[{"data":1,"prerenderedAt":8723},["ShallowReactive",2],{"lang-switch-post-\u002Fplaylists\u002Fmachine-learning-specialization\u002Fw2-lab03-feature-scaling":3,"post-pt-machine-learning-specialization-w2-lab03-feature-scaling":4},"\u002Fen\u002Fplaylists\u002Fmachine-learning-specialization\u002Fw2-lab03-feature-scaling",{"id":5,"title":6,"body":7,"cover":8708,"date":8709,"description":8710,"extension":8711,"meta":8712,"navigation":8466,"order":8485,"path":8713,"playlist":8714,"seo":8715,"status":8716,"stem":8717,"tags":8718,"__hash__":8722},"posts\u002Fpt\u002Fplaylists\u002Fmachine-learning-specialization\u002Fw2-lab03-feature-scaling.md","Normalização de Features e Taxa de Aprendizado",{"type":8,"value":9,"toc":8693},"minimark",[10,18,86,91,98,210,217,220,223,226,229,327,330,334,955,1268,1300,1304,1307,1329,1457,2349,2352,2362,2556,2600,2603,2870,3026,3577,3799,4014,4046,4052,4152,4317,4494,4498,4530,6096,6275,6308,6311,6318,6322,6451,6457,6722,6789,6793,6796,6811,6817,6824,6899,7790,7956,7960,7976,7988,7993,7997,8002,8005,8008,8011,8016,8022,8027,8035,8038,8093,8097,8192,8195,8274,8278,8452,8506,8516,8680,8683,8686,8689],[11,12,13],"p",{},[14,15],"img",{"alt":16,"src":17},"Meme de uma criança confusa com as mãos abertas e a legenda \"como assim eu preciso de scaling?\"","\u002Fimages\u002Fposts\u002Fmachine-learning-specialization\u002Fw2-lab03-feature-scaling\u002Fmeme-feature-scaling.jpeg",[11,19,20,21,25,26,31,32,36,37,85],{},"Peguei o lab opcional do Curso 1, Semana 2 (o ",[22,23,24],"code",{},"C1_W2_Lab03",") esperando só mais um post de continuação, e acabei achando o assunto que mais me irritou até agora na especialização: o gradiente descendente que eu construí no ",[27,28,30],"a",{"href":29},"\u002Fplaylists\u002Fmachine-learning-specialization\u002Flab04-gradient-descent","post passado",", o mesmo algoritmo, sem nenhuma mudança de código, simplesmente ",[33,34,35],"strong",{},"para de funcionar"," dependendo da unidade que eu uso pra medir uma feature. Troquei metro por centímetro e o algoritmo divergiu. Isso não é bug, é matemática, e depois desse post você nunca mais vai escolher um ",[38,39,42,65],"span",{"className":40},[41],"katex",[38,43,46],{"className":44},[45],"katex-mathml",[47,48,50],"math",{"xmlns":49},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[51,52,53,60],"semantics",{},[54,55,56],"mrow",{},[57,58,59],"mi",{},"α",[61,62,64],"annotation",{"encoding":63},"application\u002Fx-tex","\\alpha",[38,66,70],{"className":67,"ariaHidden":69},[68],"katex-html","true",[38,71,74,79],{"className":72},[73],"base",[38,75],{"className":76,"style":78},[77],"strut","height:0.4306em;",[38,80,59],{"className":81,"style":84},[82,83],"mord","mathnormal","margin-right:0.0037em;"," no escuro sem entender por quê.",[87,88,90],"h2",{"id":89},"o-problema-4-features-escalas-bem-diferentes","O problema: 4 features, escalas bem diferentes",[11,92,93,94,97],{},"O notebook original usa um dataset de 100 casas (derivado do Ames Housing dataset, o mesmo usado no curso) com 4 features: tamanho em pés², número de quartos, número de andares e idade em anos. A meta é prever o preço de uma casa de ",[33,95,96],{},"1200 pés², 3 quartos, 1 andar, 40 anos",".",[99,100,101,128],"table",{},[102,103,104],"thead",{},[105,106,107,112,116,119,122,125],"tr",{},[108,109,111],"th",{"align":110},"left","Feature",[108,113,115],{"align":114},"right","Mínimo",[108,117,118],{"align":114},"Máximo",[108,120,121],{"align":114},"Média",[108,123,124],{"align":114},"Desvio-padrão",[108,126,127],{"align":114},"Amplitude",[129,130,131,152,171,190],"tbody",{},[105,132,133,137,140,143,146,149],{},[134,135,136],"td",{"align":110},"tamanho (pés²)",[134,138,139],{"align":114},"788",[134,141,142],{"align":114},"3194",[134,144,145],{"align":114},"1413.7",[134,147,148],{"align":114},"412.2",[134,150,151],{"align":114},"2406",[105,153,154,157,160,163,166,169],{},[134,155,156],{"align":110},"quartos",[134,158,159],{"align":114},"0",[134,161,162],{"align":114},"4",[134,164,165],{"align":114},"2.7",[134,167,168],{"align":114},"0.7",[134,170,162],{"align":114},[105,172,173,176,179,182,185,188],{},[134,174,175],{"align":110},"andares",[134,177,178],{"align":114},"1",[134,180,181],{"align":114},"2",[134,183,184],{"align":114},"1.4",[134,186,187],{"align":114},"0.5",[134,189,178],{"align":114},[105,191,192,195,198,201,204,207],{},[134,193,194],{"align":110},"idade (anos)",[134,196,197],{"align":114},"12",[134,199,200],{"align":114},"107",[134,202,203],{"align":114},"38.6",[134,205,206],{"align":114},"25.8",[134,208,209],{"align":114},"95",[11,211,212,213,216],{},"Reparei nesse número assim que rodei a célula: a feature de maior amplitude (tamanho) é ",[33,214,215],{},"2406 vezes"," maior que a de menor amplitude (andares). Guardei esse número, porque ele é a raiz de tudo que vem a seguir.",[11,218,219],{},"Antes de sair modelando, olhei cada feature contra o preço, uma de cada vez:",[221,222],"feature-scaling-overview",{},[11,224,225],{},"Tamanho carrega sinal de verdade (a nuvem de pontos sobe claramente da esquerda pra direita). Quartos e andares são discretos e ruidosos, dá pra ver casas de 2 quartos custando mais que casas de 3. Idade puxa o preço pra baixo, mas com bastante dispersão. Isso não significa que quartos e andares são inúteis, só que sozinhas elas explicam pouco, o valor delas aparece quando entram no modelo junto com as outras (o coeficiente negativo de quartos que eu calculo mais na frente é exatamente esse tipo de efeito escondido).",[11,227,228],{},"Calculei também a correlação entre cada par de features (e entre cada feature e o preço):",[99,230,231,250],{},[102,232,233],{},[105,234,235,237,240,242,244,247],{},[108,236],{"align":110},[108,238,239],{"align":114},"tamanho",[108,241,156],{"align":114},[108,243,175],{"align":114},[108,245,246],{"align":114},"idade",[108,248,249],{"align":114},"preço",[129,251,252,273,292,310],{},[105,253,254,258,261,264,267,270],{},[134,255,256],{"align":110},[33,257,239],{},[134,259,260],{"align":114},"1.00",[134,262,263],{"align":114},"0.56",[134,265,266],{"align":114},"0.60",[134,268,269],{"align":114},"-0.27",[134,271,272],{"align":114},"0.86",[105,274,275,279,281,283,286,289],{},[134,276,277],{"align":110},[33,278,156],{},[134,280,263],{"align":114},[134,282,260],{"align":114},[134,284,285],{"align":114},"0.38",[134,287,288],{"align":114},"-0.06",[134,290,291],{"align":114},"0.29",[105,293,294,298,300,302,304,307],{},[134,295,296],{"align":110},[33,297,175],{},[134,299,266],{"align":114},[134,301,285],{"align":114},[134,303,260],{"align":114},[134,305,306],{"align":114},"-0.20",[134,308,309],{"align":114},"0.32",[105,311,312,316,318,320,322,324],{},[134,313,314],{"align":110},[33,315,246],{},[134,317,269],{"align":114},[134,319,288],{"align":114},[134,321,306],{"align":114},[134,323,260],{"align":114},[134,325,326],{"align":114},"-0.58",[11,328,329],{},"Tamanho é de longe a feature mais correlacionada com o preço (0.86), o que bate com o gráfico de cima. E repara que tamanho e quartos não são independentes (0.56), casas maiores tendem a ter mais quartos, o que é meio óbvio quando você para pra pensar, mas é sempre bom confirmar com número em vez de achismo.",[87,331,333],{"id":332},"recapitulando-rápido-com-uma-feature-só","Recapitulando rápido, com uma feature só",[11,335,336,337,527,528,950,951,954],{},"Pra manter a base que a gente já construiu, vou simplificar pra 1 feature só (tamanho) nessa primeira parte, exatamente como nos posts anteriores: ",[38,338,340,392],{"className":339},[41],[38,341,343],{"className":342},[45],[47,344,345],{"xmlns":49},[51,346,347,389],{},[54,348,349,367,371,374,377,380,382,384,387],{},[350,351,352,355],"msub",{},[57,353,354],{},"f",[54,356,357,360,364],{},[57,358,359],{},"w",[361,362,363],"mo",{"separator":69},",",[57,365,366],{},"b",[361,368,370],{"stretchy":369},"false","(",[57,372,373],{},"x",[361,375,376],{"stretchy":369},")",[361,378,379],{},"=",[57,381,359],{},[57,383,373],{},[361,385,386],{},"+",[57,388,366],{},[61,390,391],{"encoding":63},"f_{w,b}(x) = wx + b",[38,393,395,493,517],{"className":394,"ariaHidden":69},[68],[38,396,398,402,470,474,477,481,486,490],{"className":397},[73],[38,399],{"className":400,"style":401},[77],"height:1.0361em;vertical-align:-0.2861em;",[38,403,405,409],{"className":404},[82],[38,406,354],{"className":407,"style":408},[82,83],"margin-right:0.1076em;",[38,410,413],{"className":411},[412],"msupsub",[38,414,418,461],{"className":415},[416,417],"vlist-t","vlist-t2",[38,419,422,456],{"className":420},[421],"vlist-r",[38,423,427],{"className":424,"style":426},[425],"vlist","height:0.3361em;",[38,428,430,435],{"style":429},"top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;",[38,431],{"className":432,"style":434},[433],"pstrut","height:2.7em;",[38,436,442],{"className":437},[438,439,440,441],"sizing","reset-size6","size3","mtight",[38,443,445,449,453],{"className":444},[82,441],[38,446,359],{"className":447,"style":448},[82,83,441],"margin-right:0.0269em;",[38,450,363],{"className":451},[452,441],"mpunct",[38,454,366],{"className":455},[82,83,441],[38,457,460],{"className":458},[459],"vlist-s","​",[38,462,464],{"className":463},[421],[38,465,468],{"className":466,"style":467},[425],"height:0.2861em;",[38,469],{},[38,471,370],{"className":472},[473],"mopen",[38,475,373],{"className":476},[82,83],[38,478,376],{"className":479},[480],"mclose",[38,482],{"className":483,"style":485},[484],"mspace","margin-right:0.2778em;",[38,487,379],{"className":488},[489],"mrel",[38,491],{"className":492,"style":485},[484],[38,494,496,500,503,506,510,514],{"className":495},[73],[38,497],{"className":498,"style":499},[77],"height:0.6667em;vertical-align:-0.0833em;",[38,501,359],{"className":502,"style":448},[82,83],[38,504,373],{"className":505},[82,83],[38,507],{"className":508,"style":509},[484],"margin-right:0.2222em;",[38,511,386],{"className":512},[513],"mbin",[38,515],{"className":516,"style":509},[484],[38,518,520,524],{"className":519},[73],[38,521],{"className":522,"style":523},[77],"height:0.6944em;",[38,525,366],{"className":526},[82,83],", com custo ",[38,529,531,628],{"className":530},[41],[38,532,534],{"className":533},[45],[47,535,536],{"xmlns":49},[51,537,538,625],{},[54,539,540,543,545,547,549,551,553,555,568,571,573,585,587,601,603,606,619],{},[57,541,542],{},"J",[361,544,370],{"stretchy":369},[57,546,359],{},[361,548,363],{"separator":69},[57,550,366],{},[361,552,376],{"stretchy":369},[361,554,379],{},[556,557,558,561],"mfrac",{},[559,560,178],"mn",{},[54,562,563,565],{},[559,564,181],{},[57,566,567],{},"m",[361,569,570],{},"∑",[361,572,370],{"stretchy":369},[350,574,575,577],{},[57,576,354],{},[54,578,579,581,583],{},[57,580,359],{},[361,582,363],{"separator":69},[57,584,366],{},[361,586,370],{"stretchy":369},[588,589,590,592],"msup",{},[57,591,373],{},[54,593,594,596,599],{},[361,595,370],{"stretchy":369},[57,597,598],{},"i",[361,600,376],{"stretchy":369},[361,602,376],{"stretchy":369},[361,604,605],{},"−",[588,607,608,611],{},[57,609,610],{},"y",[54,612,613,615,617],{},[361,614,370],{"stretchy":369},[57,616,598],{},[361,618,376],{"stretchy":369},[588,620,621,623],{},[361,622,376],{"stretchy":369},[559,624,181],{},[61,626,627],{"encoding":63},"J(w,b) = \\frac{1}{2m}\\sum(f_{w,b}(x^{(i)}) - y^{(i)})^2",[38,629,631,670,874],{"className":630,"ariaHidden":69},[68],[38,632,634,638,642,645,648,651,655,658,661,664,667],{"className":633},[73],[38,635],{"className":636,"style":637},[77],"height:1em;vertical-align:-0.25em;",[38,639,542],{"className":640,"style":641},[82,83],"margin-right:0.0962em;",[38,643,370],{"className":644},[473],[38,646,359],{"className":647,"style":448},[82,83],[38,649,363],{"className":650},[452],[38,652],{"className":653,"style":654},[484],"margin-right:0.1667em;",[38,656,366],{"className":657},[82,83],[38,659,376],{"className":660},[480],[38,662],{"className":663,"style":485},[484],[38,665,379],{"className":666},[489],[38,668],{"className":669,"style":485},[484],[38,671,673,677,757,760,767,770,819,822,862,865,868,871],{"className":672},[73],[38,674],{"className":675,"style":676},[77],"height:1.233em;vertical-align:-0.345em;",[38,678,680,684,754],{"className":679},[82],[38,681],{"className":682},[473,683],"nulldelimiter",[38,685,687],{"className":686},[556],[38,688,690,745],{"className":689},[416,417],[38,691,693,742],{"className":692},[421],[38,694,697,716,727],{"className":695,"style":696},[425],"height:0.8451em;",[38,698,700,704],{"style":699},"top:-2.655em;",[38,701],{"className":702,"style":703},[433],"height:3em;",[38,705,707],{"className":706},[438,439,440,441],[38,708,710,713],{"className":709},[82,441],[38,711,181],{"className":712},[82,441],[38,714,567],{"className":715},[82,83,441],[38,717,719,722],{"style":718},"top:-3.23em;",[38,720],{"className":721,"style":703},[433],[38,723],{"className":724,"style":726},[725],"frac-line","border-bottom-width:0.04em;",[38,728,730,733],{"style":729},"top:-3.394em;",[38,731],{"className":732,"style":703},[433],[38,734,736],{"className":735},[438,439,440,441],[38,737,739],{"className":738},[82,441],[38,740,178],{"className":741},[82,441],[38,743,460],{"className":744},[459],[38,746,748],{"className":747},[421],[38,749,752],{"className":750,"style":751},[425],"height:0.345em;",[38,753],{},[38,755],{"className":756},[480,683],[38,758],{"className":759,"style":654},[484],[38,761,570],{"className":762,"style":766},[763,764,765],"mop","op-symbol","small-op","position:relative;top:0em;",[38,768,370],{"className":769},[473],[38,771,773,776],{"className":772},[82],[38,774,354],{"className":775,"style":408},[82,83],[38,777,779],{"className":778},[412],[38,780,782,811],{"className":781},[416,417],[38,783,785,808],{"className":784},[421],[38,786,788],{"className":787,"style":426},[425],[38,789,790,793],{"style":429},[38,791],{"className":792,"style":434},[433],[38,794,796],{"className":795},[438,439,440,441],[38,797,799,802,805],{"className":798},[82,441],[38,800,359],{"className":801,"style":448},[82,83,441],[38,803,363],{"className":804},[452,441],[38,806,366],{"className":807},[82,83,441],[38,809,460],{"className":810},[459],[38,812,814],{"className":813},[421],[38,815,817],{"className":816,"style":467},[425],[38,818],{},[38,820,370],{"className":821},[473],[38,823,825,828],{"className":824},[82],[38,826,373],{"className":827},[82,83],[38,829,831],{"className":830},[412],[38,832,834],{"className":833},[416],[38,835,837],{"className":836},[421],[38,838,841],{"className":839,"style":840},[425],"height:0.888em;",[38,842,844,847],{"style":843},"top:-3.063em;margin-right:0.05em;",[38,845],{"className":846,"style":434},[433],[38,848,850],{"className":849},[438,439,440,441],[38,851,853,856,859],{"className":852},[82,441],[38,854,370],{"className":855},[473,441],[38,857,598],{"className":858},[82,83,441],[38,860,376],{"className":861},[480,441],[38,863,376],{"className":864},[480],[38,866],{"className":867,"style":509},[484],[38,869,605],{"className":870},[513],[38,872],{"className":873,"style":509},[484],[38,875,877,881,920],{"className":876},[73],[38,878],{"className":879,"style":880},[77],"height:1.138em;vertical-align:-0.25em;",[38,882,884,888],{"className":883},[82],[38,885,610],{"className":886,"style":887},[82,83],"margin-right:0.0359em;",[38,889,891],{"className":890},[412],[38,892,894],{"className":893},[416],[38,895,897],{"className":896},[421],[38,898,900],{"className":899,"style":840},[425],[38,901,902,905],{"style":843},[38,903],{"className":904,"style":434},[433],[38,906,908],{"className":907},[438,439,440,441],[38,909,911,914,917],{"className":910},[82,441],[38,912,370],{"className":913},[473,441],[38,915,598],{"className":916},[82,83,441],[38,918,376],{"className":919},[480,441],[38,921,923,926],{"className":922},[480],[38,924,376],{"className":925},[480],[38,927,929],{"className":928},[412],[38,930,932],{"className":931},[416],[38,933,935],{"className":934},[421],[38,936,939],{"className":937,"style":938},[425],"height:0.8141em;",[38,940,941,944],{"style":843},[38,942],{"className":943,"style":434},[433],[38,945,947],{"className":946},[438,439,440,441],[38,948,181],{"className":949},[82,441],". O gradiente descendente que eu montei ",[27,952,953],{"href":29},"dois posts atrás"," não muda uma linha:",[11,956,957],{},[38,958,960,1025],{"className":959},[41],[38,961,963],{"className":962},[45],[47,964,965],{"xmlns":49},[51,966,967,1022],{},[54,968,969,971,973,975,977,979,995,998,1000,1002,1004,1006,1008],{},[57,970,359],{},[361,972,379],{},[57,974,359],{},[361,976,605],{},[57,978,59],{},[556,980,981,989],{},[54,982,983,987],{},[57,984,986],{"mathvariant":985},"normal","∂",[57,988,542],{},[54,990,991,993],{},[57,992,986],{"mathvariant":985},[57,994,359],{},[484,996],{"width":997},"2em",[57,999,366],{},[361,1001,379],{},[57,1003,366],{},[361,1005,605],{},[57,1007,59],{},[556,1009,1010,1016],{},[54,1011,1012,1014],{},[57,1013,986],{"mathvariant":985},[57,1015,542],{},[54,1017,1018,1020],{},[57,1019,986],{"mathvariant":985},[57,1021,366],{},[61,1023,1024],{"encoding":63},"w = w - \\alpha \\frac{\\partial J}{\\partial w} \\qquad b = b - \\alpha \\frac{\\partial J}{\\partial