PythonML

purpose:梯度下降的实例

利用梯度下降求函数y=(x+5)^2的最小值

梯度下降算法

import numpy as np

def func(x):return (x+5)**2
x_points=np.linspace(-10,5,num=100)
y_points=func(x_points)

import matplotlib.pyplot as plt
%matplotlib inline
plt.figure()
plt.plot(x_points,y_points)
plt.show()

#Initialize Parameters
import random
start_x=random.uniform(-20,10)
alpha=0.01 #learning rate
max_iter=1000 #迭代次数#梯度
def gradient(x):return 2*(x+5)

start_x

9.847889809556982

n_iter=0
cur_x=start_x
x_list=[]
y_list=[]
while n_iter<max_iter:pre_x=cur_xcur_x=pre_x-alpha*gradient(pre_x)n_iter+=1cur_y=func(cur_x)x_list.append(cur_x)y_list.append(cur_y)print('迭代次数={} , x={} , y={}'.format(n_iter,cur_x,cur_y))

迭代次数=1 , x=9.550932013365841 , y=211.7296224575949
迭代次数=2 , x=9.259913373098524 , y=203.34512940827412
迭代次数=3 , x=8.974715105636554 , y=195.29266228370648
迭代次数=4 , x=8.695220803523823 , y=187.5590728572717
迭代次数=5 , x=8.421316387453347 , y=180.13173357212375
迭代次数=6 , x=8.15289005970428 , y=172.99851692266765
迭代次数=7 , x=7.889832258510195 , y=166.14777565253002
迭代次数=8 , x=7.6320356133399905 , y=159.56832373668982
迭代次数=9 , x=7.379394901073191 , y=153.24941811671692
迭代次数=10 , x=7.131807003051727 , y=147.1807411592949
迭代次数=11 , x=6.889170862990692 , y=141.35238380938685
迭代次数=12 , x=6.651387445730879 , y=135.75482941053514
迭代次数=13 , x=6.418359696816261 , y=130.3789381658779
迭代次数=14 , x=6.189992502879936 , y=125.21593221450915
迭代次数=15 , x=5.966192652822337 , y=120.25738129881458
迭代次数=16 , x=5.74686879976589 , y=115.49518899938154
迭代次数=17 , x=5.531931423770573 , y=110.92157951500604
迭代次数=18 , x=5.321292795295161 , y=106.5290849662118
迭代次数=19 , x=5.114866939389258 , y=102.31053320154983
迭代次数=20 , x=4.912569600601473 , y=98.25903608676843
迭代次数=21 , x=4.714318208589443 , y=94.3679782577324
迭代次数=22 , x=4.520031844417654 , y=90.63100631872622
迭代次数=23 , x=4.329631207529301 , y=87.04201846850465
迭代次数=24 , x=4.143038583378715 , y=83.59515453715186
迭代次数=25 , x=3.960177811711141 , y=80.28478641748065
迭代次数=26 , x=3.780974255476918 , y=77.10550887534842
迭代次数=27 , x=3.6053547703673794 , y=74.0521307238846
迭代次数=28 , x=3.4332476749600316 , y=71.1196663472188
迭代次数=29 , x=3.264582721460831 , y=68.30332755986892
迭代次数=30 , x=3.0992910670316145 , y=65.59851578849812
迭代次数=31 , x=2.937305245690982 , y=63.000814563273586
迭代次数=32 , x=2.7785591407771624 , y=60.50598230656794
迭代次数=33 , x=2.622987957961619 , y=58.10994540722786
迭代次数=34 , x=2.470528198802387 , y=55.808791569101636
迭代次数=35 , x=2.321117634826339 , y=53.59876342296521
迭代次数=36 , x=2.1746952821298122 , y=51.47625239141579
迭代次数=37 , x=2.031201376487216 , y=49.43779279671572
迭代次数=38 , x=1.8905773489574715 , y=47.48005620196577
迭代次数=39 , x=1.752765801978322 , y=45.59984597636793
迭代次数=40 , x=1.6177104859387557 , y=43.79409207570376
迭代次数=41 , x=1.4853562762199806 , y=42.05984602950589
迭代次数=42 , x=1.355649150695581 , y=40.39427612673746
迭代次数=43 , x=1.2285361676816693 , y=38.79466279211866
迭代次数=44 , x=1.1039654443280358 , y=37.25839414555076
