[{"data":1,"prerenderedAt":2394},["ShallowReactive",2],{"lang-switch-post-\u002Fplaylists\u002Fmachine-learning-specialization\u002Fw3-lab06-gradient-descent":3,"post-pt-machine-learning-specialization-w3-lab06-gradient-descent":4},"\u002Fen\u002Fplaylists\u002Fmachine-learning-specialization\u002Fw3-lab06-gradient-descent",{"id":5,"title":6,"body":7,"cover":2378,"date":2379,"description":2380,"extension":2381,"meta":2382,"navigation":1871,"order":2383,"path":2384,"playlist":2385,"seo":2386,"status":2387,"stem":2388,"tags":2389,"__hash__":2393},"posts\u002Fpt\u002Fplaylists\u002Fmachine-learning-specialization\u002Fw3-lab06-gradient-descent.md","O Gradiente Descendente Não Precisou Aprender Nada de Novo",{"type":8,"value":9,"toc":2370},"minimark",[10,38,43,606,619,1807,1828,1832,1932,1981,1992,2012,2028,2032,2099,2122,2129,2151,2162,2174,2181,2185,2197,2212,2227,2235,2242,2246,2299,2303,2319,2325,2366],[11,12,13,14,19,20,24,25,28,29,33,34,37],"p",{},"Lá na Semana 1, gradiente descendente pra regressão linear era: calcula o gradiente, dá um passo, repete. ",[15,16,18],"a",{"href":17},"\u002Fplaylists\u002Fmachine-learning-specialization\u002Fw3-lab05-cost-function","O código dessa vez"," já tem sigmoide, fronteira de decisão, perda e custo logísticos prontos. Falta uma coisa: rodar gradiente descendente em cima desse custo novo pra achar o ",[21,22,23],"code",{},"w"," e o ",[21,26,27],{},"b"," que minimizam ele. E aqui vem a parte satisfeita do capítulo: o loop que dá os passos é ",[30,31,32],"strong",{},"literalmente o mesmo",". Só a conta de dentro do loop (o gradiente) muda, porque ",[21,35,36],{},"f"," trocou de reta pra sigmoide.",[39,40,42],"h2",{"id":41},"o-loop-não-muda-só-o-que-ele-deriva","O loop não muda, só o que ele deriva",[11,44,45],{},[46,47,50,170],"span",{"className":48},[49],"katex",[46,51,54],{"className":52},[53],"katex-mathml",[55,56,58],"math",{"xmlns":57},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[59,60,61,165],"semantics",{},[62,63,64,73,77,83,86,89,127,131,133,135,137,139,141],"mrow",{},[65,66,67,70],"msub",{},[68,69,23],"mi",{},[68,71,72],{},"j",[74,75,76],"mo",{},"=",[65,78,79,81],{},[68,80,23],{},[68,82,72],{},[74,84,85],{},"−",[68,87,88],{},"α",[90,91,92,117],"mfrac",{},[62,93,94,98,101,105,108,112,114],{},[68,95,97],{"mathvariant":96},"normal","∂",[68,99,100],{},"J",[74,102,104],{"stretchy":103},"false","(",[68,106,23],{"mathvariant":107},"bold",[74,109,111],{"separator":110},"true",",",[68,113,27],{},[74,115,116],{"stretchy":103},")",[62,118,119,121],{},[68,120,97],{"mathvariant":96},[65,122,123,125],{},[68,124,23],{},[68,126,72],{},[128,129],"mspace",{"width":130},"2em",[68,132,27],{},[74,134,76],{},[68,136,27],{},[74,138,85],{},[68,140,88],{},[90,142,143,159],{},[62,144,145,147,149,151,153,155,157],{},[68,146,97],{"mathvariant":96},[68,148,100],{},[74,150,104],{"stretchy":103},[68,152,23],{"mathvariant":107},[74,154,111],{"separator":110},[68,156,27],{},[74,158,116],{"stretchy":103},[62,160,161,163],{},[68,162,97],{"mathvariant":96},[68,164,27],{},[166,167,169],"annotation",{"encoding":168},"application\u002Fx-tex","w_j = w_j - \\alpha \\frac{\\partial J(\\mathbf{w},b)}{\\partial w_j} \\qquad b = b - \\alpha \\frac{\\partial J(\\mathbf{w},b)}{\\partial b}",[46,171,174,254,312,487,506],{"className":172,"ariaHidden":110},[173],"katex-html",[46,175,178,183,243,247,251],{"className":176},[177],"base",[46,179],{"className":180,"style":182},[181],"strut","height:0.7167em;vertical-align:-0.2861em;",[46,184,187,192],{"className":185},[186],"mord",[46,188,23],{"className":189,"style":191},[186,190],"mathnormal","margin-right:0.0269em;",[46,193,196],{"className":194},[195],"msupsub",[46,197,201,234],{"className":198},[199,200],"vlist-t","vlist-t2",[46,202,205,229],{"className":203},[204],"vlist-r",[46,206,210],{"className":207,"style":209},[208],"vlist","height:0.3117em;",[46,211,213,218],{"style":212},"top:-2.55em;margin-left:-0.0269em;margin-right:0.05em;",[46,214],{"className":215,"style":217},[216],"pstrut","height:2.7em;",[46,219,225],{"className":220},[221,222,223,224],"sizing","reset-size6","size3","mtight",[46,226,72],{"className":227,"style":228},[186,190,224],"margin-right:0.0572em;",[46,230,233],{"className":231},[232],"vlist-s","​",[46,235,237],{"className":236