b}",[38,1026,1028,1046,1064,1166,1185],{"className":1027,"ariaHidden":69},[68],[38,1029,1031,1034,1037,1040,1043],{"className":1030},[73],[38,1032],{"className":1033,"style":78},[77],[38,1035,359],{"className":1036,"style":448},[82,83],[38,1038],{"className":1039,"style":485},[484],[38,1041,379],{"className":1042},[489],[38,1044],{"className":1045,"style":485},[484],[38,1047,1049,1052,1055,1058,1061],{"className":1048},[73],[38,1050],{"className":1051,"style":499},[77],[38,1053,359],{"className":1054,"style":448},[82,83],[38,1056],{"className":1057,"style":509},[484],[38,1059,605],{"className":1060},[513],[38,1062],{"className":1063,"style":509},[484],[38,1065,1067,1071,1074,1150,1154,1157,1160,1163],{"className":1066},[73],[38,1068],{"className":1069,"style":1070},[77],"height:1.2251em;vertical-align:-0.345em;",[38,1072,59],{"className":1073,"style":84},[82,83],[38,1075,1077,1080,1147],{"className":1076},[82],[38,1078],{"className":1079},[473,683],[38,1081,1083],{"className":1082},[556],[38,1084,1086,1139],{"className":1085},[416,417],[38,1087,1089,1136],{"className":1088},[421],[38,1090,1093,1111,1119],{"className":1091,"style":1092},[425],"height:0.8801em;",[38,1094,1095,1098],{"style":699},[38,1096],{"className":1097,"style":703},[433],[38,1099,1101],{"className":1100},[438,439,440,441],[38,1102,1104,1108],{"className":1103},[82,441],[38,1105,986],{"className":1106,"style":1107},[82,441],"margin-right:0.0556em;",[38,1109,359],{"className":1110,"style":448},[82,83,441],[38,1112,1113,1116],{"style":718},[38,1114],{"className":1115,"style":703},[433],[38,1117],{"className":1118,"style":726},[725],[38,1120,1121,1124],{"style":729},[38,1122],{"className":1123,"style":703},[433],[38,1125,1127],{"className":1126},[438,439,440,441],[38,1128,1130,1133],{"className":1129},[82,441],[38,1131,986],{"className":1132,"style":1107},[82,441],[38,1134,542],{"className":1135,"style":641},[82,83,441],[38,1137,460],{"className":1138},[459],[38,1140,1142],{"className":1141},[421],[38,1143,1145],{"className":1144,"style":751},[425],[38,1146],{},[38,1148],{"className":1149},[480,683],[38,1151],{"className":1152,"style":1153},[484],"margin-right:2em;",[38,1155,366],{"className":1156},[82,83],[38,1158],{"className":1159,"style":485},[484],[38,1161,379],{"className":1162},[489],[38,1164],{"className":1165,"style":485},[484],[38,1167,1169,1173,1176,1179,1182],{"className":1168},[73],[38,1170],{"className":1171,"style":1172},[77],"height:0.7778em;vertical-align:-0.0833em;",[38,1174,366],{"className":1175},[82,83],[38,1177],{"className":1178,"style":509},[484],[38,1180,605],{"className":1181},[513],[38,1183],{"className":1184,"style":509},[484],[38,1186,1188,1191,1194],{"className":1187},[73],[38,1189],{"className":1190,"style":1070},[77],[38,1192,59],{"className":1193,"style":84},[82,83],[38,1195,1197,1200,1265],{"className":1196},[82],[38,1198],{"className":1199},[473,683],[38,1201,1203],{"className":1202},[556],[38,1204,1206,1257],{"className":1205},[416,417],[38,1207,1209,1254],{"className":1208},[421],[38,1210,1212,1229,1237],{"className":1211,"style":1092},[425],[38,1213,1214,1217],{"style":699},[38,1215],{"className":1216,"style":703},[433],[38,1218,1220],{"className":1219},[438,439,440,441],[38,1221,1223,1226],{"className":1222},[82,441],[38,1224,986],{"className":1225,"style":1107},[82,441],[38,1227,366],{"className":1228},[82,83,441],[38,1230,1231,1234],{"style":718},[38,1232],{"className":1233,"style":703},[433],[38,1235],{"className":1236,"style":726},[725],[38,1238,1239,1242],{"style":729},[38,1240],{"className":1241,"style":703},[433],[38,1243,1245],{"className":1244},[438,439,440,441],[38,1246,1248,1251],{"className":1247},[82,441],[38,1249,986],{"className":1250,"style":1107},[82,441],[38,1252,542],{"className":1253,"style":641},[82,83,441],[38,1255,460],{"className":1256},[459],[38,1258,1260],{"className":1259},[421],[38,1261,1263],{"className":1262,"style":751},[425],[38,1264],{},[38,1266],{"className":1267},[480,683],[11,1269,1270,1271,1299],{},"O notebook de verdade roda isso com as 4 features ao mesmo tempo (um ",[38,1272,1274,1287],{"className":1273},[41],[38,1275,1277],{"className":1276},[45],[47,1278,1279],{"xmlns":49},[51,1280,1281,1285],{},[54,1282,1283],{},[57,1284,359],{},[61,1286,359],{"encoding":63},[38,1288,1290],{"className":1289,"ariaHidden":69},[68],[38,1291,1293,1296],{"className":1292},[73],[38,1294],{"className":1295,"style":78},[77],[38,1297,359],{"className":1298,"style":448},[82,83]," pra cada), mas a lição central aparece inteira já com uma feature só, e fica bem mais fácil de visualizar num gráfico 2D. Guarda essa simplificação na cabeça, ela volta lá na frente quando eu mostrar o resultado com as 4 features de verdade.",[87,1301,1303],{"id":1302},"o-gradiente-descendente-diverge-sozinho","O gradiente descendente diverge sozinho",[11,1305,1306],{},"Peguei 8 casas reais do dataset do notebook (tamanho em pés², preço em milhares de dólares) pra testar isso na mão:",[1308,1309,1314],"pre",{"className":1310,"code":1311,"language":1312,"meta":1313,"style":1313},"language-python shiki shiki-themes github-light github-dark","x_train = [952, 1244, 1947, 1725, 1959, 1314, 864, 1836]\ny_train = [271.5, 300, 509.8, 394, 540, 415, 230, 560]\n","python","",[22,1315,1316,1323],{"__ignoreMap":1313},[38,1317,1320],{"class":1318,"line":1319},"line",1,[38,1321,1322],{},"x_train = [952, 1244, 1947, 1725, 1959, 1314, 864, 1836]\n",[38,1324,1326],{"class":1318,"line":1325},2,[38,1327,1328],{},"y_train = [271.5, 300, 509.8, 394, 540, 415, 230, 560]\n",[11,1330,1331,1332,1335,1336,1338,1339,1367,1368,1456],{},"Rodei o mesmo ",[22,1333,1334],{},"gradient_descent"," do ",[27,1337,30],{"href":29}," com três valores de ",[38,1340,1342,1355],{"className":1341},[41],[38,1343,1345],{"className":1344},[45],[47,1346,1347],{"xmlns":49},[51,1348,1349,1353],{},[54,1350,1351],{},[57,1352,59],{},[61,1354,64],{"encoding":63},[38,1356,1358],{"className":1357,"ariaHidden":69},[68],[38,1359,1361,1364],{"className":1360},[73],[38,1362],{"className":1363,"style":78},[77],[38,1365,59],{"className":1366,"style":84},[82,83]," diferentes, só 10 iterações, começando em ",[38,1369,1371,1397],{"className":1370},[41],[38,1372,1374],{"className":1373},[45],[47,1375,1376],{"xmlns":49},[51,1377,1378,1394],{},[54,1379,1380,1382,1384,1386,1388,1390,1392],{},[57,1381,359],{},[361,1383,379],{},[559,1385,159],{},[361,1387,363],{"separator":69},[57,1389,366],{},[361,1391,379],{},[559,1393,159],{},[61,1395,1396],{"encoding":63},"w=0, b=0",[38,1398,1400,1418,1446],{"className":1399,"ariaHidden":69},[68],[38,1401,1403,1406,1409,1412,1415],{"className":1402},[73],[38,1404],{"className":1405,"style":78},[77],[38,1407,359],{"className":1408,"style":448},[82,83],[38,1410],{"className":1411,"style":485},[484],[38,1413,379],{"className":1414},[489],[38,1416],{"className":1417,"style":485},[484],[38,1419,1421,1425,1428,1431,1434,1437,1440,1443],{"className":1420},[73],[38,1422],{"className":1423,"style":1424},[77],"height:0.8889em;vertical-align:-0.1944em;",[38,1426,159],{"className":1427},[82],[38,1429,363],{"className":1430},[452],[38,1432],{"className":1433,"style":654},[484],[38,1435,366],{"className":1436},[82,83],[38,1438],{"className":1439,"style":485},[484],[38,1441,379],{"className":1442},[489],[38,1444],{"className":1445,"style":485},[484],[38,1447,1449,1453],{"className":1448},[73],[38,1450],{"className":1451,"style":1452},[77],"height:0.6444em;",[38,1454,159],{"className":1455},[82],":",[1458,1459,1460,1763,2057],"blockquote",{},[11,1461,1462,1586,1587,1672,1673,1759,1760,1762],{},[33,1463,1464,1465,1456],{},"Saída com ",[38,1466,1468,1502],{"className":1467},[41],[38,1469,1471],{"className":1470},[45],[47,1472,1473],{"xmlns":49},[51,1474,1475,1499],{},[54,1476,1477,1479,1481,1484,1487],{},[57,1478,59],{},[361,1480,379],{},[559,1482,1483],{},"9",[361,1485,1486],{},"×",[588,1488,1489,1492],{},[559,1490,1491],{},"10",[54,1493,1494,1496],{},[361,1495,605],{},[559,1497,1498],{},"7",[61,1500,1501],{"encoding":63},"\\alpha = 9\\times10^{-7}",[38,1503,1505,1523,1542],{"className":1504,"ariaHidden":69},[68],[38,1506,1508,1511,1514,1517,1520],{"className":1507},[73],[38,1509],{"className":1510,"style":78},[77],[38,1512,59],{"className":1513,"style":84},[82,83],[38,1515],{"className":1516,"style":485},[484],[38,1518,379],{"className":1519},[489],[38,1521],{"className":1522,"style":485},[484],[38,1524,1526,1530,1533,1536,1539],{"className":1525},[73],[38,1527],{"className":1528,"style":1529},[77],"height:0.7278em;vertical-align:-0.0833em;",[38,1531,1483],{"className":1532},[82],[38,1534],{"className":1535,"style":509},[484],[38,1537,1486],{"className":1538},[513],[38,1540],{"className":1541,"style":509},[484],[38,1543,1545,1548,1551],{"className":1544},[73],[38,1546],{"className":1547,"style":938},[77],[38,1549,178],{"className":1550},[82],[38,1552,1554,1557],{"className":1553},[82],[38,1555,159],{"className":1556},[82],[38,1558,1560],{"className":1559},[412],[38,1561,1563],{"className":1562},[416],[38,1564,1566],{"className":1565},[421],[38,1567,1569],{"className":1568,"style":938},[425],[38,1570,1571,1574],{"style":843},[38,1572],{"className":1573,"style":434},[433],[38,1575,1577],{"className":1576},[438,439,440,441],[38,1578,1580,1583],{"className":1579},[82,441],[38,1581,605],{"className":1582},[82,441],[38,1584,1498],{"className":1585},[82,441]," custo sobe a cada iteração, de ",[38,1588,1590,1613],{"className":1589},[41],[38,1591,1593],{"className":1592},[45],[47,1594,1595],{"xmlns":49},[51,1596,1597,1610],{},[54,1598,1599,1602,1604],{},[559,1600,1601],{},"8.8",[361,1603,1486],{},[588,1605,1606,1608],{},[559,1607,1491],{},[559,1609,162],{},[61,1611,1612],{"encoding":63},"8.8\\times10^4",[38,1614,1616,1634],{"className":1615,"ariaHidden":69},[68],[38,1617,1619,1622,1625,1628,1631],{"className":1618},[73],[38,1620],{"className":1621,"style":1529},[77],[38,1623,1601],{"className":1624},[82],[38,1626],{"className":1627,"style":509},[484],[38,1629,1486],{"className":1630},[513],[38,1632],{"className":1633,"style":509},[484],[38,1635,1637,1640,1643],{"className":1636},[73],[38,1638],{"className":1639,"style":938},[77],[38,1641,178],{"className":1642},[82],[38,1644,1646,1649],{"className":1645},[82],[38,1647,159],{"className":1648},[82],[38,1650,1652],{"className":1651},[412],[38,1653,1655],{"className":1654},[416],[38,1656,1658],{"className":1657},[421],[38,1659,1661],{"className":1660,"style":938},[425],[38,1662,1663,1666],{"style":843},[38,1664],{"className":1665,"style":434},[433],[38,1667,1669],{"className":1668},[438,439,440,441],[38,1670,162],{"className":1671},[82,441]," pra ",[38,1674,1676,1700],{"className":1675},[41],[38,1677,1679],{"className":1678},[45],[47,1680,1681],{"xmlns":49},[51,1682,1683,1697],{},[54,1684,1685,1688,1690],{},[559,1686,1687],{},"9.4",[361,1689,1486],{},[588,1691,1692,1694],{},[559,1693,1491],{},[559,1695,1696],{},"5",[61,1698,1699],{"encoding":63},"9.4\\times10^5",[38,1701,1703,1721],{"className":1702,"ariaHidden":69},[68],[38,1704,1706,1709,1712,1715,1718],{"className":1705},[73],[38,1707],{"className":1708,"style":1529},[77],[38,1710,1687],{"className":1711},[82],[38,1713],{"className":1714,"style":509},[484],[38,1716,1486],{"className":1717},[513],[38,1719],{"className":1720,"style":509},[484],[38,1722,1724,1727,1730],{"className":1723},[73],[38,1725],{"className":1726,"style":938},[77],[38,1728,178],{"className":1729},[82],[38,1731,1733,1736],{"className":1732},[82],[38,1734,159],{"className":1735},[82],[38,1737,1739],{"className":1738},[412],[38,1740,1742],{"className":1741},[416],[38,1743,1745],{"className":1744},[421],[38,1746,1748],{"className":1747,"style":938},[425],[38,1749,1750,1753],{"style":843},[38,1751],{"className":1752,"style":434},[433],[38,1754,1756],{"className":1755},[438,439,440,441],[38,1757,1696],{"className":1758},[82,441],". ",[22,1761,359],{}," troca de sinal toda hora.",[11,1764,1765,1884,1885,1672,1968,2053,2054,2056],{},[33,1766,1464,1767,1456],{},[38,1768,1770,1801],{"className":1769},[41],[38,1771,1773],{"className":1772},[45],[47,1774,1775],{"xmlns":49},[51,1776,1777,1798],{},[54,1778,1779,1781,1783,1786,1788],{},[57,1780,59],{},[361,1782,379],{},[559,1784,1785],{},"8",[361,1787,1486],{},[588,1789,1790,1792],{},[559,1791,1491],{},[54,1793,1794,1796],{},[361,1795,605],{},[559,1797,1498],{},[61,1799,1800],{"encoding":63},"\\alpha = 8\\times10^{-7}",[38,1802,1804,1822,1840],{"className":1803,"ariaHidden":69},[68],[38,1805,1807,1810,1813,1816,1819],{"className":1806},[73],[38,1808],{"className":1809,"style":78},[77],[38,1811,59],{"className":1812,"style":84},[82,83],[38,1814],{"className":1815,"style":485},[484],[38,1817,379],{"className":1818},[489],[38,1820],{"className":1821,"style":485},[484],[38,1823,1825,1828,1831,1834,1837],{"className":1824},[73],[38,1826],{"className":1827,"style":1529},[77],[38,1829,1785],{"className":1830},[82],[38,1832],{"className":1833,"style":509},[484],[38,1835,1486],{"className":1836},[513],[38,1838],{"className":1839,"style":509},[484],[38,1841,1843,1846,1849],{"className":1842},[73],[38,1844],{"className":1845,"style":938},[77],[38,1847,178],{"className":1848},[82],[38,1850,1852,1855],{"className":1851},[82],[38,1853,159],{"className":1854},[82],[38,1856,1858],{"className":1857},[412],[38,1859,1861],{"className":1860},[416],[38,1862,1864],{"className":1863},[421],[38,1865,1867],{"className":1866,"style":938},[425],[38,1868,1869,1872],{"style":843},[38,1870],{"className":1871,"style":434},[433],[38,1873,1875],{"className":1874},[438,439,440,441],[38,1876,1878,1881],{"className":1877},[82,441],[38,1879,605],{"className":1880},[82,441],[38,1882,1498],{"className":1883},[82,441]," custo cai de ",[38,1886,1888,1909],{"className":1887},[41],[38,1889,1891],{"className":1890},[45],[47,1892,1893],{"xmlns":49},[51,1894,1895,1907],{},[54,1896,1897,1899,1901],{},[559,1898,1601],{},[361,1900,1486],{},[588,1902,1903,1905],{},[559,1904,1491],{},[559,1906,162],{},[61,1908,1612],{"encoding":63},[38,1910,1912,1930],{"className":1911,"ariaHidden":69},[68],[38,1913,1915,1918,1921,1924,1927],{"className":1914},[73],[38,1916],{"className":1917,"style":1529},[77],[38,1919,1601],{"className":1920},[82],[38,1922],{"className":1923,"style":509},[484],[38,1925,1486],{"className":1926},[513],[38,1928],{"className":1929,"style":509},[484],[38,1931,1933,1936,1939],{"className":1932},[73],[38,1934],{"className":1935,"style":938},[77],[38,1937,178],{"className":1938},[82],[38,1940,1942,1945],{"className":1941},[82],[38,1943,159],{"className":1944},[82],[38,1946,1948],{"className":1947},[412],[38,1949,1951],{"className":1950},[416],[38,1952,1954],{"className":1953},[421],[38,1955,1957],{"className":1956,"style":938},[425],[38,1958,1959,1962],{"style":843},[38,1960],{"className":1961,"style":434},[433],[38,1963,1965],{"className":1964},[438,439,440,441],[38,1966,162],{"className":1967},[82,441],[38,1969,1971,1994],{"className":1970},[41],[38,1972,1974],{"className":1973},[45],[47,1975,1976],{"xmlns":49},[51,1977,1978,1991],{},[54,1979,1980,1982,1984],{},[559,1981,1687],{},[361,1983,1486],{},[588,1985,1986,1988],{},[559,1987,1491],{},[559,1989,1990],{},"3",[61,1992,1993],{"encoding":63},"9.4\\times10^3",[38,1995,1997,2015],{"className":1996,"ariaHidden":69},[68],[38,1998,2000,2003,2006,2009,2012],{"className":1999},[73],[38,2001],{"className":2002,"style":1529},[77],[38,2004,1687],{"className":2005},[82],[38,2007],{"className":2008,"style":509},[484],[38,2010,1486],{"className":2011},[513],[38,2013],{"className":2014,"style":509},[484],[38,2016,2018,2021,2024],{"className":2017},[73],[38,2019],{"className":2020,"style":938},[77],[38,2022,178],{"className":2023},[82],[38,2025,2027,2030],{"className":2026},[82],[38,2028,159],{"className":2029},[82],[38,2031,2033],{"className":2032},[412],[38,2034,2036],{"className":2035},[416],[38,2037,2039],{"className":2038},[421],[38,2040,2042],{"className":2041,"style":938},[425],[38,2043,2044,2047],{"style":843},[38,2045],{"className":2046,"style":434},[433],[38,2048,2050],{"className":2049},[438,439,440,441],[38,2051,1990],{"className":2052},[82,441],", mas ",[22,2055,359],{}," ainda oscila (0 → 0.51 → 0.06 → 0.46 → 0.10...).",[11,2058,2059,1884,2177,1672,2260,2345,2346,2348],{},[33,2060,1464,2061,1456],{},[38,2062,2064,2094],{"className":2063},[41],[38,2065,2067],{"className":2066},[45],[47,2068,2069],{"xmlns":49},[51,2070,2071,2091],{},[54,2072,2073,2075,2077,2079,2081],{},[57,2074,59],{},[361,2076,379],{},[559,2078,178],{},[361,2080,1486],{},[588,2082,2083,2085],{},[559,2084,1491],{},[54,2086,2087,2089],{},[361,2088,605],{},[559,2090,1498],{},[61,2092,2093],{"encoding":63},"\\alpha = 