迭代次数=45 , x=0.9818861354414751 , y=35.78296173738695
迭代次数=46 , x=0.8622484127326455 , y=34.365956452586424
迭代次数=47 , x=0.7450034444779926 , y=33.005064577064
迭代次数=48 , x=0.6301033755884328 , y=31.69806401981226
迭代次数=49 , x=0.5175013080766642 , y=30.442820684627698
迭代次数=50 , x=0.4071512819151309 , y=29.237284985516446
迭代次数=51 , x=0.29900825627682825 , y=28.079488500089987
迭代次数=52 , x=0.19302809115129171 , y=26.96754075548643
迭代次数=53 , x=0.08916752932826588 , y=25.899626141569172
迭代次数=54 , x=-0.01261582125829945 , y=24.874000946363022
迭代次数=55 , x=-0.11236350483313345 , y=23.888990508887048
迭代次数=56 , x=-0.21011623473647079 , y=22.942986484735126
迭代次数=57 , x=-0.3059139100417414 , y=22.03444421993961
迭代次数=58 , x=-0.39979563184090655 , y=21.16188022883
迭代次数=59 , x=-0.49179971920408844 , y=20.323869771768333
迭代次数=60 , x=-0.5819637248200067 , y=19.519044528806308
迭代次数=61 , x=-0.6703244503236065 , y=18.746090365465577
迭代次数=62 , x=-0.7569179613171344 , y=18.003745186993147
迭代次数=63 , x=-0.8417796020907917 , y=17.29079687758821
迭代次数=64 , x=-0.9249440100489759 , y=16.60608132123572
迭代次数=65 , x=-1.0064451298479964 , y=15.948480500914785
迭代次数=66 , x=-1.0863162272510365 , y=15.31692067307856
迭代次数=67 , x=-1.1645899027060158 , y=14.710370614424647
迭代次数=68 , x=-1.2412981046518956 , y=14.127839938093432
迭代次数=69 , x=-1.3164721425588577 , y=13.568377476544933
迭代次数=70 , x=-1.3901426997076805 , y=13.031069728473755
迭代次数=71 , x=-1.4623398457135268 , y=12.515039367226192
迭代次数=72 , x=-1.5330930487992562 , y=12.019443808284036
迭代次数=73 , x=-1.602431187823271 , y=11.54347383347599
迭代次数=74 , x=-1.6703825640668057 , y=11.08635226967034
迭代次数=75 , x=-1.7369749127854697 , y=10.647332719791395
迭代次数=76 , x=-1.8022354145297603 , y=10.225698344087654
迭代次数=77 , x=-1.8661907062391652 , y=9.820760689661782
迭代次数=78 , x=-1.928866892114382 , y=9.431858566351178
迭代次数=79 , x=-1.9902895542720942 , y=9.05835696712367
迭代次数=80 , x=-2.0504837631866524 , y=8.699646031225571
迭代次数=81 , x=-2.1094740879229192 , y=8.355140048389039
迭代次数=82 , x=-2.167284606164461 , y=8.024276502472834
迭代次数=83 , x=-2.2239389140411716 , y=7.70651515297491
迭代次数=84 , x=-2.279460135760348 , y=7.401337152917104
迭代次数=85 , x=-2.3338709330451413 , y=7.108244201661585
迭代次数=86 , x=-2.387193514384238 , y=6.826757731275788
迭代次数=87 , x=-2.4394496440965536 , y=6.556418125117266
迭代次数=88 , x=-2.4906606512146228 , y=6.296783967362621
迭代次数=89 , x=-2.54084743819033 , y=6.047431322255062
迭代次数=90 , x=-2.5900304894265234 , y=5.807953041893763
迭代次数=91 , x=-2.638229879637993 , y=5.577958101434769
迭代次数=92 , x=-2.685465282045233 , y=5.3570709606179525
迭代次数=93 , x=-2.7317559764043287 , y=5.14493095057748
迭代次数=94 , x=-2.777120856876242 , y=4.941191684934613
迭代次数=95 , x=-2.821578439738717 , y=4.745520494211202
迭代次数=96 , x=-2.865146870943943 , y=4.5575978826404375
迭代次数=97 , x=-2.907843933525064 , y=4.377117006487876
迭代次数=98 , x=-2.9496870548545626 , y=4.203783173030957
迭代次数=99 , x=-2.9906933137574714 , y=4.037313359378931
迭代次数=100 , x=-3.030879447482322 , y=3.8774357503475256