},[204],[46,238,241],{"className":239,"style":240},[208],"height:0.2861em;",[46,242],{},[46,244],{"className":245,"style":246},[128],"margin-right:0.2778em;",[46,248,76],{"className":249},[250],"mrel",[46,252],{"className":253,"style":246},[128],[46,255,257,261,301,305,309],{"className":256},[177],[46,258],{"className":259,"style":260},[181],"height:0.8694em;vertical-align:-0.2861em;",[46,262,264,267],{"className":263},[186],[46,265,23],{"className":266,"style":191},[186,190],[46,268,270],{"className":269},[195],[46,271,273,293],{"className":272},[199,200],[46,274,276,290],{"className":275},[204],[46,277,279],{"className":278,"style":209},[208],[46,280,281,284],{"style":212},[46,282],{"className":283,"style":217},[216],[46,285,287],{"className":286},[221,222,223,224],[46,288,72],{"className":289,"style":228},[186,190,224],[46,291,233],{"className":292},[232],[46,294,296],{"className":295},[204],[46,297,299],{"className":298,"style":240},[208],[46,300],{},[46,302],{"className":303,"style":304},[128],"margin-right:0.2222em;",[46,306,85],{"className":307},[308],"mbin",[46,310],{"className":311,"style":304},[128],[46,313,315,319,323,471,475,478,481,484],{"className":314},[177],[46,316],{"className":317,"style":318},[181],"height:1.5523em;vertical-align:-0.5423em;",[46,320,88],{"className":321,"style":322},[186,190],"margin-right:0.0037em;",[46,324,326,331,468],{"className":325},[186],[46,327],{"className":328},[329,330],"mopen","nulldelimiter",[46,332,334],{"className":333},[90],[46,335,337,459],{"className":336},[199,200],[46,338,340,456],{"className":339},[204],[46,341,344,407,418],{"className":342,"style":343},[208],"height:1.01em;",[46,345,347,351],{"style":346},"top:-2.655em;",[46,348],{"className":349,"style":350},[216],"height:3em;",[46,352,354],{"className":353},[221,222,223,224],[46,355,357,361],{"className":356},[186,224],[46,358,97],{"className":359,"style":360},[186,224],"margin-right:0.0556em;",[46,362,364,367],{"className":363},[186,224],[46,365,23],{"className":366,"style":191},[186,190,224],[46,368,370],{"className":369},[195],[46,371,373,398],{"className":372},[199,200],[46,374,376,395],{"className":375},[204],[46,377,380],{"className":378,"style":379},[208],"height:0.3281em;",[46,381,383,387],{"style":382},"top:-2.357em;margin-left:-0.0269em;margin-right:0.0714em;",[46,384],{"className":385,"style":386},[216],"height:2.5em;",[46,388,392],{"className":389},[221,390,391,224],"reset-size3","size1",[46,393,72],{"className":394,"style":228},[186,190,224],[46,396,233],{"className":397},[232],[46,399,401],{"className":400},[204],[46,402,405],{"className":403,"style":404},[208],"height:0.2819em;",[46,406],{},[46,408,410,413],{"style":409},"top:-3.23em;",[46,411],{"className":412,"style":350},[216],[46,414],{"className":415,"style":417},[416],"frac-line","border-bottom-width:0.04em;",[46,419,421,424],{"style":420},"top:-3.485em;",[46,422],{"className":423,"style":350},[216],[46,425,427],{"className":426},[221,222,223,224],[46,428,430,433,437,440,445,449,452],{"className":429},[186,224],[46,431,97],{"className":432,"style":360},[186,224],[46,434,100],{"className":435,"style":436},[186,190,224],"margin-right:0.0962em;",[46,438,104],{"className":439},[329,224],[46,441,23],{"className":442,"style":444},[186,443,224],"mathbf","margin-right:0.016em;",[46,446,111],{"className":447},[448,224],"mpunct",[46,450,27],{"className":451},[186,190,224],[46,453,116],{"className":454},[455,224],"mclose",[46,457,233],{"className":458},[232],[46,460,462],{"className":461},[204],[46,463,466],{"className":464,"style":465},[208],"height:0.5423em;",[46,467],{},[46,469],{"className":470},[455,330],[46,472],{"className":473,"style":474},[128],"margin-right:2em;",[46,476,27],{"className":477},[186,190],[46,479],{"className":480,"style":246},[128],[46,482,76],{"className":483},[250],[46,485],{"className":486,"style":246},[128],[46,488,490,494,497,500,503],{"className":489},[177],[46,491],{"className":492,"style":493},[181],"height:0.7778em;vertical-align:-0.0833em;",[46,495,27],{"className":496},[186,190],[46,498],{"className":499,"style":304},[128],[46,501,85],{"className":502},[308],[46,504],{"className":505,"style":304},[128],[46,507,509,513,516],{"className":508},[177],[46,510],{"className":511,"style":512},[181],"height:1