1\\times10^{-7}",[38,2095,2097,2115,2133],{"className":2096,"ariaHidden":69},[68],[38,2098,2100,2103,2106,2109,2112],{"className":2099},[73],[38,2101],{"className":2102,"style":78},[77],[38,2104,59],{"className":2105,"style":84},[82,83],[38,2107],{"className":2108,"style":485},[484],[38,2110,379],{"className":2111},[489],[38,2113],{"className":2114,"style":485},[484],[38,2116,2118,2121,2124,2127,2130],{"className":2117},[73],[38,2119],{"className":2120,"style":1529},[77],[38,2122,178],{"className":2123},[82],[38,2125],{"className":2126,"style":509},[484],[38,2128,1486],{"className":2129},[513],[38,2131],{"className":2132,"style":509},[484],[38,2134,2136,2139,2142],{"className":2135},[73],[38,2137],{"className":2138,"style":938},[77],[38,2140,178],{"className":2141},[82],[38,2143,2145,2148],{"className":2144},[82],[38,2146,159],{"className":2147},[82],[38,2149,2151],{"className":2150},[412],[38,2152,2154],{"className":2153},[416],[38,2155,2157],{"className":2156},[421],[38,2158,2160],{"className":2159,"style":938},[425],[38,2161,2162,2165],{"style":843},[38,2163],{"className":2164,"style":434},[433],[38,2166,2168],{"className":2167},[438,439,440,441],[38,2169,2171,2174],{"className":2170},[82,441],[38,2172,605],{"className":2173},[82,441],[38,2175,1498],{"className":2176},[82,441],[38,2178,2180,2201],{"className":2179},[41],[38,2181,2183],{"className":2182},[45],[47,2184,2185],{"xmlns":49},[51,2186,2187,2199],{},[54,2188,2189,2191,2193],{},[559,2190,1601],{},[361,2192,1486],{},[588,2194,2195,2197],{},[559,2196,1491],{},[559,2198,162],{},[61,2200,1612],{"encoding":63},[38,2202,2204,2222],{"className":2203,"ariaHidden":69},[68],[38,2205,2207,2210,2213,2216,2219],{"className":2206},[73],[38,2208],{"className":2209,"style":1529},[77],[38,2211,1601],{"className":2212},[82],[38,2214],{"className":2215,"style":509},[484],[38,2217,1486],{"className":2218},[513],[38,2220],{"className":2221,"style":509},[484],[38,2223,2225,2228,2231],{"className":2224},[73],[38,2226],{"className":2227,"style":938},[77],[38,2229,178],{"className":2230},[82],[38,2232,2234,2237],{"className":2233},[82],[38,2235,159],{"className":2236},[82],[38,2238,2240],{"className":2239},[412],[38,2241,2243],{"className":2242},[416],[38,2244,2246],{"className":2245},[421],[38,2247,2249],{"className":2248,"style":938},[425],[38,2250,2251,2254],{"style":843},[38,2252],{"className":2253,"style":434},[433],[38,2255,2257],{"className":2256},[438,439,440,441],[38,2258,162],{"className":2259},[82,441],[38,2261,2263,2286],{"className":2262},[41],[38,2264,2266],{"className":2265},[45],[47,2267,2268],{"xmlns":49},[51,2269,2270,2283],{},[54,2271,2272,2275,2277],{},[559,2273,2274],{},"1.3",[361,2276,1486],{},[588,2278,2279,2281],{},[559,2280,1491],{},[559,2282,1990],{},[61,2284,2285],{"encoding":63},"1.3\\times10^3",[38,2287,2289,2307],{"className":2288,"ariaHidden":69},[68],[38,2290,2292,2295,2298,2301,2304],{"className":2291},[73],[38,2293],{"className":2294,"style":1529},[77],[38,2296,2274],{"className":2297},[82],[38,2299],{"className":2300,"style":509},[484],[38,2302,1486],{"className":2303},[513],[38,2305],{"className":2306,"style":509},[484],[38,2308,2310,2313,2316],{"className":2309},[73],[38,2311],{"className":2312,"style":938},[77],[38,2314,178],{"className":2315},[82],[38,2317,2319,2322],{"className":2318},[82],[38,2320,159],{"className":2321},[82],[38,2323,2325],{"className":2324},[412],[38,2326,2328],{"className":2327},[416],[38,2329,2331],{"className":2330},[421],[38,2332,2334],{"className":2333,"style":938},[425],[38,2335,2336,2339],{"style":843},[38,2337],{"className":2338,"style":434},[433],[38,2340,2342],{"className":2341},[438,439,440,441],[38,2343,1990],{"className":2344},[82,441],", ",[22,2347,359],{}," sobe direto sem oscilar (0 → 0.06 → 0.11 → 0.15...).",[11,2350,2351],{},"Testa você mesmo no simulador abaixo. Comecei com um alpha bem pequeno de propósito, arrasta o slider pra cima devagar e repara exatamente onde o custo vira de \"cai\" pra \"sobe\":",[2353,2354],"gradient-descent-simulator",{":b-range":2355,":initial-b":159,":initial-w":159,":w-range":2356,":x-train":2357,":y-train":2358,"b-label":366,"w-label":359,":alpha-presets":2359,":alpha-slider-max":2360,":alpha-slider-min":2361,":alpha-slider-step":2361},"[-250, 250]","[0, 0.6]","[952, 1244, 1947, 1725, 1959, 1314, 864, 1836]","[271.5, 300, 509.8, 394, 540, 415, 230, 560]","[1e-7, 3e-7, 8e-7, 9e-7, 9.9e-7]","2e-6","1e-8",[11,2363,2364,2365,2459,2460,2555],{},"O padrão que eu vi bate certinho com o que o notebook descreve pro dataset completo (com as 100 casas e as 4 features, o limite fica entre ",[38,2366,2368,2394],{"className":2367},[41],[38,2369,2371],{"className":2370},[45],[47,2372,2373],{"xmlns":49},[51,2374,2375,2391],{},[54,2376,2377,2379,2381],{},[559,2378,1483],{},[361,2380,1486],{},[588,2382,2383,2385],{},[559,2384,1491],{},[54,2386,2387,2389],{},[361,2388,605],{},[559,2390,1498],{},[61,2392,2393],{"encoding":63},"9\\times10^{-7}",[38,2395,2397,2415],{"className":2396,"ariaHidden":69},[68],[38,2398,2400,2403,2406,2409,2412],{"className":2399},[73],[38,2401],{"className":2402,"style":1529},[77],[38,2404,1483],{"className":2405},[82],[38,2407],{"className":2408,"style":509},[484],[38,2410,1486],{"className":2411},[513],[38,2413],{"className":2414,"style":509},[484],[38,2416,2418,2421,2424],{"className":2417},[73],[38,2419],{"className":2420,"style":938},[77],[38,2422,178],{"className":2423},[82],[38,2425,2427,2430],{"className":2426},[82],[38,2428,159],{"className":2429},[82],[38,2431,2433],{"className":2432},[412],[38,2434,2436],{"className":2435},[416],[38,2437,2439],{"className":2438},[421],[38,2440,2442],{"className":2441,"style":938},[425],[38,2443,2444,2447],{"style":843},[38,2445],{"className":2446,"style":434},[433],[38,2448,2450],{"className":2449},[438,439,440,441],[38,2451,2453,2456],{"className":2452},[82,441],[38,2454,605],{"className":2455},[82,441],[38,2457,1498],{"className":2458},[82,441]," e ",[38,2461,2463,2490],{"className":2462},[41],[38,2464,2466],{"className":2465},[45],[47,2467,2468],{"xmlns":49},[51,2469,2470,2487],{},[54,2471,2472,2475,2477],{},[559,2473,2474],{},"9.9",[361,2476,1486],{},[588,2478,2479,2481],{},[559,2480,1491],{},[54,2482,2483,2485],{},[361,2484,605],{},[559,2486,1498],{},[61,2488,2489],{"encoding":63},"9.9\\times10^{-7}",[38,2491,2493,2511],{"className":2492,"ariaHidden":69},[68],[38,2494,2496,2499,2502,2505,2508],{"className":2495},[73],[38,2497],{"className":2498,"style":1529},[77],[38,2500,2474],{"className":2501},[82],[38,2503],{"className":2504,"style":509},[484],[38,2506,1486],{"className":2507},[513],[38,2509],{"className":2510,"style":509},[484],[38,2512,2514,2517,2520],{"className":2513},[73],[38,2515],{"className":2516,"style":938},[77],[38,2518,178],{"className":2519},[82],[38,2521,2523,2526],{"className":2522},[82],[38,2524,159],{"className":2525},[82],[38,2527,2529],{"className":2528},[412],[38,2530,2532],{"className":2531},[416],[38,2533,2535],{"className":2534},[421],[38,2536,2538],{"className":2537,"style":938},[425],[38,2539,2540,2543],{"style":843},[38,2541],{"className":2542,"style":434},[433],[38,2544,2546],{"className":2545},[438,439,440,441],[38,2547,2549,2552],{"className":2548},[82,441],[38,2550,605],{"className":2551},[82,441],[38,2553,1498],{"className":2554},[82,441],", bem parecido com o que eu achei aqui só com tamanho e um subconjunto menor). Não é coincidência, e a próxima seção explica exatamente por quê.",[87,2557,2559,2560,2599],{"id":2558},"por-que-exatamente-ali-o-limite-2l2l2l-e-o-número-de-condição","Por que exatamente ali: o limite ",[38,2561,2563,2583],{"className":2562},[41],[38,2564,2566],{"className":2565},[45],[47,2567,2568],{"xmlns":49},[51,2569,2570,2580],{},[54,2571,2572,2574,2577],{},[559,2573,181],{},[57,2575,2576],{"mathvariant":985},"\u002F",[57,2578,2579],{},"L",[61,2581,2582],{"encoding":63},"2\u002FL",[38,2584,2586],{"className":2585,"ariaHidden":69},[68],[38,2587,2589,2592,2596],{"className":2588},[73],[38,2590],{"className":2591,"style":637},[77],[38,2593,2595],{"className":2594},[82],"2\u002F",[38,2597,2579],{"className":2598},[82,83]," e o número de condição",[11,2601,2602],{},"Isso não é tentativa e erro, dá pra calcular. Pra uma função de custo quadrática como a nossa, o gradiente descendente converge se e somente se",[11,2604,2605],{},[38,2606,2608,2662],{"className":2607},[41],[38,2609,2611],{"className":2610},[45],[47,2612,2613],{"xmlns":49},[51,2614,2615,2659],{},[54,2616,2617,2619,2622,2624,2626,2632,2634,2636,2638,2640,2651,2653,2657],{},[559,2618,159],{},[361,2620,2621],{},"\u003C",[57,2623,59],{},[361,2625,2621],{},[556,2627,2628,2630],{},[559,2629,181],{},[57,2631,2579],{},[361,2633,363],{"separator":69},[484,2635],{"width":997},[57,2637,2579],{},[361,2639,379],{},[350,2641,2642,2645,2648],{},[57,2643,2644],{},"λ",[57,2646,2647],{},"max",[361,2649,2650],{},"⁡",[361,2652,370],{"stretchy":369},[57,2654,2656],{"mathvariant":2655},"bold","H",[361,2658,376],{"stretchy":369},[61,2660,2661],{"encoding":63},"0 \u003C \\alpha \u003C \\frac{2}{L}, \\qquad L = \\lambda_{\\max}(\\mathbf{H})",[38,2663,2665,2684,2703,2799],{"className":2664,"ariaHidden":69},[68],[38,2666,2668,2672,2675,2678,2681],{"className":2667},[73],[38,2669],{"className":2670,"style":2671},[77],"height:0.6835em;vertical-align:-0.0391em;",[38,2673,159],{"className":2674},[82],[38,2676],{"className":2677,"style":485},[484],[38,2679,2621],{"className":2680},[489],[38,2682],{"className":2683,"style":485},[484],[38,2685,2687,2691,2694,2697,2700],{"className":2686},[73],[38,2688],{"className":2689,"style":2690},[77],"height:0.5782em;vertical-align:-0.0391em;",[38,2692,59],{"className":2693,"style":84},[82,83],[38,2695],{"className":2696,"style":485},[484],[38,2698,2621],{"className":2699},[489],[38,2701],{"className":2702,"style":485},[484],[38,2704,2706,2710,2778,2781,2784,2787,2790,2793,2796],{"className":2705},[73],[38,2707],{"className":2708,"style":2709},[77],"height:1.1901em;vertical-align:-0.345em;",[38,2711,2713,2716,2775],{"className":2712},[82],[38,2714],{"className":2715},[473,683],[38,2717,2719],{"className":2718},[556],[38,2720,2722,2767],{"className":2721},[416,417],[38,2723,2725,2764],{"className":2724},[421],[38,2726,2728,2742,2750],{"className":2727,"style":696},[425],[38,2729,2730,2733],{"style":699},[38,2731],{"className":2732,"style":703},[433],[38,2734,2736],{"className":2735},[438,439,440,441],[38,2737,2739],{"className":2738},[82,441],[38,2740,2579],{"className":2741},[82,83,441],[38,2743,2744,2747],{"style":718},[38,2745],{"className":2746,"style":703},[433],[38,2748],{"className":2749,"style":726},[725],[38,2751,2752,2755],{"style":729},[38,2753],{"className":2754,"style":703},[433],[38,2756,2758],{"className":2757},[438,439,440,441],[38,2759,2761],{"className":2760},[82,441],[38,2762,181],{"className":2763},[82,441],[38,2765,460],{"className":2766},[459],[38,2768,2770],{"className":2769},[421],[38,2771,2773],{"className":2772,"style":751},[425],[38,2774],{},[38,2776],{"className":2777},[480,683],[38,2779,363],{"className":2780},[452],[38,2782],{"className":2783,"style":1153},[484],[38,2785],{"className":2786,"style":654},[484],[38,2788,2579],{"className":2789},[82,83],[38,2791],{"className":2792,"style":485},[484],[38,2794,379],{"className":2795},[489],[38,2797],{"className":2798,"style":485},[484],[38,2800,2802,2805,2860,2863,2867],{"className":2801},[73],[38,2803],{"className":2804,"style":637},[77],[38,2806,2808,2811],{"className":2807},[82],[38,2809,2644],{"className":2810},[82,83],[38,2812,2814],{"className":2813},[412],[38,2815,2817,2851],{"className":2816},[416,417],[38,2818,2820,2848],{"className":2819},[421],[38,2821,2824],{"className":2822,"style":2823},[425],"height:0.1514em;",[38,2825,2827,2830],{"style":2826},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[38,2828],{"className":2829,"style":434},[433],[38,2831,2833],{"className":2832},[438,439,440,441],[38,2834,2836],{"className":2835},[82,441],[38,2837,2839,2842,2845],{"className":2838},[763,441],[38,2840,567],{"className":2841},[441],[38,2843,27],{"className":2844},[441],[38,2846,373],{"className":2847},[441],[38,2849,460],{"className":2850},[459],[38,2852,2854],{"className":2853},[421],[38,2855,2858],{"className":2856,"style":2857},[425],"height:0.15em;",[38,2859],{},[38,2861,370],{"className":2862},[473],[38,2864,2656],{"className":2865},[82,2866],"mathbf",[38,2868,376],{"className":2869},[480],[11,2871,2872,2873,2903,2904,2909,2910,2995,2996,3025],{},"onde ",[38,2874,2876,2890],{"className":2875},[41],[38,2877,2879],{"className":2878},[45],[47,2880,2881],{"xmlns":49},[51,2882,2883,2887],{},[54,2884,2885],{},[57,2886,2656],{"mathvariant":2655},[61,2888,2889],{"encoding":63},"\\mathbf{H}",[38,2891,2893],{"className":2892,"ariaHidden":69},[68],[38,2894,2896,2900],{"className":2895},[73],[38,2897],{"className":2898,"style":2899},[77],"height:0.6861em;",[38,2901,2656],{"className":2902},[82,2866]," é a ",[2905,2906,2908],"glossary-term",{"definition":2907},"a matriz das segundas derivadas da função de custo, ela descreve a curvatura da tigela em cada direção","Hessiana"," do custo e ",[38,2911,2913,2933],{"className":2912},[41],[38,2914,2916],{"className":2915},[45],[47,2917,2918],{"xmlns":49},[51,2919,2920,2930],{},[54,2921,2922],{},[350,2923,2924,2926,2928],{},[57,2925,2644],{},[57,2927,2647],{},[361,2929,2650],{},[61,2931,2932],{"encoding":63},"\\lambda_{\\max}",[38,2934,2936],{"className":2935,"ariaHidden":69},[68],[38,2937,2939,2943],{"className":2938},[73],[38,2940],{"className":2941,"style":2942},[77],"height:0.8444em;vertical-align:-0.15em;",[38,2944,2946,2949],{"className":2945},[82],[38,2947,2644],{"className":2948},[82,83],[38,2950,2952],{"className":2951},[412],[38,2953,2955,2987],{"className":2954},[416,417],[38,2956,2958,2984],{"className":2957},[421],[38,2959,2961],{"className":2960,"style":2823},[425],[38,2962,2963,2966],{"style":2826},[38,2964],{"className":2965,"style":434},[433],[38,2967,2969],{"className":2968},[438,439,440,441],[38,2970,2972],{"className":2971},[82,441],[38,2973,2975,2978,2981],{"className":2974},[763,441],[38,2976,567],{"className":2977},[441],[38,2979,27],{"className":2980},[441],[38,2982,373],{"className":2983},[441],[38,2985,460],{"className":2986},[459],[38,2988,2990],{"className":2989},[421],[38,2991,2993],{"className":2992,"style":2857},[425],[38,2994],{}," é o maior autovalor dela. ",[38,2997,2999,3012],{"className":2998},[41],[38,3000,3002],{"className":3001},[45],[47,3003,3004],{"xmlns":49},[51,3005,3006,3010],{},[54,3007,3008],{},[57,3009,2579],{},[61,3011,2579],{"encoding":63},[38,3013,3015],{"className":3014,"ariaHidden":69},[68],[38,3016,3018,3022],{"className":3017},[73],[38,3019],{"className":3020,"style":3021},[77],"height:0.6833em;",[38,3023,2579],{"className":3024},[82,83]," é a curvatura máxima da tigela: quanto mais afiada a direção mais curva, menor o passo permitido antes de pular pro outro lado do vale.",[11,3027,3028,3029,3080,3081,2459,3109,3137,3138,3247,3248,3276,3277,3387,3388,3481,3482,3576],{},"Calculei a Hessiana pro nosso exemplo de 8 casas (uma conta de ",[38,3030,3032,3050],{"className":3031},[41],[38,3033,3035],{"className":3034},[45],[47,3036,3037],{"xmlns":49},[51,3038,3039,3047],{},[54,3040,3041,3043,3045],{},[559,3042,181],{},[361,3044,1486],{},[559,3046,181],{},[61,3048,3049],{"encoding":63},"2\\times2",[38,3051,3053,3071],{"className":3052,"ariaHidden":69},[68],[38,3054,3056,3059,3062,3065,3068],{"className":3055},[73],[38,3057],{"className":3058,"style":1529},[77],[38,3060,181],{"className":3061},[82],[38,3063],{"className":3064,"style":509},[484],[38,3066,1486],{"className":3067},[513],[38,3069],{"className":3070,"style":509},[484],[38,3072,3074,3077],{"className":3073},[73],[38,3075],{"className":3076,"style":1452},[77],[38,3078,181],{"className":3079},[82],", já que é só ",[38,3082,3084,3097],{"className":3083},[41],[38,3085,3087],{"className":3086},[45],[47,3088,3089],{"xmlns":49},[51,3090,3091,3095],{},[54,3092,3093],{},[57,3094,359],{},[61,3096,359],{"encoding":63},[38,3098,3100],{"className":3099,"ariaHidden":69},[68],[38,3101,3103,3106],{"className":3102},[73],[38,3104],{"className":3105,"style":78},[77],[38,3107,359],{"className":3108,"style":448},[82,83],[38,3110,3112,3125],{"className":3111},[41],[38,3113,3115],{"className":3114},[45],[47,3116,3117],{"xmlns":49},[51,3118,3119,3123],{},[54,3120,3121],{},[57,3122,366],{},[61,3124,366],{"encoding":63},[38,3126,3128],{"className":3127,"ariaHidden":69},[68],[38,3129,3131,3134],{"className":3130},[73],[38,3132],{"className":3133,"style":523},[77],[38,3135,366],{"className":3136},[82,83],") e achei ",[38,3139,3141,3170],{"className":3140},[41],[38,3142,3144],{"className":3143},[45],[47,3145,3146],{"xmlns":49},[51,3147,3148,3167],{},[54,3149,3150,3152,3155,3158,3160],{},[57,3151,2579],{},[361,3153,3154],{},"≈",[559,3156,3157],{},"2.36",[361,3159,1486],{},[588,3161,3162,3164],{},[559,3163,1491],{},[559,3165,3166],{},"6",[61,3168,3169],{"encoding":63},"L \\approx 2.36\\times10^6",[38,3171,3173,3191,3209],{"className":3172,"ariaHidden":69},[68],[38,3174,3176,3179,3182,3185,3188],{"className":3175},[73],[38,3177],{"className":3178,"style":3021},[77],[38,3180,2579],{"className":3181},[82,83],[38,3183],{"className":3184,"style":485},[484],[38,3186,3154],{"className":3187},[489],[38,3189],{"className":3190,"style":485},[484],[38,3192,3194,3197,3200,3203,3206],{"className":3193},[73],[38,3195],{"className":3196,"style":1529},[77],[38,3198,3157],{"className":3199},[82],[38,3201],{"className":3202,"style":509},[484],[38,3204,1486],{"className":3205},[513],[38,3207],{"className":3208,"style":509},[484],[38,3210,3212,3215,3218],{"className":3211},[73],[38,3213],{"className":3214,"style":938},[77],[38,3216,178],{"className":3217},[82],[38,3219,3221,3224],{"className":3220},[82],[38,3222,159],{"className":3223},[82],[38,3225,3227],{"className":3226},[412],[38,3228,3230],{"className":3229},[416],[38,3231,3233],{"className":3232},[421],[38,3234,3236],{"className":3235,"style":938},[425],[38,3237,3238,3241],{"style":843},[38,3239],{"className":3240,"style":434},[433],[38,3242,3244],{"className":3243},[438,439,440,441],[38,3245,3166],{"className":3246},[82,441],", o