迭代次数=101 , x=-3.0702618585326755 , y=3.7238892946337634
迭代次数=102 , x=-3.108856621362022 , y=3.576423278566266
迭代次数=103 , x=-3.1466794889347818 , y=3.4347969167350416
迭代次数=104 , x=-3.183745899156086 , y=3.2987789588323344
迭代次数=105 , x=-3.2200709811729644 , y=3.1681473120625734
迭代次数=106 , x=-3.2556695615495053 , y=3.042688678504895
迭代次数=107 , x=-3.2905561703185153 , y=2.922198206836101
迭代次数=108 , x=-3.324745046912145 , y=2.806479157845392
迭代次数=109 , x=-3.358250145973902 , y=2.695342583194715
迭代次数=110 , x=-3.3910851430544238 , y=2.588607016900204
迭代次数=111 , x=-3.4232634401933355 , y=2.486098179030955
迭代次数=112 , x=-3.4547981713894687 , y=2.38764869114133
迭代次数=113 , x=-3.4857022079616793 , y=2.293097802972133
迭代次数=114 , x=-3.5159881638024455 , y=2.202291129974437
迭代次数=115 , x=-3.5456684005263965 , y=2.11508040122745
迭代次数=116 , x=-3.574755032515869 , y=2.031323217338842
迭代次数=117 , x=-3.6032599318655514 , y=1.9508828179322242
迭代次数=118 , x=-3.63119473322824 , y=1.8736278583421087
迭代次数=119 , x=-3.6585708385636755 , y=1.7994321951517607
迭代次数=120 , x=-3.6853994217924018 , y=1.7281746802237516
迭代次数=121 , x=-3.711691433356554 , y=1.6597389628868908
迭代次数=122 , x=-3.7374576046894226 , y=1.5940132999565704
迭代次数=123 , x=-3.7627084525956342 , y=1.53089037327829
迭代次数=124 , x=-3.7874542835437217 , y=1.4702671144964692
迭代次数=125 , x=-3.8117051978728473 , y=1.4120445367624088
迭代次数=126 , x=-3.83547109391539 , y=1.356127573106618
迭代次数=127 , x=-3.8587616720370823 , y=1.3024249212115961
迭代次数=128 , x=-3.8815864385963406 , y=1.2508488943316172
迭代次数=129 , x=-3.9039547098244136 , y=1.2013152781160854
迭代次数=130 , x=-3.9258756156279255 , y=1.153743193102688
迭代次数=131 , x=-3.947358103315367 , y=1.1080549626558218
迭代次数=132 , x=-3.9684109412490596 , y=1.0641759861346511
迭代次数=133 , x=-3.9890427224240783 , y=1.0220346170837191
迭代次数=134 , x=-4.009261867975597 , y=0.981562046247204
迭代次数=135 , x=-4.029076630616085 , y=0.9426921892158151
迭代次数=136 , x=-4.048495098003763 , y=0.905361578522868
迭代次数=137 , x=-4.067525196043688 , y=0.8695092600133629
迭代次数=138 , x=-4.086174692122814 , y=0.8350766933168339
迭代次数=139 , x=-4.1044511982803575 , y=0.8020076562614875
迭代次数=140 , x=-4.12236217431475 , y=0.7702481530735332
迭代次数=141 , x=-4.139914930828455 , y=0.7397463262118208
迭代次数=142 , x=-4.157116632211886 , y=0.7104523716938335
迭代次数=143 , x=-4.173974299567648 , y=0.6823184577747572
迭代次数=144 , x=-4.190494813576295 , y=0.6552986468468768
迭代次数=145 , x=-4.206684917304769 , y=0.629348820431741
迭代次数=146 , x=-4.222551218958674 , y=0.6044266071426443
迭代次数=147 , x=-4.2381001945795 , y=0.5804913134997953
迭代次数=148 , x=-4.25333819068791 , y=0.5575038574852036
迭代次数=149 , x=-4.268271426874152 , y=0.535426704728789
迭代次数=150 , x=-4.28290599833667 , y=0.5142238072215283
迭代次数=151 , x=-4.297247878369936 , y=0.4938605444555557
迭代次数=152 , x=-4.311302920802538 , y=0.4743036668951156
迭代次数=153 , x=-4.325076862386487 , y=0.4555212416860693
迭代次数=154 , x=-4.338575325138757 , y=0.4374826005153006