.355em;vertical-align:-0.345em;",[46,514,88],{"className":515,"style":322},[186,190],[46,517,519,522,603],{"className":518},[186],[46,520],{"className":521},[329,330],[46,523,525],{"className":524},[90],[46,526,528,594],{"className":527},[199,200],[46,529,531,591],{"className":530},[204],[46,532,534,551,559],{"className":533,"style":343},[208],[46,535,536,539],{"style":346},[46,537],{"className":538,"style":350},[216],[46,540,542],{"className":541},[221,222,223,224],[46,543,545,548],{"className":544},[186,224],[46,546,97],{"className":547,"style":360},[186,224],[46,549,27],{"className":550},[186,190,224],[46,552,553,556],{"style":409},[46,554],{"className":555,"style":350},[216],[46,557],{"className":558,"style":417},[416],[46,560,561,564],{"style":420},[46,562],{"className":563,"style":350},[216],[46,565,567],{"className":566},[221,222,223,224],[46,568,570,573,576,579,582,585,588],{"className":569},[186,224],[46,571,97],{"className":572,"style":360},[186,224],[46,574,100],{"className":575,"style":436},[186,190,224],[46,577,104],{"className":578},[329,224],[46,580,23],{"className":581,"style":444},[186,443,224],[46,583,111],{"className":584},[448,224],[46,586,27],{"className":587},[186,190,224],[46,589,116],{"className":590},[455,224],[46,592,233],{"className":593},[232],[46,595,597],{"className":596},[204],[46,598,601],{"className":599,"style":600},[208],"height:0.345em;",[46,602],{},[46,604],{"className":605},[455,330],[11,607,608,609,612,613,615,616,618],{},"Essa é a receita de sempre: pega o gradiente, dá um passo do tamanho ",[21,610,611],{},"alpha"," na direção contrária (atualizando ",[21,614,23],{}," e ",[21,617,27],{}," ao mesmo tempo), repete até parar de melhorar. E as derivadas parciais também têm a mesma cara de sempre:",[11,620,621],{},[46,622,624,864],{"className":623},[49],[46,625,627],{"className":626},[53],[55,628,629],{"xmlns":57},[59,630,631,861],{},[62,632,633,661,663,672,696,698,710,712,726,728,730,743,745,759,761,763,787,789,795,815,817,829,831,843,845,847,859],{},[90,634,635,651],{},[62,636,637,639,641,643,645,647,649],{},[68,638,97],{"mathvariant":96},[68,640,100],{},[74,642,104],{"stretchy":103},[68,644,23],{"mathvariant":107},[74,646,111],{"separator":110},[68,648,27],{},[74,650,116],{"stretchy":103},[62,652,653,655],{},[68,654,97],{"mathvariant":96},[65,656,657,659],{},[68,658,23],{},[68,660,72],{},[74,662,76],{},[90,664,665,669],{},[666,667,668],"mn",{},"1",[68,670,671],{},"m",[673,674,675,678,688],"msubsup",{},[74,676,677],{},"∑",[62,679,680,683,685],{},[68,681,682],{},"i",[74,684,76],{},[666,686,687],{},"0",[62,689,690,692,694],{},[68,691,671],{},[74,693,85],{},[666,695,668],{},[74,697,104],{"stretchy":103},[65,699,700,702],{},[68,701,36],{},[62,703,704,706,708],{},[68,705,23],{"mathvariant":107},[74,707,111],{"separator":110},[68,709,27],{},[74,711,104],{"stretchy":103},[713,714,715,718],"msup",{},[68,716,717],{"mathvariant":107},"x",[62,719,720,722,724],{},[74,721,104],{"stretchy":103},[68,723,682],{},[74,725,116],{"stretchy":103},[74,727,116],{"stretchy":103},[74,729,85],{},[713,731,732,735],{},[68,733,734],{},"y",[62,736,737,739,741],{},[74,738,104],{"stretchy":103},[68,740,682],{},[74,742,116],{"stretchy":103},[74,744,116],{"stretchy":103},[673,746,747,749,751],{},[68,748,717],{},[68,750,72],{},[62,752,753,755,757],{},[74,754,104],{"stretchy":103},[68,756,682],{},[74,758,116],{"stretchy":103},[74,760,111],{"separator":110},[128,762],{"width":130},[90,764,765,781],{},[62,766,767,769,771,773,775,777,779],{},[68,768,97],{"mathvariant":96},[68,770,100],{},[74,772,104],{"stretchy":103},[68,774,23],{"mathvariant":107},[74,776,111],{"separator":110},[68,778,27],{},[74,780,116],{"stretchy":103},[62,782,783,785],{},[68,784,97],{"mathvariant":96},[68,786,27],{},[74,788,76],{},[90,790,791,793],{},[666,792,668],{},[68,794,671],{},[673,796,797,799,807],{},[74,798,677],{},[62,800,801,803,805],{},[68,802,682],{},[74,804,76],{},[666,806,687],{},[62,808,809,811,813],{},[68,810,671],{},[74,812,85],{},[666,814,668],{},[74,816,104],{"stretchy":103},[65,818,819,821],{},[68,820,36],{},[62,822,823,825,827],{},[68,824,23],{"mathvariant":107},[74,826,111],{"separator":110},[68,828,27],{},[74,830,104],{"stretchy":103},[713,832,833,835],{},[68,834,717],{"mathvariant":107},[62,836,837,839,841],{},[74,838,104],{"stretchy":103},[68,840,682],{},[74,842,116],{"stretchy":103},[74,844,116],{"stretchy":103},[74,846,85],{},[713,848,849,851],{},[68,850,734],{},[62,852,853,855,857],{},[74,854,104],{"stretchy":103},[68,856,682],{},[74,858,116],{"stretchy":103},[74,860,116],{"stretchy":103},[166,862,863],{"encoding":168},"\\frac{\\partial