que dá ",[38,3249,3251,3264],{"className":3250},[41],[38,3252,3254],{"className":3253},[45],[47,3255,3256],{"xmlns":49},[51,3257,3258,3262],{},[54,3259,3260],{},[57,3261,59],{},[61,3263,64],{"encoding":63},[38,3265,3267],{"className":3266,"ariaHidden":69},[68],[38,3268,3270,3273],{"className":3269},[73],[38,3271],{"className":3272,"style":78},[77],[38,3274,59],{"className":3275,"style":84},[82,83]," crítico ",[38,3278,3280,3309],{"className":3279},[41],[38,3281,3283],{"className":3282},[45],[47,3284,3285],{"xmlns":49},[51,3286,3287,3306],{},[54,3288,3289,3291,3294,3296],{},[361,3290,3154],{},[559,3292,3293],{},"8.46",[361,3295,1486],{},[588,3297,3298,3300],{},[559,3299,1491],{},[54,3301,3302,3304],{},[361,3303,605],{},[559,3305,1498],{},[61,3307,3308],{"encoding":63},"\\approx 8.46\\times10^{-7}",[38,3310,3312,3325,3343],{"className":3311,"ariaHidden":69},[68],[38,3313,3315,3319,3322],{"className":3314},[73],[38,3316],{"className":3317,"style":3318},[77],"height:0.4831em;",[38,3320,3154],{"className":3321},[489],[38,3323],{"className":3324,"style":485},[484],[38,3326,3328,3331,3334,3337,3340],{"className":3327},[73],[38,3329],{"className":3330,"style":1529},[77],[38,3332,3293],{"className":3333},[82],[38,3335],{"className":3336,"style":509},[484],[38,3338,1486],{"className":3339},[513],[38,3341],{"className":3342,"style":509},[484],[38,3344,3346,3349,3352],{"className":3345},[73],[38,3347],{"className":3348,"style":938},[77],[38,3350,178],{"className":3351},[82],[38,3353,3355,3358],{"className":3354},[82],[38,3356,159],{"className":3357},[82],[38,3359,3361],{"className":3360},[412],[38,3362,3364],{"className":3363},[416],[38,3365,3367],{"className":3366},[421],[38,3368,3370],{"className":3369,"style":938},[425],[38,3371,3372,3375],{"style":843},[38,3373],{"className":3374,"style":434},[433],[38,3376,3378],{"className":3377},[438,439,440,441],[38,3379,3381,3384],{"className":3380},[82,441],[38,3382,605],{"className":3383},[82,441],[38,3385,1498],{"className":3386},[82,441],". Bate com o que eu vi no simulador: ",[38,3389,3391,3416],{"className":3390},[41],[38,3392,3394],{"className":3393},[45],[47,3395,3396],{"xmlns":49},[51,3397,3398,3414],{},[54,3399,3400,3402,3404],{},[559,3401,1483],{},[361,3403,1486],{},[588,3405,3406,3408],{},[559,3407,1491],{},[54,3409,3410,3412],{},[361,3411,605],{},[559,3413,1498],{},[61,3415,2393],{"encoding":63},[38,3417,3419,3437],{"className":3418,"ariaHidden":69},[68],[38,3420,3422,3425,3428,3431,3434],{"className":3421},[73],[38,3423],{"className":3424,"style":1529},[77],[38,3426,1483],{"className":3427},[82],[38,3429],{"className":3430,"style":509},[484],[38,3432,1486],{"className":3433},[513],[38,3435],{"className":3436,"style":509},[484],[38,3438,3440,3443,3446],{"className":3439},[73],[38,3441],{"className":3442,"style":938},[77],[38,3444,178],{"className":3445},[82],[38,3447,3449,3452],{"className":3448},[82],[38,3450,159],{"className":3451},[82],[38,3453,3455],{"className":3454},[412],[38,3456,3458],{"className":3457},[416],[38,3459,3461],{"className":3460},[421],[38,3462,3464],{"className":3463,"style":938},[425],[38,3465,3466,3469],{"style":843},[38,3467],{"className":3468,"style":434},[433],[38,3470,3472],{"className":3471},[438,439,440,441],[38,3473,3475,3478],{"className":3474},[82,441],[38,3476,605],{"className":3477},[82,441],[38,3479,1498],{"className":3480},[82,441]," passa desse limite e diverge, ",[38,3483,3485,3511],{"className":3484},[41],[38,3486,3488],{"className":3487},[45],[47,3489,3490],{"xmlns":49},[51,3491,3492,3508],{},[54,3493,3494,3496,3498],{},[559,3495,1785],{},[361,3497,1486],{},[588,3499,3500,3502],{},[559,3501,1491],{},[54,3503,3504,3506],{},[361,3505,605],{},[559,3507,1498],{},[61,3509,3510],{"encoding":63},"8\\times10^{-7}",[38,3512,3514,3532],{"className":3513,"ariaHidden":69},[68],[38,3515,3517,3520,3523,3526,3529],{"className":3516},[73],[38,3518],{"className":3519,"style":1529},[77],[38,3521,1785],{"className":3522},[82],[38,3524],{"className":3525,"style":509},[484],[38,3527,1486],{"className":3528},[513],[38,3530],{"className":3531,"style":509},[484],[38,3533,3535,3538,3541],{"className":3534},[73],[38,3536],{"className":3537,"style":938},[77],[38,3539,178],{"className":3540},[82],[38,3542,3544,3547],{"className":3543},[82],[38,3545,159],{"className":3546},[82],[38,3548,3550],{"className":3549},[412],[38,3551,3553],{"className":3552},[416],[38,3554,3556],{"className":3555},[421],[38,3557,3559],{"className":3558,"style":938},[425],[38,3560,3561,3564],{"style":843},[38,3562],{"className":3563,"style":434},[433],[38,3565,3567],{"className":3566},[438,439,440,441],[38,3568,3570,3573],{"className":3569},[82,441],[38,3571,605],{"className":3572},[82,441],[38,3574,1498],{"className":3575},[82,441]," fica embaixo e converge.",[11,3578,3579,3580,3631,3632,3660,3661,3276,3689,3798],{},"O notebook original faz essa mesma conta com as 4 features de verdade (a Hessiana vira ",[38,3581,3583,3601],{"className":3582},[41],[38,3584,3586],{"className":3585},[45],[47,3587,3588],{"xmlns":49},[51,3589,3590,3598],{},[54,3591,3592,3594,3596],{},[559,3593,1696],{},[361,3595,1486],{},[559,3597,1696],{},[61,3599,3600],{"encoding":63},"5\\times5",[38,3602,3604,3622],{"className":3603,"ariaHidden":69},[68],[38,3605,3607,3610,3613,3616,3619],{"className":3606},[73],[38,3608],{"className":3609,"style":1529},[77],[38,3611,1696],{"className":3612},[82],[38,3614],{"className":3615,"style":509},[484],[38,3617,1486],{"className":3618},[513],[38,3620],{"className":3621,"style":509},[484],[38,3623,3625,3628],{"className":3624},[73],[38,3626],{"className":3627,"style":1452},[77],[38,3629,1696],{"className":3630},[82],", contando o ",[38,3633,3635,3648],{"className":3634},[41],[38,3636,3638],{"className":3637},[45],[47,3639,3640],{"xmlns":49},[51,3641,3642,3646],{},[54,3643,3644],{},[57,3645,366],{},[61,3647,366],{"encoding":63},[38,3649,3651],{"className":3650,"ariaHidden":69},[68],[38,3652,3654,3657],{"className":3653},[73],[38,3655],{"className":3656,"style":523},[77],[38,3658,366],{"className":3659},[82,83],") e chega em ",[38,3662,3664,3677],{"className":3663},[41],[38,3665,3667],{"className":3666},[45],[47,3668,3669],{"xmlns":49},[51,3670,3671,3675],{},[54,3672,3673],{},[57,3674,59],{},[61,3676,64],{"encoding":63},[38,3678,3680],{"className":3679,"ariaHidden":69},[68],[38,3681,3683,3686],{"className":3682},[73],[38,3684],{"className":3685,"style":78},[77],[38,3687,59],{"className":3688,"style":84},[82,83],[38,3690,3692,3721],{"className":3691},[41],[38,3693,3695],{"className":3694},[45],[47,3696,3697],{"xmlns":49},[51,3698,3699,3718],{},[54,3700,3701,3703,3706,3708],{},[361,3702,3154],{},[559,3704,3705],{},"9.22",[361,3707,1486],{},[588,3709,3710,3712],{},[559,3711,1491],{},[54,3713,3714,3716],{},[361,3715,605],{},[559,3717,1498],{},[61,3719,3720],{"encoding":63},"\\approx 9.22\\times10^{-7}",[38,3722,3724,3736,3754],{"className":3723,"ariaHidden":69},[68],[38,3725,3727,3730,3733],{"className":3726},[73],[38,3728],{"className":3729,"style":3318},[77],[38,3731,3154],{"className":3732},[489],[38,3734],{"className":3735,"style":485},[484],[38,3737,3739,3742,3745,3748,3751],{"className":3738},[73],[38,3740],{"className":3741,"style":1529},[77],[38,3743,3705],{"className":3744},[82],[38,3746],{"className":3747,"style":509},[484],[38,3749,1486],{"className":3750},[513],[38,3752],{"className":3753,"style":509},[484],[38,3755,3757,3760,3763],{"className":3756},[73],[38,3758],{"className":3759,"style":938},[77],[38,3761,178],{"className":3762},[82],[38,3764,3766,3769],{"className":3765},[82],[38,3767,159],{"className":3768},[82],[38,3770,3772],{"className":3771},[412],[38,3773,3775],{"className":3774},[416],[38,3776,3778],{"className":3777},[421],[38,3779,3781],{"className":3780,"style":938},[425],[38,3782,3783,3786],{"style":843},[38,3784],{"className":3785,"style":434},[433],[38,3787,3789],{"className":3788},[438,439,440,441],[38,3790,3792,3795],{"className":3791},[82,441],[38,3793,605],{"className":3794},[82,441],[38,3796,1498],{"className":3797},[82,441],", praticamente o mesmo número que eu achei simplificando pra 1 feature. Faz sentido: o tamanho é a feature com a escala mais absurda, então ela sozinha já domina quase toda a curvatura da tigela.",[11,3800,3801,3802,3806,3807,3870,3871,3901,3902,4009,4010,4013],{},"Tem um segundo número que sai dessa mesma conta, o ",[2905,3803,3805],{"definition":3804},"a razão entre a maior e a menor curvatura da tigela de custo, kappa = L\u002Fmu, e quanto maior ele é, mais alongada a tigela e mais lento o gradiente descendente","número de condição"," ",[38,3808,3810,3834],{"className":3809},[41],[38,3811,3813],{"className":3812},[45],[47,3814,3815],{"xmlns":49},[51,3816,3817,3831],{},[54,3818,3819,3822,3824,3826,3828],{},[57,3820,3821],{},"κ",[361,3823,379],{},[57,3825,2579],{},[57,3827,2576],{"mathvariant":985},[57,3829,3830],{},"μ",[61,3832,3833],{"encoding":63},"\\kappa = L\u002F\\mu",[38,3835,3837,3855],{"className":3836,"ariaHidden":69},[68],[38,3838,3840,3843,3846,3849,3852],{"className":3839},[73],[38,3841],{"className":3842,"style":78},[77],[38,3844,3821],{"className":3845},[82,83],[38,3847],{"className":3848,"style":485},[484],[38,3850,379],{"className":3851},[489],[38,3853],{"className":3854,"style":485},[484],[38,3856,3858,3861,3864,3867],{"className":3857},[73],[38,3859],{"className":3860,"style":637},[77],[38,3862,2579],{"className":3863},[82,83],[38,3865,2576],{"className":3866},[82],[38,3868,3830],{"className":3869},[82,83]," (onde ",[38,3872,3874,3888],{"className":3873},[41],[38,3875,3877],{"className":3876},[45],[47,3878,3879],{"xmlns":49},[51,3880,3881,3885],{},[54,3882,3883],{},[57,3884,3830],{},[61,3886,3887],{"encoding":63},"\\mu",[38,3889,3891],{"className":3890,"ariaHidden":69},[68],[38,3892,3894,3898],{"className":3893},[73],[38,3895],{"className":3896,"style":3897},[77],"height:0.625em;vertical-align:-0.1944em;",[38,3899,3830],{"className":3900},[82,83]," é a menor curvatura). Ele mede o quão \"esticada\" é a tigela. Achei ",[38,3903,3905,3932],{"className":3904},[41],[38,3906,3908],{"className":3907},[45],[47,3909,3910],{"xmlns":49},[51,3911,3912,3929],{},[54,3913,3914,3916,3918,3921,3923],{},[57,3915,3821],{},[361,3917,3154],{},[559,3919,3920],{},"5.8",[361,3922,1486],{},[588,3924,3925,3927],{},[559,3926,1491],{},[559,3928,1498],{},[61,3930,3931],{"encoding":63},"\\kappa \\approx 5.8\\times10^7",[38,3933,3935,3953,3971],{"className":3934,"ariaHidden":69},[68],[38,3936,3938,3941,3944,3947,3950],{"className":3937},[73],[38,3939],{"className":3940,"style":3318},[77],[38,3942,3821],{"className":3943},[82,83],[38,3945],{"className":3946,"style":485},[484],[38,3948,3154],{"className":3949},[489],[38,3951],{"className":3952,"style":485},[484],[38,3954,3956,3959,3962,3965,3968],{"className":3955},[73],[38,3957],{"className":3958,"style":1529},[77],[38,3960,3920],{"className":3961},[82],[38,3963],{"className":3964,"style":509},[484],[38,3966,1486],{"className":3967},[513],[38,3969],{"className":3970,"style":509},[484],[38,3972,3974,3977,3980],{"className":3973},[73],[38,3975],{"className":3976,"style":938},[77],[38,3978,178],{"className":3979},[82],[38,3981,3983,3986],{"className":3982},[82],[38,3984,159],{"className":3985},[82],[38,3987,3989],{"className":3988},[412],[38,3990,3992],{"className":3991},[416],[38,3993,3995],{"className":3994},[421],[38,3996,3998],{"className":3997,"style":938},[425],[38,3999,4000,4003],{"style":843},[38,4001],{"className":4002,"style":434},[433],[38,4004,4006],{"className":4005},[438,439,440,441],[38,4007,1498],{"className":4008},[82,441]," pro dataset completo de 100 casas com as 4 features. Calculei também quantas iterações seriam necessárias, no melhor caso, só pra reduzir o erro em 10 vezes: ",[33,4011,4012],{},"cerca de 67 milhões",". Foi aí que eu entendi de vez por que \"diminui um pouco o alpha e espera mais\" não é uma solução de verdade aqui.",[87,4015,4017,4018],{"id":4016},"receita-prática-pra-escolher-αalphaα","Receita prática pra escolher ",[38,4019,4021,4034],{"className":4020},[41],[38,4022,4024],{"className":4023},[45],[47,4025,4026],{"xmlns":49},[51,4027,4028,4032],{},[54,4029,4030],{},[57,4031,59],{},[61,4033,64],{"encoding":63},[38,4035,4037],{"className":4036,"ariaHidden":69},[68],[38,4038,4040,4043],{"className":4039},[73],[38,4041],{"className":4042,"style":78},[77],[38,4044,59],{"className":4045,"style":84},[82,83],[11,4047,4048,4049,1456],{},"Nem sempre eu vou querer calcular autovalor. A receita que o Andrew Ng ensina, e que eu já uso desde ",[27,4050,4051],{"href":29},"o post de gradiente descendente",[4053,4054,4055,4111,4117,4120],"ol",{},[4056,4057,4058,4059,97],"li",{},"Começa pequeno, tipo ",[38,4060,4062,4081],{"className":4061},[41],[38,4063,4065],{"className":4064},[45],[47,4066,4067],{"xmlns":49},[51,4068,4069,4078],{},[54,4070,4071,4073,4075],{},[57,4072,59],{},[361,4074,379],{},[559,4076,4077],{},"0.001",[61,4079,4080],{"encoding":63},"\\alpha = 0.001",[38,4082,4084,4102],{"className":4083,"ariaHidden":69},[68],[38,4085,4087,4090,4093,4096,4099],{"className":4086},[73],[38,4088],{"className":4089,"style":78},[77],[38,4091,59],{"className":4092,"style":84},[82,83],[38,4094],{"className":4095,"style":485},[484],[38,4097,379],{"className":4098},[489],[38,4100],{"className":4101,"style":485},[484],[38,4103,4105,4108],{"className":4104},[73],[38,4106],{"className":4107,"style":1452},[77],[38,4109,4077],{"className":4110},[82],[4056,4112,4113,4114],{},"Multiplica por uns 3 a cada tentativa: ",[22,4115,4116],{},"0.001, 0.003, 0.01, 0.03, 0.1...",[4056,4118,4119],{},"Roda poucas iterações e olha o gráfico de custo por iteração.",[4056,4121,4122,4123,4151],{},"Pega o maior ",[38,4124,4126,4139],{"className":4125},[41],[38,4127,4129],{"className":4128},[45],[47,4130,4131],{"xmlns":49},[51,4132,4133,4137],{},[54,4134,4135],{},[57,4136,59],{},[61,4138,64],{"encoding":63},[38,4140,4142],{"className":4141,"ariaHidden":69},[68],[38,4143,4145,4148],{"className":4144},[73],[38,4146],{"className":4147,"style":78},[77],[38,4149,59],{"className":4150,"style":84},[82,83]," que ainda dá uma curva decrescente e suave.",[99,4153,4154,4164],{},[102,4155,4156],{},[105,4157,4158,4161],{},[108,4159,4160],{"align":110},"O que aparece no gráfico",[108,4162,4163],{"align":110},"Diagnóstico",[129,4165,4166,4208,4244,4280],{},[105,4167,4168,4177],{},[134,4169,4170,4171,2576,4174],{"align":110},"Custo sobe, ou vira ",[22,4172,4173],{},"nan",[22,4175,4176],{},"inf",[134,4178,4179,4207],{"align":110},[38,4180,4182,4195],{"className":4181},[41],[38,4183,4185],{"className":4184},[45],[47,4186,4187],{"xmlns":49},[51,4188,4189,4193],{},[54,4190,4191],{},[57,4192,59],{},[61,4194,64],{"encoding":63},[38,4196,4198],{"className":4197,"ariaHidden":69},[68],[38,4199,4201,4204],{"className":4200},[73],[38,4202],{"className":4203,"style":78},[77],[38,4205,59],{"className":4206,"style":84},[82,83]," grande demais",[105,4209,4210,4213],{},[134,4211,4212],{"align":110},"Custo desce mas serrilhado, com picos",[134,4214,4215,4243],{"align":110},[38,4216,4218,4231],{"className":4217},[41],[38,4219,4221],{"className":4220},[45],[47,4222,4223],{"xmlns":49},[51,4224,4225,4229],{},[54,4226,4227],{},[57,4228,59],{},[61,4230,64],{"encoding":63},[38,4232,4234],{"className":4233,"ariaHidden":69},[68],[38,4235,4237,4240],{"className":4236},[73],[38,4238],{"className":4239,"style":78},[77],[38,4241,59],{"className":4242,"style":84},[82,83]," no limite, reduz",[105,4245,4246,4249],{},[134,4247,4248],{"align":110},"Custo desce numa linha quase reta e devagar",[134,4250,4251,4279],{"align":110},[38,4252,4254,4267],{"className":4253},[41],[38,4255,4257],{"className":4256},[45],[47,4258,4259],{"xmlns":49},[51,4260,4261,4265],{},[54,4262,4263],{},[57,4264,59],{},[61,4266,64],{"encoding":63},[38,4268,4270],{"className":4269,"ariaHidden":69},[68],[38,4271,4273,4276],{"className":4272},[73],[38,4274],{"className":4275,"style":78},[77],[38,4277,59],{"className":4278,"style":84},[82,83]," pequeno demais",[105,4281,4282,4285],{},[134,4283,4284],{"align":110},"Custo desce rápido e depois achata",[134,4286,4287,4288,4316],{"align":110},"bom ",[38,4289,4291,4304],{"className":4290},[41],[38,4292,4294],{"className":4293},[45],[47,4295,4296],{"xmlns":49},[51,4297,4298,4302],{},[54,4299,4300],{},[57,4301,59],{},[61,4303,64],{"encoding":63},[38,4305,4307],{"className":4306,"ariaHidden":69},[68],[38,4308,4310,4313],{"className":4309},[73],[38,4311],{"className":4312,"style":78},[77],[38,4314,59],{"className":4315,"style":84},[82,83],", convergiu",[11,4318,4319,4320,4348,4349,2459,4421,4493],{},"Rodei essa varredura no dataset completo e achei que a faixa utilizável de ",[38,4321,4323,4336],{"className":4322},[41],[38,4324,4326],{"className":4325},[45],[47,4327,4328],{"xmlns":49},[51,4329,4330,4334],{},[54,4331,4332],{},[57,4333,59],{},[61,4335,64],{"encoding":63},[38,4337,4339],{"className":4338,"ariaHidden":69},[68],[38,4340,4342,4345],{"className":4341},[73],[38,4343],{"className":4344,"style":78},[77],[38,4346,59],{"className":4347,"style":84},[82,83]," (sem