25448,8620],{},[1128,25450,8631],{"mathvariant":1450},[1204,25452,2342],{},[1208,25454,1685],{},[1132,25456,25457],{"encoding":1134},"\\|x_k\\|+1",[1109,25459,25461,25522],{"className":25460,"ariaHidden":1139},[1138],[1109,25462,25464,25467,25470,25510,25513,25516,25519],{"className":25463},[1143],[1109,25465],{"className":25466,"style":1498},[1147],[1109,25468,8631],{"className":25469},[1152],[1109,25471,25473,25476],{"className":25472},[1152],[1109,25474,1268],{"className":25475},[1152,1153],[1109,25477,25479],{"className":25478},[1293],[1109,25480,25482,25502],{"className":25481},[1297,1298],[1109,25483,25485,25499],{"className":25484},[1302],[1109,25486,25488],{"className":25487,"style":8735},[1306],[1109,25489,25490,25493],{"style":1310},[1109,25491],{"className":25492,"style":1315},[1314],[1109,25494,25496],{"className":25495},[1319,1320,1321,1322],[1109,25497,8620],{"className":25498,"style":8750},[1152,1153,1322],[1109,25500,1330],{"className":25501},[1329],[1109,25503,25505],{"className":25504},[1302],[1109,25506,25508],{"className":25507,"style":1337},[1306],[1109,25509],{},[1109,25511,8631],{"className":25512},[1152],[1109,25514],{"className":25515,"style":2497},[1231],[1109,25517,2342],{"className":25518},[2501],[1109,25520],{"className":25521,"style":2497},[1231],[1109,25523,25525,25528],{"className":25524},[1143],[1109,25526],{"className":25527,"style":1246},[1147],[1109,25529,1685],{"className":25530},[1152],")。",[12252,25533,25534,25537],{},[1104,25535,25536],{},"ADMM",":增广拉格朗日 = 拉格朗日 + 二次罚项;交替更新 x、z、对偶 λ。Lasso 的 z 更新是软阈值算子;分位数回归用 pinball 损失。",[12252,25539,25540,25543,25546],{},[1104,25541,25542],{},"GD vs SGD",[25544,25545],"gd",{}," 全样本、稳但慢;SGD 单样本、快但抖、迭代多;mini-batch 折中。",[12252,25548,25549,25552],{},[1104,25550,25551],{},"BFGS \u002F L-BFGS",":用矩阵迭代逼近 Hessian 逆,避免直接求逆;L-BFGS 只存最近 m 次信息省内存。",{"title":25554,"searchDepth":25555,"depth":25555,"links":25556},"",4,[25557,25568,25574,25578,25584],{"id":1093,"depth":25558,"text":1094,"children":25559},2,[25560,25561,25562,25563,25564,25565,25566,25567],{"id":1098,"depth":12885,"text":1099},{"id":4027,"depth":12885,"text":4028},{"id":5498,"depth":12885,"text":5499},{"id":5747,"depth":12885,"text":5747},{"id":6734,"depth":12885,"text":6734},{"id":7183,"depth":12885,"text":7183},{"id":8561,"depth":12885,"text":8561},{"id":9383,"depth":12885,"text":9384},{"id":10218,"depth":25558,"text":10219,"children":25569},[25570,25571,25572,25573],{"id":12246,"depth":12885,"text":12247},{"id":12363,"depth":12885,"text":12364},{"id":13190,"depth":12885,"text":13190},{"id":14082,"depth":12885,"text":14083},{"id":16211,"depth":25558,"text":16212,"children":25575},[25576,25577],{"id":18031,"depth":12885,"text":18032},{"id":20362,"depth":12885,"text":20363},{"id":21648,"depth":25558,"text":21649,"children":25579},[25580,25581,25582,25583],{"id":21652,"depth":12885,"text":21653},{"id":23384,"depth":12885,"text":23385},{"id":23952,"depth":12885,"text":23953},{"id":24001,"depth":12885,"text":24002},{"id":24422,"depth":25558,"text":24423},"https:\u002F\u002Fpica.zhimg.com\u002Fv2-98fc1ffc26fe3137f57afefd13dc7d8c_1440w.jpg","牛顿迭代法与收敛性理论基础","md",true,{"uuid":25590,"slots":25591},"f3561dd0-e86a-11f0-8fd1-914b1152d8ed",{},{"title":1085,"description":25586},"posts\u002F分布式处理与计算\u002F分布式处理与计算：数值优化方法",[323,334,335,336],"BYwICb04RowXk99BJcb1a1dC2EyUvB7IrKX53IrkKKI",1790443283500]