J(\\mathbf{w},b)}{\\partial w_j}  = \\frac{1}{m} \\sum\\limits_{i = 0}^{m-1} (f_{\\mathbf{w},b}(\\mathbf{x}^{(i)}) - y^{(i)})x_{j}^{(i)}, \\qquad \\frac{\\partial J(\\mathbf{w},b)}{\\partial b}  = \\frac{1}{m} \\sum\\limits_{i = 0}^{m-1} (f_{\\mathbf{w},b}(\\mathbf{x}^{(i)}) - y^{(i)})",[46,865,867,1008,1281,1504,1759],{"className":866,"ariaHidden":110},[173],[46,868,870,873,999,1002,1005],{"className":869},[177],[46,871],{"className":872,"style":318},[181],[46,874,876,879,996],{"className":875},[186],[46,877],{"className":878},[329,330],[46,880,882],{"className":881},[90],[46,883,885,988],{"className":884},[199,200],[46,886,888,985],{"className":887},[204],[46,889,891,945,953],{"className":890,"style":343},[208],[46,892,893,896],{"style":346},[46,894],{"className":895,"style":350},[216],[46,897,899],{"className":898},[221,222,223,224],[46,900,902,905],{"className":901},[186,224],[46,903,97],{"className":904,"style":360},[186,224],[46,906,908,911],{"className":907},[186,224],[46,909,23],{"className":910,"style":191},[186,190,224],[46,912,914],{"className":913},[195],[46,915,917,937],{"className":916},[199,200],[46,918,920,934],{"className":919},[204],[46,921,923],{"className":922,"style":379},[208],[46,924,925,928],{"style":382},[46,926],{"className":927,"style":386},[216],[46,929,931],{"className":930},[221,390,391,224],[46,932,72],{"className":933,"style":228},[186,190,224],[46,935,233],{"className":936},[232],[46,938,940],{"className":939},[204],[46,941,943],{"className":942,"style":404},[208],[46,944],{},[46,946,947,950],{"style":409},[46,948],{"className":949,"style":350},[216],[46,951],{"className":952,"style":417},[416],[46,954,955,958],{"style":420},[46,956],{"className":957,"style":350},[216],[46,959,961],{"className":960},[221,222,223,224],[46,962,964,967,970,973,976,979,982],{"className":963},[186,224],[46,965,97],{"className":966,"style":360},[186,224],[46,968,100],{"className":969,"style":436},[186,190,224],[46,971,104],{"className":972},[329,224],[46,974,23],{"className":975,"style":444},[186,443,224],[46,977,111],{"className":978},[448,224],[46,980,27],{"className":981},[186,190,224],[46,983,116],{"className":984},[455,224],[46,986,233],{"className":987},[232],[46,989,991],{"className":990},[204],[46,992,994],{"className":993,"style":465},[208],[46,995],{},[46,997],{"className":998},[455,330],[46,1000],{"className":1001,"style":246},[128],[46,1003,76],{"className":1004},[250],[46,1006],{"className":1007,"style":246},[128],[46,1009,1011,1015,1085,1089,1171,1174,1226,1229,1269,1272,1275,1278],{"className":1010},[177],[46,1012],{"className":1013,"style":1014},[181],"height:2.4788em;vertical-align:-0.9777em;",[46,1016,1018,1021,1082],{"className":1017},[186],[46,1019],{"className":1020},[329,330],[46,1022,1024],{"className":1023},[90],[46,1025,1027,1074],{"className":1026},[199,200],[46,1028,1030,1071],{"className":1029},[204],[46,1031,1034,1048,1056],{"className":1032,"style":1033},[208],"height:0.8451em;",[46,1035,1036,1039],{"style":346},[46,1037],{"className":1038,"style":350},[216],[46,1040,1042],{"className":1041},[221,222,223,224],[46,1043,1045],{"className":1044},[186,224],[46,1046,671],{"className":1047},[186,190,224],[46,1049,1050,1053],{"style":409},[46,1051],{"className":1052,"style":350},[216],[46,1054],{"className":1055,"style":417},[416],[46,1057,1059,1062],{"style":1058},"top:-3.394em;",[46,1060],{"className":1061,"style":350},[216],[46,1063,1065],{"className":1064},[221,222,223,224],[46,1066,1068],{"className":1067},[186,224],[46,1069,668],{"className":1070},[186,224],[46,1072,233],{"className":1073},[232],[46,1075,1077],{"className":1076},[204],[46,1078,1080],{"className":1079,"style":600},[208],[46,1081],{},[46,1083],{"className":1084},[455,330],[46,1086],{"className":1087,"style":1088},[128],"margin-right:0.1667em;",[46,1090,1094],{"className":1091},[1092,1093],"mop","op-limits",[46,1095,1097,1162],{"className":1096},[199,200],[46,1098,1100,1159