normalizar nada) fica entre ",[38,4350,4352,4374],{"className":4351},[41],[38,4353,4355],{"className":4354},[45],[47,4356,4357],{"xmlns":49},[51,4358,4359,4371],{},[54,4360,4361],{},[588,4362,4363,4365],{},[559,4364,1491],{},[54,4366,4367,4369],{},[361,4368,605],{},[559,4370,1785],{},[61,4372,4373],{"encoding":63},"10^{-8}",[38,4375,4377],{"className":4376,"ariaHidden":69},[68],[38,4378,4380,4383,4386],{"className":4379},[73],[38,4381],{"className":4382,"style":938},[77],[38,4384,178],{"className":4385},[82],[38,4387,4389,4392],{"className":4388},[82],[38,4390,159],{"className":4391},[82],[38,4393,4395],{"className":4394},[412],[38,4396,4398],{"className":4397},[416],[38,4399,4401],{"className":4400},[421],[38,4402,4404],{"className":4403,"style":938},[425],[38,4405,4406,4409],{"style":843},[38,4407],{"className":4408,"style":434},[433],[38,4410,4412],{"className":4411},[438,439,440,441],[38,4413,4415,4418],{"className":4414},[82,441],[38,4416,605],{"className":4417},[82,441],[38,4419,1785],{"className":4420},[82,441],[38,4422,4424,4446],{"className":4423},[41],[38,4425,4427],{"className":4426},[45],[47,4428,4429],{"xmlns":49},[51,4430,4431,4443],{},[54,4432,4433],{},[588,4434,4435,4437],{},[559,4436,1491],{},[54,4438,4439,4441],{},[361,4440,605],{},[559,4442,3166],{},[61,4444,4445],{"encoding":63},"10^{-6}",[38,4447,4449],{"className":4448,"ariaHidden":69},[68],[38,4450,4452,4455,4458],{"className":4451},[73],[38,4453],{"className":4454,"style":938},[77],[38,4456,178],{"className":4457},[82],[38,4459,4461,4464],{"className":4460},[82],[38,4462,159],{"className":4463},[82],[38,4465,4467],{"className":4466},[412],[38,4468,4470],{"className":4469},[416],[38,4471,4473],{"className":4472},[421],[38,4474,4476],{"className":4475,"style":938},[425],[38,4477,4478,4481],{"style":843},[38,4479],{"className":4480,"style":434},[433],[38,4482,4484],{"className":4483},[438,439,440,441],[38,4485,4487,4490],{"className":4486},[82,441],[38,4488,605],{"className":4489},[82,441],[38,4491,3166],{"className":4492},[82,441],". Uma faixa estreitíssima, e que depende inteiramente da unidade que eu usei pra medir tamanho. Se eu tivesse medido em metros quadrados em vez de pés², todos esses números mudariam. Isso é frágil, e não é assim que eu quero treinar modelo nenhum.",[87,4495,4497],{"id":4496},"as-técnicas-de-normalização","As técnicas de normalização",[11,4499,4500,4501,4529],{},"A solução não é caçar um ",[38,4502,4504,4517],{"className":4503},[41],[38,4505,4507],{"className":4506},[45],[47,4508,4509],{"xmlns":49},[51,4510,4511,4515],{},[54,4512,4513],{},[57,4514,59],{},[61,4516,64],{"encoding":63},[38,4518,4520],{"className":4519,"ariaHidden":69},[68],[38,4521,4523,4526],{"className":4522},[73],[38,4524],{"className":4525,"style":78},[77],[38,4527,59],{"className":4528,"style":84},[82,83]," melhor, é consertar a escala das features antes de rodar qualquer coisa. Existem algumas formas de fazer isso:",[99,4531,4532,4545],{},[102,4533,4534],{},[105,4535,4536,4539,4542],{},[108,4537,4538],{"align":110},"Técnica",[108,4540,4541],{"align":110},"Fórmula",[108,4543,4544],{"align":110},"Faixa resultante",[129,4546,4547,4819,5318,5788],{},[105,4548,4549,4552,4763],{},[134,4550,4551],{"align":110},"Dividir pelo máximo",[134,4553,4554],{"align":110},[38,4555,4557,4601],{"className":4556},[41],[38,4558,4560],{"className":4559},[45],[47,4561,4562],{"xmlns":49},[51,4563,4564,4598],{},[54,4565,4566,4573,4576,4582,4584,4586,4588,4590,4596],{},[350,4567,4568,4570],{},[57,4569,373],{},[57,4571,4572],{},"j",[361,4574,4575],{},"←",[350,4577,4578,4580],{},[57,4579,373],{},[57,4581,4572],{},[57,4583,2576],{"mathvariant":985},[57,4585,2647],{},[361,4587,2650],{},[361,4589,370],{"stretchy":369},[350,4591,4592,4594],{},[57,4593,373],{},[57,4595,4572],{},[361,4597,376],{"stretchy":369},[61,4599,4600],{"encoding":63},"x_j \\leftarrow x_j \u002F \\max(x_j)",[38,4602,4604,4662],{"className":4603,"ariaHidden":69},[68],[38,4605,4607,4611,4653,4656,4659],{"className":4606},[73],[38,4608],{"className":4609,"style":4610},[77],"height:0.7167em;vertical-align:-0.2861em;",[38,4612,4614,4617],{"className":4613},[82],[38,4615,373],{"className":4616},[82,83],[38,4618,4620],{"className":4619},[412],[38,4621,4623,4645],{"className":4622},[416,417],[38,4624,4626,4642],{"className":4625},[421],[38,4627,4630],{"className":4628,"style":4629},[425],"height:0.3117em;",[38,4631,4632,4635],{"style":2826},[38,4633],{"className":4634,"style":434},[433],[38,4636,4638],{"className":4637},[438,439,440,441],[38,4639,4572],{"className":4640,"style":4641},[82,83,441],"margin-right:0.0572em;",[38,4643,460],{"className":4644},[459],[38,4646,4648],{"className":4647},[421],[38,4649,4651],{"className":4650,"style":467},[425],[38,4652],{},[38,4654],{"className":4655,"style":485},[484],[38,4657,4575],{"className":4658},[489],[38,4660],{"className":4661,"style":485},[484],[38,4663,4665,4668,4708,4711,4714,4717,4720,4760],{"className":4664},[73],[38,4666],{"className":4667,"style":401},[77],[38,4669,4671,4674],{"className":4670},[82],[38,4672,373],{"className":4673},[82,83],[38,4675,4677],{"className":4676},[412],[38,4678,4680,4700],{"className":4679},[416,417],[38,4681,4683,4697],{"className":4682},[421],[38,4684,4686],{"className":4685,"style":4629},[425],[38,4687,4688,4691],{"style":2826},[38,4689],{"className":4690,"style":434},[433],[38,4692,4694],{"className":4693},[438,439,440,441],[38,4695,4572],{"className":4696,"style":4641},[82,83,441],[38,4698,460],{"className":4699},[459],[38,4701,4703],{"className":4702},[421],[38,4704,4706],{"className":4705,"style":467},[425],[38,4707],{},[38,4709,2576],{"className":4710},[82],[38,4712],{"className":4713,"style":654},[484],[38,4715,2647],{"className":4716},[763],[38,4718,370],{"className":4719},[473],[38,4721,4723,4726],{"className":4722},[82],[38,4724,373],{"className":4725},[82,83],[38,4727,4729],{"className":4728},[412],[38,4730,4732,4752],{"className":4731},[416,417],[38,4733,4735,4749],{"className":4734},[421],[38,4736,4738],{"className":4737,"style":4629},[425],[38,4739,4740,4743],{"style":2826},[38,4741],{"className":4742,"style":434},[433],[38,4744,4746],{"className":4745},[438,439,440,441],[38,4747,4572],{"className":4748,"style":4641},[82,83,441],[38,4750,460],{"className":4751},[459],[38,4753,4755],{"className":4754},[421],[38,4756,4758],{"className":4757,"style":467},[425],[38,4759],{},[38,4761,376],{"className":4762},[480],[134,4764,4765],{"align":110},[38,4766,4768,4792],{"className":4767},[41],[38,4769,4771],{"className":4770},[45],[47,4772,4773],{"xmlns":49},[51,4774,4775,4789],{},[54,4776,4777,4780,4782,4784,4786],{},[361,4778,4779],{"stretchy":369},"[",[559,4781,159],{},[361,4783,363],{"separator":69},[559,4785,178],{},[361,4787,4788],{"stretchy":369},"]",[61,4790,4791],{"encoding":63},"[0, 1]",[38,4793,4795],{"className":4794,"ariaHidden":69},[68],[38,4796,4798,4801,4804,4807,4810,4813,4816],{"className":4797},[73],[38,4799],{"className":4800,"style":637},[77],[38,4802,4779],{"className":4803},[473],[38,4805,159],{"className":4806},[82],[38,4808,363],{"className":4809},[452],[38,4811],{"className":4812,"style":654},[484],[38,4814,178],{"className":4815},[82],[38,4817,4788],{"className":4818},[480],[105,4820,4821,4824,5265],{},[134,4822,4823],{"align":110},"Min-max",[134,4825,4826],{"align":110},[38,4827,4829,4908],{"className":4828},[41],[38,4830,4832],{"className":4831},[45],[47,4833,4834],{"xmlns":49},[51,4835,4836,4905],{},[54,4837,4838,4844,4846],{},[350,4839,4840,4842],{},[57,4841,373],{},[57,4843,4572],{},[361,4845,4575],{},[556,4847,4848,4873],{},[54,4849,4850,4856,4858,4861,4863,4865,4871],{},[350,4851,4852,4854],{},[57,4853,373],{},[57,4855,4572],{},[361,4857,605],{},[57,4859,4860],{},"min",[361,4862,2650],{},[361,4864,370],{"stretchy":369},[350,4866,4867,4869],{},[57,4868,373],{},[57,4870,4572],{},[361,4872,376],{"stretchy":369},[54,4874,4875,4877,4879,4881,4887,4889,4891,4893,4895,4897,4903],{},[57,4876,2647],{},[361,4878,2650],{},[361,4880,370],{"stretchy":369},[350,4882,4883,4885],{},[57,4884,373],{},[57,4886,4572],{},[361,4888,376],{"stretchy":369},[361,4890,605],{},[57,4892,4860],{},[361,4894,2650],{},[361,4896,370],{"stretchy":369},[350,4898,4899,4901],{},[57,4900,373],{},[57,4902,4572],{},[361,4904,376],{"stretchy":369},[61,4906,4907],{"encoding":63},"x_j \\leftarrow \\frac{x_j - \\min(x_j)}{\\max(x_j) - \\min(x_j)}",[38,4909,4911,4966],{"className":4910,"ariaHidden":69},[68],[38,4912,4914,4917,4957,4960,4963],{"className":4913},[73],[38,4915],{"className":4916,"style":4610},[77],[38,4918,4920,4923],{"className":4919},[82],[38,4921,373],{"className":4922},[82,83],[38,4924,4926],{"className":4925},[412],[38,4927,4929,4949],{"className":4928},[416,417],[38,4930,4932,4946],{"className":4931},[421],[38,4933,4935],{"className":4934,"style":4629},[425],[38,4936,4937,4940],{"style":2826},[38,4938],{"className":4939,"style":434},[433],[38,4941,4943],{"className":4942},[438,439,440,441],[38,4944,4572],{"className":4945,"style":4641},[82,83,441],[38,4947,460],{"className":4948},[459],[38,4950,4952],{"className":4951},[421],[38,4953,4955],{"className":4954,"style":467},[425],[38,4956],{},[38,4958],{"className":4959,"style":485},[484],[38,4961,4575],{"className":4962},[489],[38,4964],{"className":4965,"style":485},[484],[38,4967,4969,4973],{"className":4968},[73],[38,4970],{"className":4971,"style":4972},[77],"height:1.5746em;vertical-align:-0.5423em;",[38,4974,4976,4979,5262],{"className":4975},[82],[38,4977],{"className":4978},[473,683],[38,4980,4982],{"className":4981},[556],[38,4983,4985,5253],{"className":4984},[416,417],[38,4986,4988,5250],{"className":4987},[421],[38,4989,4992,5129,5137],{"className":4990,"style":4991},[425],"height:1.0323em;",[38,4993,4994,4997],{"style":699},[38,4995],{"className":4996,"style":703},[433],[38,4998,5000],{"className":4999},[438,439,440,441],[38,5001,5003,5015,5018,5064,5067,5070,5083,5086,5126],{"className":5002},[82,441],[38,5004,5006,5009,5012],{"className":5005},[763,441],[38,5007,567],{"className":5008},[441],[38,5010,27],{"className":5011},[441],[38,5013,373],{"className":5014},[441],[38,5016,370],{"className":5017},[473,441],[38,5019,5021,5024],{"className":5020},[82,441],[38,5022,373],{"className":5023},[82,83,441],[38,5025,5027],{"className":5026},[412],[38,5028,5030,5055],{"className":5029},[416,417],[38,5031,5033,5052],{"className":5032},[421],[38,5034,5037],{"className":5035,"style":5036},[425],"height:0.3281em;",[38,5038,5040,5044],{"style":5039},"top:-2.357em;margin-left:0em;margin-right:0.0714em;",[38,5041],{"className":5042,"style":5043},[433],"height:2.5em;",[38,5045,5049],{"className":5046},[438,5047,5048,441],"reset-size3","size1",[38,5050,4572],{"className":5051,"style":4641},[82,83,441],[38,5053,460],{"className":5054},[459],[38,5056,5058],{"className":5057},[421],[38,5059,5062],{"className":5060,"style":5061},[425],"height:0.2819em;",[38,5063],{},[38,5065,376],{"className":5066},[480,441],[38,5068,605],{"className":5069},[513,441],[38,5071,5073,5076,5079],{"className":5072},[763,441],[38,5074,567],{"className":5075},[441],[38,5077,598],{"className":5078},[441],[38,5080,5082],{"className":5081},[441],"n",[38,5084,370],{"className":5085},[473,441],[38,5087,5089,5092],{"className":5088},[82,441],[38,5090,373],{"className":5091},[82,83,441],[38,5093,5095],{"className":5094},[412],[38,5096,5098,5118],{"className":5097},[416,417],[38,5099,5101,5115],{"className":5100},[421],[38,5102,5104],{"className":5103,"style":5036},[425],[38,5105,5106,5109],{"style":5039},[38,5107],{"className":5108,"style":5043},[433],[38,5110,5112],{"className":5111},[438,5047,5048,441],[38,5113,4572],{"className":5114,"style":4641},[82,83,441],[38,5116,460],{"className":5117},[459],[38,5119,5121],{"className":5120},[421],[38,5122,5124],{"className":5123,"style":5061},[425],[38,5125],{},[38,5127,376],{"className":5128},[480,441],[38,5130,5131,5134],{"style":718},[38,5132],{"className":5133,"style":703},[433],[38,5135],{"className":5136,"style":726},[725],[38,5138,5140,5143],{"style":5139},"top:-3.5073em;",[38,5141],{"className":5142,"style":703},[433],[38,5144,5146],{"className":5145},[438,439,440,441],[38,5147,5149,5189,5192,5204,5207,5247],{"className":5148},[82,441],[38,5150,5152,5155],{"className":5151},[82,441],[38,5153,373],{"className":5154},[82,83,441],[38,5156,5158],{"className":5157},[412],[38,5159,5161,5181],{"className":5160},[416,417],[38,5162,5164,5178],{"className":5163},[421],[38,5165,5167],{"className":5166,"style":5036},[425],[38,5168,5169,5172],{"style":5039},[38,5170],{"className":5171,"style":5043},[433],[38,5173,5175],{"className":5174},[438,5047,5048,441],[38,5176,4572],{"className":5177,"style":4641},[82,83,441],[38,5179,460],{"className":5180},[459],[38,5182,5184],{"className":5183},[421],[38,5185,5187],{"className":5186,"style":5061},[425],[38,5188],{},[38,5190,605],{"className":5191},[513,441],[38,5193,5195,5198,5201],{"className":5194},[763,441],[38,5196,567],{"className":5197},[441],[38,5199,598],{"className":5200},[441],[38,5202,5082],{"className":5203},[441],[38,5205,370],{"className":5206},[473,441],[38,5208,5210,5213],{"className":5209},[82,441],[38,5211,373],{"className":5212},[82,83,441],[38,5214,5216],{"className":5215},[412],[38,5217,5219,5239],{"className":5218},[416,417],[38,5220,5222,5236],{"className":5221},[421],[38,5223,5225],{"className":5224,"style":5036},[425],[38,5226,5227,5230],{"style":5039},[38,5228],{"className":5229,"style":5043},[433],[38,5231,5233],{"className":5232},[438,5047,5048,441],[38,5234,4572],{"className":5235,"style":4641},[82,83,441],[38,5237,460],{"className":5238},[459],[38,5240,5242],{"className":5241},[421],[38,5243,5245],{"className":5244,"style":5061},[425],[38,5246],{},[38,5248,376],{"className":5249},[480,441],[38,5251,460],{"className":5252},[459],[38,5254,5256],{"className":5255},[421],[38,5257,5260],{"className":5258,"style":5259},[425],"height:0.5423em;",[38,5261],{},[38,5263],{"className":5264},[480,683],[134,5266,5267],{"align":110},[38,5268,5270,5291],{"className":5269},[41],[38,5271,5273],{"className":5272},[45],[47,5274,5275],{"xmlns":49},[51,5276,5277,5289],{},[54,5278,5279,5281,5283,5285,5287],{},[361,5280,4779],{"stretchy":369},[559,5282,159],{},[361,5284,363],{"separator":69},[559,5286,178],{},[361,5288,4788],{"stretchy":369},[61,5290,4791],{"encoding":63},[38,5292,5294],{"className":5293,"ariaHidden":69},[68],[38,5295,5297,5300,5303,5306,5309,5312,5315],{"className":5296},[73],[38,5298],{"className":5299,"style":637},[77],[38,5301,4779],{"className":5302},[473],[38,5304,159],{"className":5305},[82],[38,5307,363],{"className":5308},[452],[38,5310],{"className":5311,"style":654},[484],[38,5313,178],{"className":5314},[82],[38,5316,4788],{"className":5317},[480],[105,5319,5320,5323,5728],{},[134,5321,5322],{"align":110},"Normalização de média",[134,5324,5325],{"align":110},[38,5326,5328,5398],{"className":5327},[41],[38,5329,5331],{"className":5330},[45],[47,5332,5333],{"xmlns":49},[51,5334,5335,5395],{},[54,5336,5337,5343,5345],{},[350,5338,5339,5341],{},[57,5340,373],{},[57,5342,4572],{},[361,5344,4575],{},[556,5346,5347,5363],{},[54,5348,5349,5355,5357],{},[350,5350,5351,5353],{},[57,5352,373],{},[57,5354,4572],{},[361,5356,605],{},[350,5358,5359,5361],{},[57,5360,3830],{},[57,5362,4572],{},[54,5364,5365,5367,5369,5371,5377,5379,5381,5383,5385,5387,5393],{},[57,5366,2647],{},[361,5368,2650],{},[361,5370,370],{"stretchy":369},[350,5372,5373,5375],{},[57,5374,373],{},[57,5376,4572],{},[361,5378,376],{"stretchy":369},[361,5380,605],{},[57,5382,4860],{},[361,5384,2650],{},[361,5386,370],{"stretchy":369},[350,5388,5389,5391],{},[57,5390,373],{},[57,5392,4572],{},[361,5394,376],{"stretchy":369},[61,5396,5397],{"encoding":63},"x_j \\leftarrow \\frac{x_j - \\mu_j}{\\max(x_j) - \\min(x_j)}",[38,5399,5401,5456],{"className":5400,"ariaHidden":69},[68],[38,5402,5404,5407,5447,5450,5453],{"className":5403},[73],[38,5405],{"className":5406,"style":4610},[77],[38,5408,5410,5413],{"className":5409},[82],[38,5411,373],{"className":5412},[82,83],[38,5414,5416],{"className":5415},[412],[38,5417,5419,5439],{"className":5418},[416,417],[38,5420,5422,5436],{"className":5421},[421],[38,5423,5425],{"className":5424,"style":4629},[425],[38,5426,5427,5430],{"style":2826},[38,5428],{"className":5429,"style":434},[433],[38,5431,5433],{"className":5432},[438,439,440,441],[38,5434,4572],{"className":5435,"style":4641},[82,83,441],[38,5437,460],{"className":5438},[459],[38,5440,5442],{"className":5441},[421],[38,5443,5445],{"className":5444,"style":467},[425],[38,5446],{},[38,5448],{"className":5449,"style":485},[484],[38,5451,4575],{"className":5452},[489],[38,5454],{"className":5455,"style":485},[484],[38,5457,5459,5463],{"className":5458},[73],[38,5460],{"className":5461,"style":5462},[77],"height:1.458em;vertical-align:-0.5423em;",[38,5464,5466,5469,5725],{"className":5465},[82],[38,5467],{"className":5468},[473,683],[38,5470,5472],{"className":5471},[556],[38,5473,5475,5717],{"className":5474},[416,417],[38,5476,5478,5714],{"className":5477},[421],[38,5479,5482,5612,5620],{"className":5480,"style":5481},[425],"height:0.9157em;",[38,5483,5484,5487],{"style":699},[38,5485],{"className":5486,"style":703},[433],[38,5488,5490],{"className":5489},[438,439,440,441],[38,5491,5493,5505,5508,5548,5551,5554,5566,5569,5609],{"className":5492},[82,441],[38,5494,5496,5499,5502],{"className":5495},[763,441],[38,5497,567],{"className":5498},[441],[38,5500,27],{"className":5501},[441],[38,5503,373],{"className":5504},[441],[38,5506,370],{"className":5507},[473,441],[38,5509,5511,5514],{"className":5510},[82,441],[38,5512,373],{"className":5513},[82,83,441],[38,5515,5517],{"className":5516},[412],[38,5518,5520,5540],{"className":5519},[416,417],[38,5521,5523,5537],{"className":5522},[421],[38,5524,5526],{"className":5525,"style":5036},[425],[38,5527,5528,5531],{"style":5039},[38,5529],{"className":5530,"style":5043},[433],[38,5532,5534],{"className":5533},[438,5047,5048,441],[38,5535,4572],{"className":5536,"style":4641},[82,83,441],[38,5538,460],{"className":5539},[459],[38,5541,5543],{"className":5542},[421],[38,5544,5546],{"className":5545,"style":5061},[425],[38,5547],{},[38,5549,376],{"className":5550},[480,441],[38,5552,605],{"className":5553},[513,441],[38,5555,5557,5560,5563],{"className":5556},[763,441],[38,5558,567],{"className":5559},[441],[38,5561,598],{"className":5562},[441],[38,5564,5082],{"className":5565},[441],[38,5567,370],{"className":5568},[473,441],[38,5570,5572,5575],{"className":5571},[82,441],[38,5573,373],{"className":5574},[82,83,441],[38,5576,5578],{"className":5577},[412],[38,5579,5581,5601],{"className":5580},[416,417],[38,5582,5584,5598],{"className":5583},[421],[38,5585,5587],{"className":5586,"style":5036},[425],[38,5588,5589,5592],{"style":5039},[38,5590],{"className":5591,"style":5043},[433],[38,5593,5595],{"className":5594},[438,5047,5048,441],[38,5596,4572],{"className":5597,"style":4641},[82,83,441],[38,5599,460],{"className":5600},[459],[38,5602,5604],{"className":5603},[421],[38,5605,5607],{"className":5606,"style":5061},[425],[38,5608],{},[38,5610,376],{"className":5611},[480,441],[38,5613,5614,5617],{"style":718},[38,5615],{"className":5616,"style":703},[433],[38,5618],{"className":5619,"style":726},[725],[38,5621,5622,5625],{"style":5139},[38,5623],{"className":5624,"style":703},[433],[38,5626,5628],{"className":5627},[438,439,440,441],[38,5629,5631,5671,5674],{"className":5630},[82,441],[38,5632,5634,5637],{"className":5633},[82,441],[38,5635,373],{"className":5636},[82,83,441],[38,5638,5640],{"className":5639},[412],[38,5641,5643,5663],{"className":5642},[416,417],[38,5644,5646,5660],{"className":5645},[421],[38,5647,5649],{"className":5648,"style":5036},[425],[38,5650,5651,5654],{"style":5039},[38,5652],{"className":5653,"style":5043},[433],[38,5655,5657],{"className":5656},[438,5047,5048,441],[38,5658,4572],{"className":5659,"style":4641},[82,83,441],[38,5661,460],{"className":5662},[459],[38,5664,5666],{"className":5665},[421],[38,5667,5669],{"className":5668,"style":5061},[425],[38,5670],{},[38,5672,605],{"className":5673},[513,441],[38,5675,5677,5680],{"className":5676},[82,441],[38,5678,3830],{"className":5679},[82,83,441],[38,5681,5683],{"className":5682},[412],[38,5684,5686,5706],{"className":5685},[416,417],[38,5687,5689,5703],{"className":5688},[421],[38,5690,5692],{"className":5691,"style":5036},[425],[38,5693,5694,5697],{"style":5039},[38,5695],{"className":5696,"style":5043},[433],[38,5698,5700],{"className":5699},[438,5047,5048,441],[38,5701,4572],{"className":5702,"style":4641},[82,83,441],[38,5704,460],{"className":5705},[459],[38,5707,5709],{"className":5708},[421],[38,5710,5712],{"className":5711,"style":5061},[425],[38,5713],{},[38,5715,460],{"className":5716},[459],[38,5718,5720],{"className":5719},[421],[38,5721,5723],{"className":5722,"style":5259},[425],[38,5724],{},[38,5726],{"className":5727},[480,683],[134,5729,5730,5787],{"align":110},[38,5731,5733,5757],{"className":5732},[41],[38,5734,5736],{"className":5735},[45],[47,5737,5738],{"xmlns":49},[51,5739,5740,5754],{},[54,5741,5742,5744,5746,5748,5750,5752],{},[361,5743,4779],{"stretchy":369},[361,5745,605],{},[559,5747,178],{},[361,5749,363],{"separator":69},[559,5751,178],{},[361,5753,4788],{"stretchy":369},[61,5755,5756],{"encoding":63},"[-1, 1]",[38,5758,5760],{"className":5759,"ariaHidden":69},[68],[38,5761,5763,5766,5769,5772,5775,5778,5781,5784],{"className":5762},[73],[38,5764],{"className":5765,"style":637},[77],[38,5767,4779],{"className":5768},[473],[38,5770,605],{"className":5771},[82],[38,5773,178],{"className":5774},[82],[38,5776,363],{"className":5777},[452],[38,5779],{"className":5780,"style":654},[484],[38,5782,178],{"className":5783},[82],[38,5785,4788],{"className":5786},[480],", média 0",[105,5789,5790,5793,6093],{},[134,5791,5792],{"align":110},"Z-score",[134,5794,5795],{"align":110},[38,5796,5798,5843],{"className":5797},[41],[38,5799,5801],{"className":5800},[45],[47,5802,5803],{"xmlns":49},[51,5804,5805,5840],{},[54,5806,5807,5813,5815],{},[350,5808,5809,5811],{},[57,5810,373],{},[57,5812,4572],{},[361,5814,4575],{},[556,5816,5817,5833],{},[54,5818,5819,5825,5827],{},[350,5820,5821,5823],{},[57,5822,373],{},[57,5824,4572],{},[361,5826,605],{},[350,5828,5829,5831],{},[57,5830,3830],{},[57,5832,4572],{},[350,5834,5835,5838],{},[57,5836,5837],{},"σ",[57,5839,4572],{},[61,5841,5842],{"encoding":63},"x_j \\leftarrow \\frac{x_j - \\mu_j}{\\sigma_j}",[38,5844,5846,5901],{"className":5845,"ariaHidden":69},[68],[38,5847,5849,5852,5892,5895,5898],{"className":5848},[73],[38,5850],{"className":5851,"style":4610},[77],[38,5853,5855,5858],{"className":5854},[82],[38,5856,373],{"className":5857},[82,83],[38,5859,5861],{"className":5860},[412],[38,5862,5864,5884],{"className":5863},[416,417],[38,5865,5867,5881],{"className":5866},[421],[38,5868,5870],{"className":5869,"style":4629},[425],[38,5871,5872,5875],{"style":2826},[38,5873],{"className":5874,"style":434},[433],[38,5876,5878],{"className":5877},[438,439,440,441],[38,5879,4572],{"className":5880,"style":4641},[82,83,441],[38,5882,460],{"className":5883},[459],[38,5885,5887],{"className":5886},[421],[38,5888,5890],{"className":5889,"style":467},[425],[38,5891],{},[38,5893],{"className":5894,"style":485},[484],[38,5896,4575],{"className":5897},[489],[38,5899],{"className":5900,"style":485},[484],[38,5902,5904,5907],{"className":5903},[73],[38,5905],{"className":5906,"style":5462},[77],[38,5908,5910,5913,6090],{"className":5909},[82],[38,5911],{"className":5912},[473,683],[38,5914,5916],{"className":5915},[556],[38,5917,5919,6082],{"className":5918},[416,417],[38,5920,5922,6079],{"className":5921},[421],[38,5923,5925,5977,5985],{"className":5924,"style":5481},[425],[38,5926,5927,5930],{"style":699},[38,5928],{"className":5929,"style":703},[433],[38,5931,5933],{"className":5932},[438,439,440,441],[38,5934,5936],{"className":5935},[82,441],[38,5937,5939,5942],{"className":5938},[82,441],[38,5940,5837],{"className":5941,"style":887},[82,83,441],[38,5943,5945],{"className":5944},[412],[38,5946,5948,5969],{"className":5947},[416,417],[38,5949,5951,5966],{"className":5950},[421],[38,5952,5954],{"className":5953,"style":5036},[425],[38,5955,5957,5960],{"style":5956},"top:-2.357em;margin-left:-0.0359em;margin-right:0.0714em;",[38,5958],{"className":5959,"style":5043},[433],[38,5961,5963],{"className":5962},[438,5047,5048,441],[38,5964,4572],{"className":5965,"style":4641},[82,83,441],[38,5967,460],{"className":5968},[459],[38,5970,5972],{"className":5971},[421],[38,5973,5975],{"className":5974,"style":5061},[425],[38,5976],{},[38,5978,5979,5982],{"style":718},[38,5980],{"className":5981,"style":703},[433],[38,5983],{"className":5984,"style":726},[725],[38,5986,5987,5990],{"style":5139},[38,5988],{"className":5989,"style":703},[433],[38,5991,5993],{"className":5992},[438,439,440,441],[38,5994,5996,6036,6039],{"className":5995},[82,441],[38,5997,5999,6002],{"className":5998},[82,441],[38,6000,373],{"className":6001},[82,83,441],[38,6003,6005],{"className":6004},[412],[38,6006,6008,6028],{"className":6007},[416,417],[38,6009,6011,6025],{"className":6010},[421],[38,6012,6014],{"className":6013,"style":5036},[425],[38,6015,6016,6019],{"style":5039},[38,6017],{"className":6018,"style":5043},[433],[38,6020,6022],{"className":6021},[438,5047,5048,441],[38,6023,4572],{"className":6024,"style":4641},[82,83,441],[38,6026,460],{"className":6027},[459],[38,6029,6031],{"className":6030},[421],[38,6032,6034],{"className":6033,"style":5061},[425],[38,6035],{},[38,6037,605],{"className":6038},[513,441],[38,6040,6042,6045],{"className":6041},[82,441],[38,6043,3830],{"className":6044},[82,83,441],[38,6046,6048],{"className":6047},[412],[38,6049,6051,6071],{"className":6050},[416,417],[38,6052,6054,6068],{"className":6053},[421],[38,6055,6057],{"className":6056,"style":5036},[425],[38,6058,6059,6062],{"style":5039},[38,6060],{"className":6061,"style":5043},[433],[38,6063,6065],{"className":6064},[438,5047,5048,441],[38,6066,4572],{"className":6067,"style":4641},[82,83,441],[38,6069,460],{"className":6070},[459],[38,6072,6074],{"className":6073},[421],[38,6075,6077],{"className":6076,"style":5061},[425],[38,6078],{},[38,6080,460],{"className":6081},[459],[38,6083,6085],{"className":6084},[421],[38,6086,6088],{"className":6087,"style":5259},[425],[38,6089],{},[38,6091],{"className":6092},[480,683],[134,6094,6095],{"align":110},"média 0, desvio-padrão 1",[11,6097,6098,6099,2459,6169,6240,6241,6270,6271,6274],{},"Usei z-score daqui pra frente, é a opção mais robusta (não depende só de dois pontos extremos do dataset, como o min-max depende). ",[38,6100,6102,6120],{"className":6101},[41],[38,6103,6105],{"className":6104},[45],[47,6106,6107],{"xmlns":49},[51,6108,6109,6117],{},[54,6110,6111],{},[350,6112,6113,6115],{},[57,6114,3830],{},[57,6116,4572],{},[61,6118,6119],{"encoding":63},"\\mu_j",[38,6121,6123],{"className":6122,"ariaHidden":69},[68],[38,6124,6126,6129],{"className":6125},[73],[38,6127],{"className":6128,"style":4610},[77],[38,6130,6132,6135],{"className":6131},[82],[38,6133,3830],{"className":6134},[82,83],[38,6136,6138],{"className":6137},[412],[38,6139,6141,6161],{"className":6140},[416,417],[38,6142,6144,6158],{"className":6143},[421],[38,6145,6147],{"className":6146,"style":4629},[425],[38,6148,6149,6152],{"style":2826},[38,6150],{"className":6151,"style":434},[433],[38,6153,6155],{"className":6154},[438,439,440,441],[38,6156,4572],{"className":6157,"style":4641},[82,83,441],[38,6159,460],{"className":6160},[459],[38,6162,6164],{"className":6163},[421],[38,6165,6167],{"className":6166,"style":467},[425],[38,6168],{},[38,6170,6172,6190],{"className":6171},[41],[38,6173,6175],{"className":6174},[45],[47,6176,6177],{"xmlns":49},[51,6178,6179,6187],{},[54,6180,6181],{},[350,6182,6183,6185],{},[57,6184,5837],{},[57,6186,4572],{},[61,6188,6189],{"encoding":63},"\\sigma_j",[38,6191,6193],{"className":6192,"ariaHidden":69},[68],[38,6194,6196,6199],{"className":6195},[73],[38,6197],{"className":6198,"style":4610},[77],[38,6200,6202,6205],{"className":6201},[82],[38,6203,5837],{"className":6204,"style":887},[82,83],[38,6206,6208],{"className":6207},[412],[38,6209,6211,6232],{"className":6210},[416,417],[38,6212,6214,6229],{"className":6213},[421],[38,6215,6217],{"className":6216,"style":4629},[425],[38,6218,6220,6223],{"style":6219},"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;",[38,6221],{"className":6222,"style":434},[433],[38,6224,6226],{"className":6225},[438,439,440,441],[38,6227,4572],{"className":6228,"style":4641},[82,83,441],[38,6230,460],{"className":6231},[459],[38,6233,6235],{"className":6234},[421],[38,6236,6238],{"className":6237,"style":467},[425],[38,6239],{}," são a média e o desvio-padrão da feature ",[38,6242,6244,6257],{"className":6243},[41],[38,6245,6247],{"className":6246},[45],[47,6248,6249],{"xmlns":49},[51,6250,6251,6255],{},[54,6252,6253],{},[57,6254,4572],{},[61,6256,4572],{"encoding":63},[38,6258,6260],{"className":6259,"ariaHidden":69},[68],[38,6261,6263,6267],{"className":6262},[73],[38,6264],{"className":6265,"style":6266},[77],"height:0.854em;vertical-align:-0.1944em;",[38,6268,4572],{"className":6269,"style":4641},[82,83],", calculados ",[33,6272,6273],{},"só com os dados de treino",". Isso importa: eu guardo esses dois números e reuso pra qualquer dado novo que aparecer depois, nunca recalculo.",[1308,6276,6278],{"className":1310,"code":6277,"language":1312,"meta":1313,"style":1313},"def zscore_normalize_features(X):\n    mu = X.mean(axis=0)      # média de cada coluna\n    sigma = X.std(axis=0)    # desvio-padrao de cada coluna\n    X_norm = (X - mu) \u002F sigma\n    return X_norm, mu, sigma\n",[22,6279,6280,6285,6290,6296,6302],{"__ignoreMap":1313},[38,6281,6282],{"class":1318,"line":1319},[38,6283,6284],{},"def zscore_normalize_features(X):\n",[38,6286,6287],{"class":1318,"line":1325},[38,6288,6289],{},"    mu = X.mean(axis=0)      # média de cada coluna\n",[38,6291,6293],{"class":1318,"line":6292},3,[38,6294,6295],{},"    sigma = X.std(axis=0)    # desvio-padrao de cada coluna\n",[38,6297,6299],{"class":1318,"line":6298},4,[38,6300,6301],{},"    X_norm = (X - mu) \u002F sigma\n",[38,6303,6305],{"class":1318,"line":6304},5,[38,6306,6307],{},"    return X_norm, mu, sigma\n",[11,6309,6310],{},"Pro nosso exemplo de 8 casas, a média do tamanho é 1480.1 pés² e o desvio-padrão é 414.7. Normalizei e a amplitude, que antes ia de centenas a milhares, virou algo entre -1 e 2, mais ou menos.",[11,6312,6313,6314,6317],{},"Um detalhe que eu quase escorreguei: z-score ",[33,6315,6316],{},"não"," deixa o dado com distribuição normal, só reescala. A forma da distribuição continua a mesma, só o centro e a escala mudam.",[87,6319,6321],{"id":6320},"gradiente-descendente-com-features-normalizadas","Gradiente descendente com features normalizadas",[11,6323,6324,6325,6377,6378,6450],{},"Rodei o mesmo algoritmo, no mesmo dataset de 8 casas, só que agora com o tamanho normalizado por z-score, e usei ",[38,6326,6328,6347],{"className":6327},[41],[38,6329,6331],{"className":6330},[45],[47,6332,6333],{"xmlns":49},[51,6334,6335,6344],{},[54,6336,6337,6339,6341],{},[57,6338,59],{},[361,6340,379],{},[559,6342,6343],{},"0.1",[61,6345,6346],{"encoding":63},"\\alpha = 0.1",[38,6348,6350,6368],{"className":6349,"ariaHidden":69},[68],[38,6351,6353,6356,6359,6362,6365],{"className":6352},[73],[38,6354],{"className":6355,"style":78},[77],[38,6357,59],{"className":6358,"style":84},[82,83],[38,6360],{"className":6361,"style":485},[484],[38,6363,379],{"className":6364},[489],[38,6366],{"className":6367,"style":485},[484],[38,6369,6371,6374],{"className":6370},[73],[38,6372],{"className":6373,"style":1452},[77],[38,6375,6343],{"className":6376},[82],", seis ordens de grandeza maior que os ",[38,6379,6381,6403],{"className":6380},[41],[38,6382,6384],{"className":6383},[45],[47,6385,6386],{"xmlns":49},[51,6387,6388,6400],{},[54,6389,6390],{},[588,6391,6392,6394],{},[559,6393,1491],{},[54,6395,6396,6398],{},[361,6397,605],{},[559,6399,1498],{},[61,6401,6402],{"encoding":63},"10^{-7}",[38,6404,6406],{"className":6405,"ariaHidden":69},[68],[38,6407,6409,6412,6415],{"className":6408},[73],[38,6410],{"className":6411,"style":938},[77],[38,6413,178],{"className":6414},[82],[38,6416,6418,6421],{"className":6417},[82],[38,6419,159],{"className":6420},[82],[38,6422,6424],{"className":6423},[412],[38,6425,6427],{"className":6426},[416],[38,6428,6430],{"className":6429},[421],[38,6431,6433],{"className":6432,"style":938},[425],[38,6434,6435,6438],{"style":843},[38,6436],{"className":6437,"style":434},[433],[38,6439,6441],{"className":6440},[438,439,440,441],[38,6442,6444,6447],{"className":6443},[82,441],[38,6445,605],{"className":6446},[82,441],[38,6448,1498],{"className":6449},[82,441]," de antes:",[2353,6452],{":b-range":6453,":initial-b":159,":initial-w":159,":w-range":6454,":x-train":6455,":y-train":2358,"b-label":366,"w-label":6456,":initial-alpha":6343},"[300, 500]","[-50, 200]","[-1.273, -0.569, 1.125, 0.591, 1.155, -0.400, -1.486, 0.858]","w (normalizado)",[11,6458,6459,6460,6512,6513,6542,6543,6637,6638,6721],{},"Depois de umas 50 iterações o custo já achatou perto do mínimo (achei ",[38,6461,6463,6482],{"className":6462},[41],[38,6464,6466],{"className":6465},[45],[47,6467,6468],{"xmlns":49},[51,6469,6470,6479],{},[54,6471,6472,6474,6476],{},[57,6473,542],{},[361,6475,3154],{},[559,6477,6478],{},"919",[61,6480,6481],{"encoding":63},"J \\approx 919",[38,6483,6485,6503],{"className":6484,"ariaHidden":69},[68],[38,6486,6488,6491,6494,6497,6500],{"className":6487},[73],[38,6489],{"className":6490,"style":3021},[77],[38,6492,542],{"className":6493,"style":641},[82,83],[38,6495],{"className":6496,"style":485},[484],[38,6498,3154],{"className":6499},[489],[38,6501],{"className":6502,"style":485},[484],[38,6504,6506,6509],{"className":6505},[73],[38,6507],{"className":6508,"style":1452},[77],[38,6510,6478],{"className":6511},[82],", quase idêntico ao mínimo real de ",[38,6514,6516,6530],{"className":6515},[41],[38,6517,6519],{"className":6518},[45],[47,6520,6521],{"xmlns":49},[51,6522,6523,6528],{},[54,6524,6525],{},[559,6526,6527],{},"919.37",[61,6529,6527],{"encoding":63},[38,6531,6533],{"className":6532,"ariaHidden":69},[68],[38,6534,6536,6539],{"className":6535},[73],[38,6537],{"className":6538,"style":1452},[77],[38,6540,6527],{"className":6541},[82]," que