],{"className":1099},[204],[46,1101,1104,1125,1138],{"className":1102,"style":1103},[208],"height:1.5011em;",[46,1105,1107,1110],{"style":1106},"top:-2.1223em;margin-left:0em;",[46,1108],{"className":1109,"style":350},[216],[46,1111,1113],{"className":1112},[221,222,223,224],[46,1114,1116,1119,1122],{"className":1115},[186,224],[46,1117,682],{"className":1118},[186,190,224],[46,1120,76],{"className":1121},[250,224],[46,1123,687],{"className":1124},[186,224],[46,1126,1128,1131],{"style":1127},"top:-3em;",[46,1129],{"className":1130,"style":350},[216],[46,1132,1133],{},[46,1134,677],{"className":1135},[1092,1136,1137],"op-symbol","small-op",[46,1139,1141,1144],{"style":1140},"top:-3.95em;margin-left:0em;",[46,1142],{"className":1143,"style":350},[216],[46,1145,1147],{"className":1146},[221,222,223,224],[46,1148,1150,1153,1156],{"className":1149},[186,224],[46,1151,671],{"className":1152},[186,190,224],[46,1154,85],{"className":1155},[308,224],[46,1157,668],{"className":1158},[186,224],[46,1160,233],{"className":1161},[232],[46,1163,1165],{"className":1164},[204],[46,1166,1169],{"className":1167,"style":1168},[208],"height:0.9777em;",[46,1170],{},[46,1172,104],{"className":1173},[329],[46,1175,1177,1181],{"className":1176},[186],[46,1178,36],{"className":1179,"style":1180},[186,190],"margin-right:0.1076em;",[46,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16,1817,1820,1821,1824,1825,1827],{},"Se você olhar rápido, essa fórmula é indistinguível da de regressão linear. A pegadinha (boa) é que ela ",[30,1811,1812],{},"é igual",", byte a byte. O que muda é o que ",[21,1815,36],{}," significa: na regressão linear, ",[21,1818,1819],{},"f = w·x + b","; aqui, ",[21,1822,1823],{},"f = sigmoid(w·x + b)",". É essa troca, escondida dentro de ",[21,1826,36],{},", que faz o resto da conta se comportar diferente sem precisar reescrever uma linha da fórmula do gradiente.",[39,1829,1831],{"id":1830},"o-código-linha-a-linha","O código, linha a linha",[1833,1834,1839],"pre",{"className":1835,"code":1836,"language":1837,"meta":1838,"style":1838},"language-python shiki shiki-themes github-light github-dark","def compute_gradient_logistic(X, y, w, b):\n    m, n = X.shape\n    dj_dw = np.zeros((n,))\n    dj_db = 0.\n\n    for i in range(m):\n        f_wb_i = sigmoid(np.dot(X[i], w) + b)\n        err_i = f_wb_i - y[i]\n        for j in range(n):\n            dj_dw[j] = dj_dw[j] + err_i * X[i, j]\n        dj_db = dj_db + err_i\n    dj_dw = dj_dw \u002F m\n    dj_db = dj_db \u002F m\n\n    return dj_db, dj_dw\n","python","",[21,1840,1841,1848,1854,1860,1866,1873,1879,1885,1891,1897,1903,1909,1915,1921,1926],{"__ignoreMap":1838},[46,1842,1845],{"class":1843,"line":1844},"line",1,[46,1846,1847],{},"def compute_gradient_logistic(X, y, w, b):\n",[46,1849,1851],{"class":1843,"line":1850},2,[46,1852,1853],{},"    m, n = X.shape\n",[46,1855,1857],{"class":1843,"line":1856},3,[46,1858,1859],{},"    dj_dw = np.zeros((n,))\n",[46,1861,1863],{"class":1843,"line":1862},4,[46,1864,1865],{},"    dj_db = 0.\n",[46,1867,1869],{"class":1843,"line":1868},5,[46,1870,1872],{"emptyLinePlaceholder":1871},true,"\n",[46,1874,1876],{"class":1843,"line":1875},6,[46,1877,1878],{},"    for i in range(m):\n",[46,1880,1882],{"class":1843,"line":1881},7,[46,1883,1884],{},"        f_wb_i = sigmoid(np.dot(X[i], w) + b)\n",[46,1886,1888],{"class":1843,"line":1887},8,[46,1889,1890],{},"        err_i = f_wb_i - y[i]\n",[46,1892,1894],{"class":1843,"line":1893},9,[46,1895,1896],{},"        for j in range(n):\n",[46,1898,1900],{"class":1843,"line":1899},10,[46,1901,1902],{},"            dj_dw[j] = dj_dw[j] + err_i * X[i, j]\n",[46,1904,1906],{"class":1843,"line":1905},11,[46,1907,1908],{},"        dj_db = dj_db + err_i\n",[46,1910,1912],{"class":1843,"line":1911},12,[46,1913,1914],{},"    dj_dw = dj_dw \u002F m\n",[46,1916,1918],{"class":1843,"line":1917},13,[46,1919,1920],{},"    dj_db = dj_db \u002F m\n",[46,1922,1924],{"class":1843,"line":1923},14,[46,1925,1872],{"emptyLinePlaceholder":1871},[46,1927,1929],{"class":1843,"line":1928},15,[46,1930,1931],{},"    return dj_db, dj_dw\n",[1933,1934,1935,1947,1953,1968,1974],"ul",{},[1936,1937,1938,1941,1942,1946],"li",{},[21,1939,1940],{},"f_wb_i = sigmoid(...)",": ",[15,1943,1945],{"href":1944},"\u002Fplaylists\u002Fmachine-learning-specialization\u002Fw3-lab03-decision-boundary","a mesma combinação linear de sempre",", espremida pela sigmoide pra virar probabilidade.",[1936,1948,1949,1952],{},[21,1950,1951],{},"err_i = f_wb_i - y[i]",": o erro desse paciente, previsão menos rótulo real.",[1936,1954,1955,1956,1959,1960,1963,1964,1967],{},"o ",[21,1957,1958],{},"for j"," interno multiplica esse erro por cada variável ",[21,1961,1962],{},"X[i,j]"," e acumula em ",[21,1965,1966],{},"dj_dw[j]",", a mesma mecânica de \"erro vezes entrada\" que a regressão linear já fazia.",[1936,1969,1970,1973],{},[21,1971,1972],{},"dj_db"," acumula só o erro puro, sem multiplicar por nada, também igual à regressão linear.",[1936,1975,1976,1977,1980],{},"no fim, os dois viram média (",[21,1978,1979],{},"\u002F m",").",[11,1982,1983,1984,1987,1988,1991],{},"Conferindo com ",[21,1985,1986],{},"w=[2,3]",", ",[21,1989,1990],{},"b=1"," no dataset de sempre:",[1833,1993,1995],{"className":1835,"code":1994,"language":1837,"meta":1838,"style":1838},"dj_db_tmp, dj_dw_tmp = compute_gradient_logistic(X_tmp, y_tmp, w_tmp, b_tmp)\nprint(f\"dj_db: {dj_db_tmp}\")\nprint(f\"dj_dw: {dj_dw_tmp.tolist()}\")\n",[21,1996,1997,2002,2007],{"__ignoreMap":1838},[46,1998,1999],{"class":1843,"line":1844},[46,2000,2001],{},"dj_db_tmp, dj_dw_tmp = compute_gradient_logistic(X_tmp, y_tmp, w_tmp, b_tmp)\n",[46,2003,2004],{"class":1843,"line":1850},[46,2005,2006],{},"print(f\"dj_db: {dj_db_tmp}\")\n",[46,2008,2009],{"class":1843,"line":1856},[46,2010,2011],{},"print(f\"dj_dw: {dj_dw_tmp.tolist()}\")\n",[2013,2014,2015],"blockquote",{},[11,2016,2017,2020,2021,2024,2025],{},[30,2018,2019],{},"Saída:"," ",[21,2022,2023],{},"dj_db: 0.49861806546328574"," · ",[21,2026,2027],{},"dj_dw: [0.498333393278696, 0.49883942983996693]",[39,2029,2031],{"id":2030},"o-loop-de-fora-esse-sim-é-cópia-e-cola","O loop de fora, esse sim, é cópia e cola",[1833,2033,2035],{"className":1835,"code":2034,"language":1837,"meta":1838,"style":1838},"def gradient_descent(X, y, w_in, b_in, alpha, num_iters):\n    w = copy.deepcopy(w_in)\n    b = b_in\n\n    for i in range(num_iters):\n        dj_db, dj_dw = compute_gradient_logistic(X, y, w, b)\n        w = w - alpha * dj_dw\n        b = b - alpha * dj_db\n\n        if i % math.ceil(num_iters \u002F 10) == 0:\n            print(f\"Iteration {i:4d}: Cost {compute_cost_logistic(X, y, w, b)}\")\n\n    return w, b\n",[21,2036,2037,2042,2047,2052,2056,2061,2066,2071,2076,2080,2085,2090,2094],{"__ignoreMap":1838},[46,2038,2039],{"class":1843,"line":1844},[46,2040,2041],{},"def gradient_descent(X, y, w_in, b_in, alpha, num_iters):\n",[46,2043,2044],{"class":1843,"line":1850},[46,2045,2046],{},"    w = copy.deepcopy(w_in)\n",[46,2048,2049],{"class":1843,"line":1856},[46,2050,2051],{},"    b = b_in\n",[46,2053,2054],{"class":1843,"line":1862},[46,2055,1872],{"emptyLinePlaceholder":1871},[46,2057,2058],{"class":1843,"line":1868},[46,2059,2060],{},"    for i in range(num_iters):\n",[46,2062,2063],{"class":1843,"line":1875},[46,2064,2065],{},"        dj_db, dj_dw = compute_gradient_logistic(X, y, w, b)\n",[46,2067,2068],{"class":1843,"line":1881},[46,2069,2070],{},"        w = w - alpha * dj_dw\n",[46,2072,2073],{"class":1843,"line":1887},[46,2074,2075],{},"        b = b - alpha * dj_db\n",[46,2077,2078],{"class":1843,"line":1893},[46,2079,1872],{"emptyLinePlaceholder":1871},[46,2081,2082],{"class":1843,"line":1899},[46,2083,2084],{},"        if i % math.ceil(num_iters \u002F 10) == 0:\n",[46,2086,2087],{"class":1843,"line":1905},[46,2088,2089],{},"            print(f\"Iteration {i:4d}: Cost {compute_cost_logistic(X, y, w, b)}\")\n",[46,2091,2092],{"class":1843,"line":1911},[46,2093,1872],{"emptyLinePlaceholder":1871},[46,2095,2096],{"class":1843,"line":1917},[46,2097,2098],{},"    return w, b\n",[11,2100,2101,2102,2020,2105,2109,2110,2113,2114,2117,2118,2121],{},"Compara com o ",[21,2103,2104],{},"gradient_descent",[15,2106,2108],{"href":2107},"\u002Fplaylists\u002Fmachine-learning-specialization\u002Flab04-gradient-descent","da Semana 1",": a única diferença de verdade é qual função de custo entra no ",[21,2111,2112],{},"print"," de acompanhamento. O ",[21,2115,2116],{},"for",", a atualização ",[21,2119,2120],{},"w = w - alpha * dj_dw",", tudo igual.",[11,2123,2124,2125,2128],{},"Rodando no dataset de seis pacientes de sempre, ",[21,2126,2127],{},"alpha=0.1",", 10 mil iterações:",[2013,2130,2131,2135,2143],{},[11,2132,2133],{},[30,2134,2019],{},[1833,2136,2141],{"className":2137,"code":2139,"language":2140},[2138],"language-text","Iteration    0: Cost 0.6846\nIteration 1000: Cost 0.1591\nIteration 2000: Cost 0.0846\nIteration 3000: Cost 0.0571\nIteration 4000: Cost 0.0429\nIteration 5000: Cost 0.0343\nIteration 6000: Cost 0.0286\nIteration 7000: Cost 0.0245\nIteration 8000: Cost 0.0214\nIteration 9000: Cost 0.0190\n","text",[21,2142,2139],{"__ignoreMap":1838},[11,2144,2145,1987,2148],{},[21,2146,2147],{},"w: [5.2812, 5.0782]",[21,2149,2150],{},"b: -14.2224",[11,2152,2153,2154,2157,2158,2161],{},"O custo despenca no começo (de ",[21,2155,2156],{},"0.68"," pra ",[21,2159,2160],{},"0.16"," só nas primeiras mil voltas) e depois vai afinando devagar, a mesma curva de \"aprende rápido no início, refina depois\" que já apareceu em toda superfície de custo convexa que treinamos até aqui.",[2163,2164],"classification-boundary-comparison-chart",{":boundaries":2165,":classes":2166,":points":2167,":x-max":2168,":x-min":687,":y-max":2169,":y-min":687,"negative-label":2170,"positive-label":2171,"x-label":2172,"y-label":2173},"[{\"w\":[5.2812,5.0782],\"bias\":-14.2224,\"label\":\"fronteira depois de 10k passos\"}]","[-1,-1,-1,1,1,1]","[[0.5,1.5],[1,1],[1.5,0.5],[3,0.5],[2,2],[1,2.5]]","4","3.5","y=0","y=1","x0","x1",[11,2175,2176,2177,2180],{},"A fronteira separa os seis pacientes sem errar nenhum, exatamente como ",[15,2178,2179],{"href":1944},"a versão treinada à mão que eu já tinha mostrado antes",", só que dessa vez eu vi o gradiente descendente chegar lá passo a passo, em vez de receber os pesos prontos.",[39,2182,2184],{"id":2183},"andando-morro-abaixo-ao-vivo","Andando morro abaixo, ao vivo",[11,2186,2187,2188,2191,2192,2196],{},"O dataset acima tem duas variáveis, então a superfície de custo mora num espaço de 4 dimensões (",[21,2189,2190],{},"w0, w1, b, custo",") que não dá pra desenhar. Pra ver gradiente descendente andando de verdade, volto pra um dataset de uma variável só, os mesmos seis pacientes ",[15,2193,2195],{"href":2194},"\u002Fplaylists\u002Fmachine-learning-specialization\u002Fw3-lab01-classification","do primeiro lab da semana",":",[1833,2198,2200],{"className":1835,"code":2199,"language":1837,"meta":1838,"style":1838},"x_train = np.array([0., 1, 2, 3, 4, 5])\ny_train = np.array([0,  0, 0, 1, 1, 1])\n",[21,2201,2202,2207],{"__ignoreMap":1838},[46,2203,2204],{"class":1843,"line":1844},[46,2205,2206],{},"x_train = np.array([0., 1, 2, 3, 4, 5])\n",[46,2208,2209],{"class":1843,"line":1850},[46,2210,2211],{},"y_train = np.array([0,  0, 0, 1, 1, 1])\n",[11,2213,2214,2215,615,2217,2219,2220,2223,2224,2226],{},"Com só ",[21,2216,23],{},[21,2218,27],{},", o custo logístico vira uma superfície 3D de verdade, a mesma tigela convexa que ",[15,2221,2222],{"href":17},"o lab passado já tinha comparado contra a tentativa \"custo errado\""," (aquela, sim, cheia de vales falsos). Mexe no ",[21,2225,611],{},", clica em \"Rodar\" e repara a bolinha vermelha descendo até o fundo, sempre pelo mesmo caminho que o formato convexo garante:",[2228,2229],"gradient-descent-simulator",{":b-range":2230,":w-range":2231,":x-train":2232,":y-train":2233,"b-label":27,"fn":2234,"w-label":23},"[-14,1]","[-1,7]","[0,1,2,3,4,5]","[0,0,0,1,1,1]","logistic",[11,2236,2237,2238,2241],{},"Repara que os controles são os mesmos que você já usou ",[15,2239,2240],{"href":2107},"nos posts de regressão linear",": alpha, passo a passo, rodar em lote, trocar entre 2D\u002F3D\u002Fparábola. Gradiente descendente não fica mais esperto nem mais burro por trabalhar com um custo diferente, ele só desce a superfície que você entrega pra ele.",[39,2243,2245],{"id":2244},"fechando","Fechando",[2247,2248,2249,2263],"table",{},[2250,2251,2252],"thead",{},[2253,2254,2255,2260],"tr",{},[2256,2257,2259],"th",{"align":2258},"left","O que eu já sabia",[2256,2261,2262],{"align":2258},"O que essa aula assentou",[2264,2265,2266,2283,2291],"tbody",{},[2253,2267,2268,2277],{},[2269,2270,2271,2272,615,2274,2276],"td",{"align":2258},"Gradiente descendente atualiza ",[21,2273,23],{},[21,2275,27],{}," na direção contrária ao