eu calculei direto por fórmula fechada). No exemplo cru, com ",[38,6544,6546,6572],{"className":6545},[41],[38,6547,6549],{"className":6548},[45],[47,6550,6551],{"xmlns":49},[51,6552,6553,6569],{},[54,6554,6555,6557,6559],{},[57,6556,59],{},[361,6558,379],{},[588,6560,6561,6563],{},[559,6562,1491],{},[54,6564,6565,6567],{},[361,6566,605],{},[559,6568,1498],{},[61,6570,6571],{"encoding":63},"\\alpha = 10^{-7}",[38,6573,6575,6593],{"className":6574,"ariaHidden":69},[68],[38,6576,6578,6581,6584,6587,6590],{"className":6577},[73],[38,6579],{"className":6580,"style":78},[77],[38,6582,59],{"className":6583,"style":84},[82,83],[38,6585],{"className":6586,"style":485},[484],[38,6588,379],{"className":6589},[489],[38,6591],{"className":6592,"style":485},[484],[38,6594,6596,6599,6602],{"className":6595},[73],[38,6597],{"className":6598,"style":938},[77],[38,6600,178],{"className":6601},[82],[38,6603,6605,6608],{"className":6604},[82],[38,6606,159],{"className":6607},[82],[38,6609,6611],{"className":6610},[412],[38,6612,6614],{"className":6613},[416],[38,6615,6617],{"className":6616},[421],[38,6618,6620],{"className":6619,"style":938},[425],[38,6621,6622,6625],{"style":843},[38,6623],{"className":6624,"style":434},[433],[38,6626,6628],{"className":6627},[438,439,440,441],[38,6629,6631,6634],{"className":6630},[82,441],[38,6632,605],{"className":6633},[82,441],[38,6635,1498],{"className":6636},[82,441],", o custo mal tinha saído de ",[38,6639,6641,6662],{"className":6640},[41],[38,6642,6644],{"className":6643},[45],[47,6645,6646],{"xmlns":49},[51,6647,6648,6660],{},[54,6649,6650,6652,6654],{},[559,6651,2274],{},[361,6653,1486],{},[588,6655,6656,6658],{},[559,6657,1491],{},[559,6659,1990],{},[61,6661,2285],{"encoding":63},[38,6663,6665,6683],{"className":6664,"ariaHidden":69},[68],[38,6666,6668,6671,6674,6677,6680],{"className":6667},[73],[38,6669],{"className":6670,"style":1529},[77],[38,6672,2274],{"className":6673},[82],[38,6675],{"className":6676,"style":509},[484],[38,6678,1486],{"className":6679},[513],[38,6681],{"className":6682,"style":509},[484],[38,6684,6686,6689,6692],{"className":6685},[73],[38,6687],{"className":6688,"style":938},[77],[38,6690,178],{"className":6691},[82],[38,6693,6695,6698],{"className":6694},[82],[38,6696,159],{"className":6697},[82],[38,6699,6701],{"className":6700},[412],[38,6702,6704],{"className":6703},[416],[38,6705,6707],{"className":6706},[421],[38,6708,6710],{"className":6709,"style":938},[425],[38,6711,6712,6715],{"style":843},[38,6713],{"className":6714,"style":434},[433],[38,6716,6718],{"className":6717},[438,439,440,441],[38,6719,1990],{"className":6720},[82,441]," depois de 10 iterações inteiras. Normalizar não mudou onde fica o mínimo, mudou completamente a velocidade de chegar lá.",[11,6723,6724,6725,6753,6754,6782,6783,6785,6786,6788],{},"Reparei também numa coisa boba mas satisfeita de entender: quando ",[38,6726,6728,6741],{"className":6727},[41],[38,6729,6731],{"className":6730},[45],[47,6732,6733],{"xmlns":49},[51,6734,6735,6739],{},[54,6736,6737],{},[57,6738,373],{},[61,6740,373],{"encoding":63},[38,6742,6744],{"className":6743,"ariaHidden":69},[68],[38,6745,6747,6750],{"className":6746},[73],[38,6748],{"className":6749,"style":78},[77],[38,6751,373],{"className":6752},[82,83]," está centrado em zero (média zero, por causa do z-score), o ",[38,6755,6757,6770],{"className":6756},[41],[38,6758,6760],{"className":6759},[45],[47,6761,6762],{"xmlns":49},[51,6763,6764,6768],{},[54,6765,6766],{},[57,6767,366],{},[61,6769,366],{"encoding":63},[38,6771,6773],{"className":6772,"ariaHidden":69},[68],[38,6774,6776,6779],{"className":6775},[73],[38,6777],{"className":6778,"style":523},[77],[38,6780,366],{"className":6781},[82,83]," ótimo vira exatamente a média dos preços de treino. Faz sentido geometricamente: se todo ",[22,6784,373],{}," é zero na média, a melhor reta passa pela média dos ",[22,6787,610],{}," nesse ponto.",[87,6790,6792],{"id":6791},"prevendo-uma-casa-nova-e-desnormalizando","Prevendo uma casa nova e desnormalizando",[11,6794,6795],{},"Ajustei o modelo com as 4 features de verdade (não só tamanho) no dataset completo de 100 casas, por equação normal (a solução exata, sem gradiente descendente):",[1308,6797,6799],{"className":1310,"code":6798,"language":1312,"meta":1313,"style":1313},"w = [0.268, -32.90, -67.29, -1.465]   # tamanho, quartos, andares, idade\nb = 221.50\n",[22,6800,6801,6806],{"__ignoreMap":1313},[38,6802,6803],{"class":1318,"line":1319},[38,6804,6805],{},"w = [0.268, -32.90, -67.29, -1.465]   # tamanho, quartos, andares, idade\n",[38,6807,6808],{"class":1318,"line":1325},[38,6809,6810],{},"b = 221.50\n",[11,6812,6813,6814,97],{},"Previ o preço da casa-alvo (1200 pés², 3 quartos, 1 andar, 40 anos) e cheguei em ",[33,6815,6816],{},"318.9 mil dólares",[11,6818,6819,6820,6823],{},"O que mais me chamou atenção foi o coeficiente de quartos: ",[33,6821,6822],{},"negativo",". Isolado, ter mais quartos parece só coisa boa, mas controlando pelo tamanho da casa (que já está no modelo), mais quartos numa casa do mesmo tamanho normalmente significa quartos menores, o que puxa o preço um pouco pra baixo. É exatamente o tipo de coisa que só aparece quando você olha os coeficientes de um modelo com várias features ao mesmo tempo, não dá pra ver isso num gráfico de quartos-vs-preço sozinho.",[11,6825,6826,6827,6898],{},"Se eu tivesse ajustado com as features normalizadas, os pesos viriam numa escala diferente (cada ",[38,6828,6830,6848],{"className":6829},[41],[38,6831,6833],{"className":6832},[45],[47,6834,6835],{"xmlns":49},[51,6836,6837,6845],{},[54,6838,6839],{},[350,6840,6841,6843],{},[57,6842,359],{},[57,6844,4572],{},[61,6846,6847],{"encoding":63},"w_j",[38,6849,6851],{"className":6850,"ariaHidden":69},[68],[38,6852,6854,6857],{"className":6853},[73],[38,6855],{"className":6856,"style":4610},[77],[38,6858,6860,6863],{"className":6859},[82],[38,6861,359],{"className":6862,"style":448},[82,83],[38,6864,6866],{"className":6865},[412],[38,6867,6869,6890],{"className":6868},[416,417],[38,6870,6872,6887],{"className":6871},[421],[38,6873,6875],{"className":6874,"style":4629},[425],[38,6876,6878,6881],{"style":6877},"top:-2.55em;margin-left:-0.0269em;margin-right:0.05em;",[38,6879],{"className":6880,"style":434},[433],[38,6882,6884],{"className":6883},[438,439,440,441],[38,6885,4572],{"className":6886,"style":4641},[82,83,441],[38,6888,460],{"className":6889},[459],[38,6891,6893],{"className":6892},[421],[38,6894,6896],{"className":6895,"style":467},[425],[38,6897],{}," representaria \"quanto o preço muda por 1 desvio-padrão daquela feature\", não por 1 unidade real). Pra voltar aos coeficientes em unidade real, a fórmula de desnormalização é:",[11,6900,6901],{},[38,6902,6904,7098],{"className":6903},[41],[38,6905,6907],{"className":6906},[45],[47,6908,6909],{"xmlns":49},[51,6910,6911,7095],{},[54,6912,6913,6942,6944,6984,6986,7008,7010,7038,7040,7046],{},[6914,6915,6916,6918,6920],"msubsup",{},[57,6917,359],{},[57,6919,4572],{},[54,6921,6922,6925,6928,6930,6933,6935,6937,6939],{},[57,6923,6924],{},"o",[57,6926,6927],{},"r",[57,6929,598],{},[57,6931,6932],{},"g",[57,6934,598],{},[57,6936,5082],{},[57,6938,27],{},[57,6940,6941],{},"l",[361,6943,379],{},[556,6945,6946,6978],{},[6914,6947,6948,6950,6952],{},[57,6949,359],{},[57,6951,4572],{},[54,6953,6954,6956,6958,6960,6962,6964,6966,6968,6971,6973,6976],{},[57,6955,5082],{},[57,6957,6924],{},[57,6959,6927],{},[57,6961,567],{},[57,6963,27],{},[57,6965,6941],{},[57,6967,598],{},[57,6969,6970],{},"z",[57,6972,27],{},[57,6974,6975],{},"d",[57,6977,6924],{},[350,6979,6980,6982],{},[57,6981,5837],{},[57,6983,4572],{},[484,6985],{"width":997},[588,6987,6988,6990],{},[57,6989,366],{},[54,6991,6992,6994,6996,6998,7000,7002,7004,7006],{},[57,6993,6924],{},[57,6995,6927],{},[57,6997,598],{},[57,6999,6932],{},[57,7001,598],{},[57,7003,5082],{},[57,7005,27],{},[57,7007,6941],{},[361,7009,379],{},[588,7011,7012,7014],{},[57,7013,366],{},[54,7015,7016,7018,7020,7022,7024,7026,7028,7030,7032,7034,7036],{},[57,7017,5082],{},[57,7019,6924],{},[57,7021,6927],{},[57,7023,567],{},[57,7025,27],{},[57,7027,6941],{},[57,7029,598],{},[57,7031,6970],{},[57,7033,27],{},[57,7035,6975],{},[57,7037,6924],{},[361,7039,605],{},[350,7041,7042,7044],{},[361,7043,570],{},[57,7045,4572],{},[556,7047,7048,7089],{},[54,7049,7050,7080,7083],{},[6914,7051,7052,7054,7056],{},[57,7053,359],{},[57,7055,4572],{},[54,7057,7058,7060,7062,7064,7066,7068,7070,7072,7074,7076,7078],{},[57,7059,5082],{},[57,7061,6924],{},[57,7063,6927],{},[57,7065,567],{},[57,7067,27],{},[57,7069,6941],{},[57,7071,598],{},[57,7073,6970],{},[57,7075,27],{},[57,7077,6975],{},[57,7079,6924],{},[361,7081,7082],{},"⋅",[350,7084,7085,7087],{},[57,7086,3830],{},[57,7088,4572],{},[350,7090,7091,7093],{},[57,7092,5837],{},[57,7094,4572],{},[61,7096,7097],{"encoding":63},"w_j^{original} = \\frac{w_j^{normalizado}}{\\sigma_j} \\qquad b^{original} = b^{normalizado} - \\sum_j \\frac{w_j^{normalizado} \\cdot \\mu_j}{\\sigma_j}",[38,7099,7101,7191,7443,7515],{"className":7100,"ariaHidden":69},[68],[38,7102,7104,7108,7182,7185,7188],{"className":7103},[73],[38,7105],{"className":7106,"style":7107},[77],"height:1.38em;vertical-align:-0.413em;",[38,7109,7111,7114],{"className":7110},[82],[38,7112,359],{"className":7113,"style":448},[82,83],[38,7115,7117],{"className":7116},[412],[38,7118,7120,7173],{"className":7119},[416,417],[38,7121,7123,7170],{"className":7122},[421],[38,7124,7127,7139],{"className":7125,"style":7126},[425],"height:0.967em;",[38,7128,7130,7133],{"style":7129},"top:-2.4231em;margin-left:-0.0269em;margin-right:0.05em;",[38,7131],{"className":7132,"style":434},[433],[38,7134,7136],{"className":7135},[438,439,440,441],[38,7137,4572],{"className":7138,"style":4641},[82,83,441],[38,7140,7142,7145],{"style":7141},"top:-3.1809em;margin-right:0.05em;",[38,7143],{"className":7144,"style":434},[433],[38,7146,7148],{"className":7147},[438,439,440,441],[38,7149,7151,7156,7159,7162,7166],{"className":7150},[82,441],[38,7152,7155],{"className":7153,"style":7154},[82,83,441],"margin-right:0.0278em;","or",[38,7157,598],{"className":7158},[82,83,441],[38,7160,6932],{"className":7161,"style":887},[82,83,441],[38,7163,7165],{"className":7164},[82,83,441],"ina",[38,7167,6941],{"className":7168,"style":7169},[82,83,441],"margin-right:0.0197em;",[38,7171,460],{"className":7172},[459],[38,7174,7176],{"className":7175},[421],[38,7177,7180],{"className":7178,"style":7179},[425],"height:0.413em;",[38,7181],{},[38,7183],{"className":7184,"style":485},[484],[38,7186,379],{"className":7187},[489],[38,7189],{"className":7190,"style":485},[484],[38,7192,7194,7198,7386,7389,7434,7437,7440],{"className":7193},[73],[38,7195],{"className":7196,"style":7197},[77],"height:1.7987em;vertical-align:-0.5423em;",[38,7199,7201,7204,7383],{"className":7200},[82],[38,7202],{"className":7203},[473,683],[38,7205,7207],{"className":7206},[556],[38,7208,7210,7375],{"className":7209},[416,417],[38,7211,7213,7372],{"className":7212},[421],[38,7214,7217,7268,7276],{"className":7215,"style":7216},[425],"height:1.2563em;",[38,7218,7219,7222],{"style":699},[38,7220],{"className":7221,"style":703},[433],[38,7223,7225],{"className":7224},[438,439,440,441],[38,7226,7228],{"className":7227},[82,441],[38,7229,7231,7234],{"className":7230},[82,441],[38,7232,5837],{"className":7233,"style":887},[82,83,441],[38,7235,7237],{"className":7236},[412],[38,7238,7240,7260],{"className":7239},[416,417],[38,7241,7243,7257],{"className":7242},[421],[38,7244,7246],{"className":7245,"style":5036},[425],[38,7247,7248,7251],{"style":5956},[38,7249],{"className":7250,"style":5043},[433],[38,7252,7254],{"className":7253},[438,5047,5048,441],[38,7255,4572],{"className":7256,"style":4641},[82,83,441],[38,7258,460],{"className":7259},[459],[38,7261,7263],{"className":7262},[421],[38,7264,7266],{"className":7265,"style":5061},[425],[38,7267],{},[38,7269,7270,7273],{"style":718},[38,7271],{"className":7272,"style":703},[433],[38,7274],{"className":7275,"style":726},[725],[38,7277,7279,7282],{"style":7278},"top:-3.6074em;",[38,7280],{"className":7281,"style":703},[433],[38,7283,7285],{"className":7284},[438,439,440,441],[38,7286,7288],{"className":7287},[82,441],[38,7289,7291,7294],{"className":7290},[82,441],[38,7292,359],{"className":7293,"style":448},[82,83,441],[38,7295,7297],{"className":7296},[412],[38,7298,7300,7363],{"className":7299},[416,417],[38,7301,7303,7360],{"className":7302},[421],[38,7304,7307,7319],{"className":7305,"style":7306},[425],"height:0.927em;",[38,7308,7310,7313],{"style":7309},"top:-2.214em;margin-left:-0.0269em;margin-right:0.0714em;",[38,7311],{"className":7312,"style":5043},[433],[38,7314,7316],{"className":7315},[438,5047,5048,441],[38,7317,4572],{"className":7318,"style":4641},[82,83,441],[38,7320,7322,7325],{"style":7321},"top:-2.931em;margin-right:0.0714em;",[38,7323],{"className":7324,"style":5043},[433],[38,7326,7328],{"className":7327},[438,5047,5048,441],[38,7329,7331,7334,7337,7341,7344,7347,7351,7354,7357],{"className":7330},[82,441],[38,7332,5082],{"className":7333},[82,83,441],[38,7335,7155],{"className":7336,"style":7154},[82,83,441],[38,7338,7340],{"className":7339},[82,83,441],"ma",[38,7342,6941],{"className":7343,"style":7169},[82,83,441],[38,7345,598],{"className":7346},[82,83,441],[38,7348,6970],{"className":7349,"style":7350},[82,83,441],"margin-right:0.044em;",[38,7352,27],{"className":7353},[82,83,441],[38,7355,6975],{"className":7356},[82,83,441],[38,7358,6924],{"className":7359},[82,83,441],[38,7361,460],{"className":7362},[459],[38,7364,7366],{"className":7365},[421],[38,7367,7370],{"className":7368,"style":7369},[425],"height:0.4249em;",[38,7371],{},[38,7373,460],{"className":7374},[459],[38,7376,7378],{"className":7377},[421],[38,7379,7381],{"className":7380,"style":5259},[425],[38,7382],{},[38,7384],{"className":7385},[480,683],[38,7387],{"className":7388,"style":1153},[484],[38,7390,7392,7395],{"className":7391},[82],[3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assName":7694},[82,441],[38,7696,5082],{"className":7697},[82,83,441],[38,7699,7155],{"className":7700,"style":7154},[82,83,441],[38,7702,7340],{"className":7703},[82,83,441],[38,7705,6941],{"className":7706,"style":7169},[82,83,441],[38,7708,598],{"className":7709},[82,83,441],[38,7711,6970],{"className":7712,"style":7350},[82,83,441],[38,7714,27],{"className":7715},[82,83,441],[38,7717,6975],{"className":7718},[82,83,441],[38,7720,6924],{"className":7721},[82,83,441],[38,7723,460],{"className":7724},[459],[38,7726,7728],{"className":7727},[421],[38,7729,7731],{"className":7730,"style":7369},[425],[38,7732],{},[38,7734,7082],{"className":7735},[513,441],[38,7737,7739,7742],{"className":7738},[82,441],[38,7740,3830],{"className":7741},[82,83,441],[38,7743,7745],{"className":7744},[412],[38,7746,7748,7768],{"className":7747},[416,417],[38,7749,7751,7765],{"className":7750},[421],[38,7752,7754],{"className":7753,"style":5036},[425],[38,7755,7756,7759],{"style":5039},[38,7757],{"className":7758,"style":5043},[433],[38,7760,7762],{"className":7761},[438,5047,5048,441],[38,7763,4572],{"className":7764,"style":4641},[82,83,441],[38,7766,460],{"className":7767},[459],[38,7769,7771],{"className":7770},[421],[38,7772,7774],{"className":7773,"style":5061},[425],[38,7775],{},[38,7777,460],{"className":7778},[459],[38,7780,7782],{"className":7781},[421],[38,7783,7785],{"className":7784,"style":5259},[425],[38,7786],{},[38,7788],{"className":7789},[480,683],[11,7791,7792,7793,7821,7822,7850,7851,2459,7903,7955],{},"Conferi isso no nosso exemplo de 1 feature: desnormalizando o ",[38,7794,7796,7809],{"className":7795},[41],[38,7797,7799],{"className":7798},[45],[47,7800,7801],{"xmlns":49},[51,7802,7803,7807],{},[54,7804,7805],{},[57,7806,359],{},[61,7808,359],{"encoding":63},[38,7810,7812],{"className":7811,"ariaHidden":69},[68],[38,7813,7815,7818],{"className":7814},[73],[38,7816],{"className":7817,"style":78},[77],[38,7819,359],{"className":7820,"style":448},[82,83]," e o ",[38,7823,7825,7838],{"className":7824},[41],[38,7826,7828],{"className":7827},[45],[47,7829,7830],{"xmlns":49},[51,7831,7832,7836],{},[54,7833,7834],{},[57,7835,366],{},[61,7837,366],{"encoding":63},[38,7839,7841],{"className":7840,"ariaHidden":69},[68],[38,7842,7844,7847],{"className":7843},[73],[38,7845],{"className":7846,"style":523},[77],[38,7848,366],{"className":7849},[82,83]," que o gradiente descendente achou em espaço normalizado, cheguei em ",[38,7852,7854,7873],{"className":7853},[41],[38,7855,7857],{"className":7856},[45],[47,7858,7859],{"xmlns":49},[51,7860,7861,7870],{},[54,7862,7863,7865,7867],{},[57,7864,359],{},[361,7866,379],{},[559,7868,7869],{},"0.267",[61,7871,7872],{"encoding":63},"w = 0.267",[38,7874,7876,7894],{"className":7875,"ariaHidden":69},[68],[38,7877,7879,7882,7885,7888,7891],{"className":7878},[73],[38,7880],{"className":7881,"style":78},[77],[38,7883,359],{"className":7884,"style":448},[82,83],[38,7886],{"className":7887,"style":485},[484],[38,7889,379],{"className":7890},[489],[38,7892],{"className":7893,"style":485},[484],[38,7895,7897,7900],{"className":7896},[73],[38,7898],{"className":7899,"style":1452},[77],[38,7901,7869],{"className":7902},[82],[38,7904,7906,7925],{"className":7905},[41],[38,7907,7909],{"className":7908},[45],[47,7910,7911],{"xmlns":49},[51,7912,7913,7922],{},[54,7914,7915,7917,7919],{},[57,7916,366],{},[361,7918,379],{},[559,7920,7921],{},"7.16",[61,7923,7924],{"encoding":63},"b = 7.16",[38,7926,7928,7946],{"className":7927,"ariaHidden":69},[68],[38,7929,7931,7934,7937,7940,7943],{"className":7930},[73],[38,7932],{"className":7933,"style":523},[77],[38,7935,366],{"className":7936},[82,83],[38,7938],{"className":7939,"style":485},[484],[38,7941,379],{"className":7942},[489],[38,7944],{"className":7945,"style":485},[484],[38,7947,7949,7952],{"className":7948},[73],[38,7950],{"className":7951,"style":1452},[77],[38,7953,7921],{"className":7954},[82],", exatamente os mesmos números que a fórmula fechada dá direto em espaço cru. Os dois caminhos levam ao mesmo lugar, só que um deles convergiu em 50 iterações e o outro precisaria de milhões.",[87,7957,7959],{"id":7958},"os-contornos-a-visão-geométrica","Os contornos: a visão geométrica",[11,7961,7962,7963,2345,7966,2345,7969,2345,7972,7975],{},"Isso tudo que eu descrevi em número, dá pra ver de uma vez só numa imagem. A partir daqui troquei de dataset: uso o de 50 casas que já apareceu nos posts anteriores da playlist (",[22,7964,7965],{},"square_feet",[22,7967,7968],{},"num_bedrooms",[22,7970,7971],{},"nota de localização",[22,7973,7974],{},"distância do centro","), não mais o de 100 casas do notebook que usei lá em cima. É outro conjunto de casas, então os números de correlação daqui são diferentes dos que calculei na seção \"O problema\".",[11,7977,7978,7979,7983,7984,7987],{},"Antes de ir pro abstrato (o formato do ",[7980,7981,7982],"em",{},"custo","), vale ver o efeito bem mais direto e concreto: o formato dos próprios ",[7980,7985,7986],{},"pontos de dado",". Peguei tamanho e número de quartos dessas 50 casas e plotei um contra o outro, cru e depois normalizado:",[11,7989,7990],{},[33,7991,7992],{},"Pontos, sem normalizar:",[7994,7995],"housing-scaling-scatter",{"mode":7996},"raw",[11,7998,7999],{},[33,8000,8001],{},"Pontos, depois de normalizar (z-score):",[7994,8003],{"mode":8004},"normalized",[11,8006,8007],{},"Repara que a nuvem de pontos não muda de formato, ela só desloca e reescala pros dois eixos ficarem na mesma unidade (desvio-padrão em vez de pés²\u002Fquartos). E é exatamente essa mudança de escala relativa entre os dois eixos que faz toda a diferença lá na frente.",[11,8009,8010],{},"Agora o mapa de custo. Ajustei um modelo com as 4 features (tamanho, quartos, nota de localização, distância do centro), fixei duas delas nos valores ajustados, e desenhei um mapa de calor variando só o peso de tamanho e o peso de quartos, o mesmo truque que o notebook original usa. A bolinha vermelha marca exatamente onde meu ajuste caiu, e a linha logo abaixo do gráfico mostra os valores certinhos.",[11,8012,8013],{},[33,8014,8015],{},"Sem normalizar:",[8017,8018],"housing-cost-contour",{":b-max":8019,":b-min":1491,":w-max":181,":w-min":159,"b-label":8020,"mode":7996,"w-label":8021},"100","w_quartos (cru)","w_tamanho (cru)",[11,8023,8024],{},[33,8025,8026],{},"Depois de normalizar (z-score):",[8017,8028],{":b-max":8029,":b-min":8030,":w-max":8031,":w-min":8032,"b-label":8033,"mode":8004,"w-label":8034},"140","20","150","30","w_quartos (normalizado)","w_tamanho (normalizado)",[11,8036,8037],{},"Repara a diferença de forma. No contorno cru, o eixo de tamanho é muito mais sensível que o de quartos (uma mudancinha de 0.5 no peso de tamanho já dispara o custo lá em cima, enquanto o peso de quartos precisa mudar dezenas de unidades pra fazer o mesmo estrago). É um vale bem alongado, quase um corredor, e a bolinha vermelha (o ajuste real) fica espremida lá no fundo dele. No contorno normalizado, as duas direções ficam com sensibilidade parecida, o formato vira bem mais redondo, e a bolinha cai perto do centro de um círculo.",[11,8039,8040,8041,8092],{},"Não é um círculo perfeito (tamanho e número de quartos, nesse dataset de 50 casas, ainda têm uma correlaçãozinha residual, calculei e ficou em torno de 0.02, bem pertinho de zero mas não exatamente zero), mas a diferença de forma é gritante. E isso não é coincidência nem aproximação visual: pra uma única feature normalizada por z-score, dá pra provar que a Hessiana vira exatamente a matriz identidade, ",[38,8042,8044,8062],{"className":8043},[41],[38,8045,8047],{"className":8046},[45],[47,8048,8049],{"xmlns":49},[51,8050,8051,8059],{},[54,8052,8053,8055,8057],{},[57,8054,3821],{},[361,8056,379],{},[559,8058,178],{},[61,8060,8061],{"encoding":63},"\\kappa = 1",[38,8063,8065,8083],{"className":8064,"ariaHidden":69},[68],[38,8066,8068,8071,8074,8077,8080],{"className":8067},[73],[38,8069],{"className":8070,"style":78},[77],[38,8072,3821],{"className":8073},[82,83],[38,8075],{"className":8076,"style":485},[484],[38,8078,379],{"className":8079},[489],[38,8081],{"className":8082,"style":485},[484],[38,8084,8086,8089],{"className":8085},[73],[38,8087],{"className":8088,"style":1452},[77],[38,8090,178],{"className":8091},[82],", círculo perfeito, resultado exato e não aproximado. Com duas features reais, o resultado fica quase circular, e o \"quase\" é literalmente o tamanho da correlação entre elas.",[87,8094,8096],{"id":8095},"fechando","Fechando",[99,8098,8099,8109],{},[102,8100,8101],{},[105,8102,8103,8106],{},[108,8104,8105],{"align":110},"O que eu já sabia",[108,8107,8108],{"align":110},"O que esse post resolveu",[129,8110,8111,8119,8184],{},[105,8112,8113,8116],{},[134,8114,8115],{"align":110},"Gradiente descendente acha o mínimo sozinho",[134,8117,8118],{"align":110},"Ele só acha rápido se a superfície de custo não for um corredor esticado",[105,8120,8121,8152],{},[134,8122,8123,8151],{"align":110},[38,8124,8126,8139],{"className":8125},[41],[38,8127,8129],{"className":8128},[45],[47,8130,8131],{"xmlns":49},[51,8132,8133,8137],{},[54,8134,8135],{},[57,8136,59],{},[61,8138,64],{"encoding":63},[38,8140,8142],{"className":8141,"ariaHidden":69},[68],[38,8143,8145,8148],{"className":8144},[73],[38,8146],{"className":8147,"style":78},[77],[38,8149,59],{"className":8150,"style":84},[82,83]," é o parâmetro mais sensível que eu mexo",[134,8153,8154,8155,8183],{"align":110},"O ",[38,8156,8158,8171],{"className":8157},[41],[38,8159,8161],{"className":8160},[45],[47,8162,8163],{"xmlns":49},[51,8164,8165,8169],{},[54,8166,8167],{},[57,8168,59],{},[61,8170,64],{"encoding":63},[38,8172,8174],{"className":8173,"ariaHidden":69},[68],[38,8175,8177,8180],{"className":8176},[73],[38,8178],{"className":8179,"style":78},[77],[38,8181,59],{"className":8182,"style":84},[82,83]," \"certo\" depende inteiramente da escala das features, não é uma constante universal",[105,8185,8186,8189],{},[134,8187,8188],{"align":110},"Vale de custo é sempre convexo, pra regressão linear",[134,8190,8191],{"align":110},"Convexo não é o mesmo que bem-condicionado: pode ser uma tigela funda e redonda, ou um cânion raso e alongado",[11,8193,8194],{},"Três ideias pra levar:",[4053,8196,8197,8203,8268],{},[4056,8198,8199,8202],{},[33,8200,8201],{},"A escala relativa das features decide o formato da tigela de custo",", não o algoritmo de otimização em si.",[4056,8204,8205,8238,8239,8267],{},[33,8206,8207,8208,8237],{},"O número de condição ",[38,8209,8211,8225],{"className":8210},[41],[38,8212,8214],{"className":8213},[45],[47,8215,8216],{"xmlns":49},[51,8217,8218,8222],{},[54,8219,8220],{},[57,8221,3821],{},[61,8223,8224],{"encoding":63},"\\kappa",[38,8226,8228],{"className":8227,"ariaHidden":69},[68],[38,8229,8231,8234],{"className":8230},[73],[38,8232],{"className":8233,"style":78},[77],[38,8235,3821],{"className":8236},[82,83]," é a métrica que quantifica isso",": perto de 1 é uma tigela redonda e rápida de descer, muito maior que 1 é um cânion que qualquer ",[38,8240,8242,8255],{"className":8241},[41],[38,8243,8245],{"className":8244},[45],[47,8246,8247],{"xmlns":49},[51,8248,8249,8253],{},[54,8250,8251],{},[57,8252,59],{},[61,8254,64],{"encoding":63},[38,8256,8258],{"className":8257,"ariaHidden":69},[68],[38,8259,8261,8264],{"className":8260},[73],[38,8262],{"className":8263,"style":78},[77],[38,8265,59],{"className":8266,"style":84},[82,83]," fixo vai demorar uma eternidade pra atravessar.",[4056,8269,8270,8273],{},[33,8271,8272],{},"Normalizar não muda a resposta, muda o caminho até ela",": mesmo mínimo, ordens de magnitude menos iterações pra chegar perto.",[87,8275,8277],{"id":8276},"aplicação-prática","Aplicação prática",[11,8279,8280,8281,8287,8288,8291,8292,2459,8295,8298,8299,8351,8352,8451],{},"Mesmo dataset real de imóveis dos posts anteriores (",[27,8282,8286],{"href":8283,"rel":8284},"https:\u002F\u002Fwww.kaggle.com\u002Fdatasets\u002Fdenkuznetz\u002Fhousing-prices-regression",[8285],"nofollow","Housing Prices Regression, Kaggle","). Já vimos no ",[27,8289,8290],{"href":29},"post de gradiente descendente"," que o algoritmo cru, com ",[22,8293,8294],{},"square_feet\u002F100",[22,8296,8297],{},"price\u002F1000",", converge devagar, precisando de ",[38,8300,8302,8321],{"className":8301},[41],[38,8303,8305],{"className":8304},[45],[47,8306,8307],{"xmlns":49},[51,8308,8309,8318],{},[54,8310,8311,8313,8315],{},[57,8312,59],{},[361,8314,379],{},[559,8316,8317],{},"0.01",[61,8319,8320],{"encoding":63},"\\alpha=0.01",[38,8322,8324,8342],{"className":8323,"ariaHidden":69},[68],[38,8325,8327,8330,8333,8336,8339],{"className":8326},[73],[38,8328],{"className":8329,"style":78},[77],[38,8331,59],{"className":8332,"style":84},[82,83],[38,8334],{"className":8335,"style":485},[484],[38,8337,379],{"className":8338},[489],[38,8340],{"className":8341,"style":485},[484],[38,8343,8345,8348],{"className":8344},[73],[38,8346],{"className":8347,"style":1452},[77],[38,8349,8317],{"className":8350},[82]," e 4000 iterações pra chegar em ",[38,8353,8355,8391],{"className":8354},[41],[38,8356,8358],{"className":8357},[45],[47,8359,8360],{"xmlns":49},[51,8361,8362,8388],{},[54,8363,8364,8366,8368,8370,8372,8374,8376,8378,8381,8383,8386],{},[361,8365,370],{"stretchy":369},[57,8367,359],{},[361,8369,363],{"separator":69},[57,8371,366],{},[361,8373,376],{"stretchy":369},[361,8375,379],{},[361,8377,370],{"stretchy":369},[559,8379,8380],{},"116.5",[361,8382,363],{"separator":69},[559,8384,8385],{},"398.3",[361,8387,376],{"stretchy":369},[61,8389,8390],{"encoding":63},"(w,b) = (116.5, 398.3)",[38,8392,8394,8427],{"className":8393,"ariaHidden":69},[68],[38,8395,8397,8400,8403,8406,8409,8412,8415,8418,8421,8424],{"className":8396},[73],[38,8398],{"className":8399,"style":637},[77],[38,8401,370],{"className":8402},[473],[38,8404,359],{"className":8405,"style":448},[82,83],[38,8407,363],{"className":8408},[452],[38,8410],{"className":8411,"style":654},[484],[38,8413,366],{"className":8414},[82,83],[38,8416,376],{"className":8417},[480],[38,8419],{"className":8420,"style":485},[484],[38,8422,379],{"className":8423},[489],[38,8425],{"className":8426,"style":485},[484],[38,8428,8430,8433,8436,8439,8442,8445,8448],{"className":8429},[73],[38,8431],{"className":8432,"style":637},[77],[38,8434,370],{"className":8435},[473],[38,8437,8380],{"className":8438},[82],[38,8440,363],{"className":8441},[452],[38,8443],{"className":8444,"style":654},[484],[38,8446,8385],{"className":8447},[82],[38,8449,376],{"className":8450},[480],". Agora normalizei o tamanho por z-score antes de rodar:",[1308,8453,8455],{"className":1310,"code":8454,"language":1312,"meta":1313,"style":1313},"tamanho_norm, mu, sigma = zscore_normalize_features(square_feet)\n\nw, b, J_hist = gradient_descent(\n    tamanho_norm, price,\n    w_in=0, b_in=0,\n    alpha=0.3, num_iters=200,\n    cost_function=compute_cost, gradient_function=compute_gradient)\n\nprint(f\"(w, b) encontrados: ({w:.1f}, {b:.1f})\")\n",[22,8456,8457,8462,8468,8473,8478,8483,8489,8495,8500],{"__ignoreMap":1313},[38,8458,8459],{"class":1318,"line":1319},[38,8460,8461],{},"tamanho_norm, mu, sigma = zscore_normalize_features(square_feet)\n",[38,8463,8464],{"class":1318,"line":1325},[38,8465,8467],{"emptyLinePlaceholder":8466},true,"\n",[38,8469,8470],{"class":1318,"line":6292},[38,8471,8472],{},"w, b, J_hist = gradient_descent(\n",[38,8474,8475],{"class":1318,"line":6298},[38,8476,8477],{},"    tamanho_norm, price,\n",[38,8479,8480],{"class":1318,"line":6304},[38,8481,8482],{},"    w_in=0, b_in=0,\n",[38,8484,8486],{"class":1318,"line":8485},6,[38,8487,8488],{},"    alpha=0.3, num_iters=200,\n",[38,8490,8492],{"class":1318,"line":8491},7,[38,8493,8494],{},"    cost_function=compute_cost, gradient_function=compute_gradient)\n",[38,8496,8498],{"class":1318,"line":8497},8,[38,8499,8467],{"emptyLinePlaceholder":8466},[38,8501,8503],{"class":1318,"line":8502},9,[38,8504,8505],{},"print(f\"(w, b) encontrados: ({w:.1f}, {b:.1f})\")\n",[1458,8507,8508],{},[11,8509,8510,3806,8513],{},[33,8511,8512],{},"Saída:",[22,8514,8515],{},"(w, b) encontrados: (88.1, 593.0)",[11,8517,8518,8519,8571,8572,8627,8628,8679],{},"O custo final bate exatamente com o do exemplo cru (",[38,8520,8522,8541],{"className":8521},[41],[38,8523,8525],{"className":8524},[45],[47,8526,8527],{"xmlns":49},[51,8528,8529,8538],{},[54,8530,8531,8533,8535],{},[57,8532,542],{},[361,8534,3154],{},[559,8536,8537],{},"5189.6",[61,8539,8540],{"encoding":63},"J \\approx 5189.6",[38,8542,8544,8562],{"className":8543,"ariaHidden":69},[68],[38,8545,8547,8550,8553,8556,8559],{"className":8546},[73],[38,8548],{"className":8549,"style":3021},[77],[38,8551,542],{"className":8552,"style":641},[82,83],[38,8554],{"className":8555,"style":485},[484],[38,8557,3154],{"className":8558},[489],[38,8560],{"className":8561,"style":485},[484],[38,8563,8565,8568],{"className":8564},[73],[38,8566],{"className":8567,"style":1452},[77],[38,8569,8537],{"className":8570},[82]," nos dois casos, é o mesmo mínimo), só que aqui eu cheguei nele em ",[33,8573,8574,8575],{},"200 iterações com ",[38,8576,8578,8597],{"className":8577},[41],[38,8579,8581],{"className":8580},[45],[47,8582,8583],{"xmlns":49},[51,8584,8585,8594],{},[54,8586,8587,8589,8591],{},[57,8588,59],{},[361,8590,379],{},[559,8592,8593],{},"0.3",[61,8595,8596],{"encoding":63},"\\alpha = 0.3",[38,8598,8600,8618],{"className":8599,"ariaHidden":69},[68],[38,8601,8603,8606,8609,8612,8615],{"className":8602},[73],[38,8604],{"className":8605,"style":78},[77],[38,8607,59],{"className":8608,"style":84},[82,83],[38,8610],{"className":8611,"style":485},[484],[38,8613,379],{"className":8614},[489],[38,8616],{"className":8617,"style":485},[484],[38,8619,8621,8624],{"className":8620},[73],[38,8622],{"className":8623,"style":1452},[77],[38,8625,8593],{"className":8626},[82],", contra as 4000 iterações com ",[38,8629,8631,8649],{"className":8630},[41],[38,8632,8634],{"className":8633},[45],[47,8635,8636],{"xmlns":49},[51,8637,8638,8646],{},[54,8639,8640,8642,8644],{},[57,8641,59],{},[361,8643,379],{},[559,8645,8317],{},[61,8647,8648],{"encoding":63},"\\alpha = 0.01",[38,8650,8652,8670],{"className":8651,"ariaHidden":69},[68],[38,8653,8655,8658,8661,8664,8667],{"className":8654},[73],[38,8656],{"className":8657,"style":78},[77],[38,8659,59],{"className":8660,"style":84},[82,83],[38,8662],{"className":8663,"style":485},[484],[38,8665,379],{"className":8666},[489],[38,8668],{"className":8669,"style":485},[484],[38,8671,8673,8676],{"className":8672},[73],[38,8674],{"className":8675,"style":1452},[77],[38,8677,8317],{"className":8678},[82]," de antes. Compara os dois ao vivo:",[8681,8682],"housing-gradient-descent-simulator",{},[8684,8685],"housing-scaled-gradient-descent-simulator",{},[11,8687,8688],{},"Roda \"Rodar 100\" algumas vezes nos dois e cronometra na cabeça: um deles já achatou, o outro ainda tá subindo a ladeira. Mesma casa, mesmo preço final, e a única diferença entre os dois simuladores é uma linha de código a mais no começo.",[8690,8691,8692],"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":1313,"searchDepth":1325,"depth":1325,"links":8694},[8695,8696,8697,8698,8700,8702,8703,8704,8705,8706,8707],{"id":89,"depth":1325,"text":90},{"id":332,"depth":1325,"text":333},{"id":1302,"depth":1325,"text":1303},{"id":2558,"depth":1325,"text":8699},"Por que exatamente ali: o limite 2\u002FL2\u002FL2\u002FL e o número de condição",{"id":4016,"depth":1325,"text":8701},"Receita prática pra escolher α\\alphaα",{"id":4496,"depth":1325,"text":4497},{"id":6320,"depth":1325,"text":6321},{"id":6791,"depth":1325,"text":6792},{"id":7958,"depth":1325,"text":7959},{"id":8095,"depth":1325,"text":8096},{"id":8276,"depth":1325,"text":8277},null,"2026-08-19","Por que o mesmo gradiente descendente que funcionou liso até aqui quebra assim que uma feature tem escala bem diferente das outras, e como um truque de estatística básica resolve isso de vez.","md",{},"\u002Fpt\u002Fplaylists\u002Fmachine-learning-specialization\u002Fw2-lab03-feature-scaling","machine-learning-specialization",{"title":6,"description":8710},"published","pt\u002Fplaylists\u002Fmachine-learning-specialization\u002Fw2-lab03-feature-scaling",[8719,8720,8721],"normalizacao","feature-scaling","taxa-de-aprendizado","Q586WKD7h4MCdQ5DyTBSBMl-hRGhNGLtuUlUXI9THiQ",1787338983764]