gradiente",[2269,2278,2279,2280,2282],{"align":2258},"A fórmula do gradiente é idêntica à da regressão linear, só que ",[21,2281,36],{}," agora é a sigmoide, não a reta",[2253,2284,2285,2288],{},[2269,2286,2287],{"align":2258},"O custo logístico é convexo",[2269,2289,2290],{"align":2258},"Isso garante que gradiente descendente sempre acha o mesmo fundo de vale, não importa de onde começa",[2253,2292,2293,2296],{},[2269,2294,2295],{"align":2258},"Eu via a fronteira treinada pronta",[2269,2297,2298],{"align":2258},"Agora eu vi ela nascer passo a passo, o mesmo loop que já rodou pra regressão linear",[39,2300,2302],{"id":2301},"aplicação-prática","Aplicação Prática",[11,2304,2305,2306,2310,2311,2314,2315,2318],{},"Reaproveito ",[15,2307,2309],{"href":2308},"\u002Fplaylists\u002Fmachine-learning-specialization\u002Fw3-lab02-sigmoid-function","a mesma variável real que a playlist já vem usando"," (",[21,2312,2313],{},"worst concave points",", normalizada por z-score), agora com a amostra de 60 pacientes ",[15,2316,2317],{"href":1944},"do lab da fronteira de decisão",", e rodo o gradiente descendente eu mesmo em vez de receber os pesos prontos:",[2228,2320],{":b-range":2321,":w-range":2322,":x-train":2323,":y-train":2324,"b-label":27,"fn":2234,"w-label":23},"[-3,1]","[-1,6]","[-0.32,-0.488,1.029,-1.323,-1.107,0.028,1.357,-0.482,0.645,2.438,-0.216,0.423,0.592,-1.75,-0.581,-0.006,-0.409,2.699,0.097,2.365,0.635,1.917,-1.357,-0.144,1.455,0.537,-1.09,-0.456,-0.343,-0.464,-0.231,-1.29,-1.608,-0.191,0.42,-0.617,-0.135,-0.45,-1.015,-0.397,-0.473,1.894,0.495,-0.289,2.369,-1.004,2.196,-0.095,0.816,0.888,0.406,-0.785,-0.251,0.497,-1.385,-0.185,0.955,1.487,-0.842,0.986]","[1,0,1,0,0,1,1,0,1,1,0,1,1,0,0,0,0,1,0,1,1,1,0,0,1,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,1,0,1,1,0,0,0,1,0,0,1,1,0,1]",[11,2326,2327,2328,2331,2332,2335,2336,2339,2340,2343,2344,2347,2348,2353,2354,2357,2358,2361,2362,2365],{},"Rodando com ",[21,2329,2330],{},"alpha=0.5",", o custo cai de ",[21,2333,2334],{},"0.6931"," (a pior chute possível, ",[21,2337,2338],{},"-log(0.5)",") pra ",[21,2341,2342],{},"0.2372"," e converge por volta de ",[21,2345,2346],{},"w=4.175, b=-1.094",", valores bem parecidos com ",[15,2349,1955,2350],{"href":2308},[21,2351,2352],{},"w=3.9565, b=-1.1326"," que os labs anteriores citam pro conjunto de treino completo (essa amostra aqui é só 60 pacientes, o modelo cheio viu bem mais gente). Acurácia nessa amostra: ",[21,2355,2356],{},"0.90",", no mesmo patamar dos ",[21,2359,2360],{},"0.91","-",[21,2363,2364],{},"0.93"," que os labs anteriores já reportavam. O mesmo loop que rodou em cima de seis pontos de brinquedo, agora rodando em cima de pacientes reais, sem precisar mudar uma linha de código, só o dado que entra.",[2367,2368,2369],"style",{},"html .default .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .dark .shiki span {color: var(--shiki-dark);background: var(--shiki-dark-bg);font-style: var(--shiki-dark-font-style);font-weight: var(--shiki-dark-font-weight);text-decoration: var(--shiki-dark-text-decoration);}html.dark .shiki span {color: var(--shiki-dark);background: var(--shiki-dark-bg);font-style: var(--shiki-dark-font-style);font-weight: var(--shiki-dark-font-weight);text-decoration: var(--shiki-dark-text-decoration);}",{"title":1838,"searchDepth":1850,"depth":1850,"links":2371},[2372,2373,2374,2375,2376,2377],{"id":41,"depth":1850,"text":42},{"id":1830,"depth":1850,"text":1831},{"id":2030,"depth":1850,"text":2031},{"id":2183,"depth":1850,"text":2184},{"id":2244,"depth":1850,"text":2245},{"id":2301,"depth":1850,"text":2302},null,"2026-09-08","Semana 3, Lab 6: depois de sigmoide, fronteira de decisão, perda e custo logísticos, falta só uma peça pra treinar de verdade. E a notícia boa é que essa peça já existe desde a Semana 1: o mesmo loop de gradiente descendente, só que descendo um morro diferente.","md",{},17,"\u002Fpt\u002Fplaylists\u002Fmachine-learning-specialization\u002Fw3-lab06-gradient-descent","machine-learning-specialization",{"title":6,"description":2380},"published","pt\u002Fplaylists\u002Fmachine-learning-specialization\u002Fw3-lab06-gradient-descent",[2390,2391,2392],"classificacao","regressao-logistica","gradiente-descendente","Qpa6rnoMQ4TDnPzEeqNfsZ2AQUORP4tP9EMlm13SHVA",1788965095726]