迭代次数=155 , x=-4.351803818635982 , y=0.42015828953489437
迭代次数=156 , x=-4.364767742263263 , y=0.40352002126931236
迭代次数=157 , x=-4.377472387417997 , y=0.38754062842704806
迭代次数=158 , x=-4.3899229396696375 , y=0.37219401954133674
迭代次数=159 , x=-4.402124480876245 , y=0.3574551363674993
迭代次数=160 , x=-4.41408199125872 , y=0.34329991296734674
迭代次数=161 , x=-4.425800351433545 , y=0.3297052364138403
迭代次数=162 , x=-4.437284344404874 , y=0.316648909051852
迭代次数=163 , x=-4.448538657516777 , y=0.30410961225339844
迭代次数=164 , x=-4.459567884366441 , y=0.2920668716081642
迭代次数=165 , x=-4.470376526679113 , y=0.28050102349248063
迭代次数=166 , x=-4.48096899614553 , y=0.2693931829621785
迭代次数=167 , x=-4.49134961622262 , y=0.2587252129168759
迭代次数=168 , x=-4.501522623898167 , y=0.24847969448536789
迭代次数=169 , x=-4.511492171420204 , y=0.23863989858374748
迭代次数=170 , x=-4.5212623279918 , y=0.22918975859983082
迭代次数=171 , x=-4.530837081431964 , y=0.22011384415927732
迭代次数=172 , x=-4.540220339803325 , y=0.21139733593057017
迭代次数=173 , x=-4.549415933007258 , y=0.20302600142771993
迭代次数=174 , x=-4.558427614347113 , y=0.1949861717711822
迭代次数=175 , x=-4.56725906206017 , y=0.18726471936904354
迭代次数=176 , x=-4.575913880818967 , y=0.17984903648202932
迭代次数=177 , x=-4.584395603202587 , y=0.17272701463734133
迭代次数=178 , x=-4.592707691138536 , y=0.16588702485770251
迭代次数=179 , x=-4.600853537315765 , y=0.15931789867333734
迭代次数=180 , x=-4.60883646656945 , y=0.1530089098858731
迭代次数=181 , x=-4.616659737238061 , y=0.14694975705439234
迭代次数=182 , x=-4.6243265424933 , y=0.14113054667503833
迭代次数=183 , x=-4.631840011643434 , y=0.1355417770267067
迭代次数=184 , x=-4.639203211410566 , y=0.13017432265644896
迭代次数=185 , x=-4.646419147182354 , y=0.12501941947925374
迭代次数=186 , x=-4.653490764238707 , y=0.12006865046787557
迭代次数=187 , x=-4.6604209489539326 , y=0.11531393190934768
迭代次数=188 , x=-4.6672125299748535 , y=0.1107475002057378
迭代次数=189 , x=-4.673868279375356 , y=0.10636189919759063
迭代次数=190 , x=-4.680390913787849 , y=0.10214996798936618
迭代次数=191 , x=-4.686783095512092 , y=0.09810482925698746
迭代次数=192 , x=-4.6930474336018495 , y=0.09421987801841099
迭代次数=193 , x=-4.699186484929813 , y=0.09048877084888172
迭代次数=194 , x=-4.705202755231217 , y=0.08690541552326575
迭代次数=195 , x=-4.711098700126593 , y=0.08346396106854427
迭代次数=196 , x=-4.716876726124061 , y=0.08015878821022974
迭代次数=197 , x=-4.72253919160158 , y=0.07698450019710462
迭代次数=198 , x=-4.728088407769548 , y=0.07393591398929937
迭代次数=199 , x=-4.7335266396141575 , y=0.07100805179532307
迭代次数=200 , x=-4.738856106821874 , y=0.06819613294422838
迭代次数=201 , x=-4.744078984685436 , y=0.06549556607963714
迭代次数=202 , x=-4.749197404991728 , y=0.06290194166288339
迭代次数=203 , x=-4.754213456891893 , y=0.06041102477303323
迭代次数=204 , x=-4.759129187754056 , y=0.05801874819202101
迭代次数=205 , x=-4.763946603998974 , y=0.05572120576361711
迭代次数=206 , x=-4.768667671918995 , y=0.05351464601537779
迭代次数=207 , x=-4.773294318480615 , y=0.05139546603316876
迭代次数=208 , x=-4.7778284321110025 , y=0.04936020557825543
迭代次数=209 , x=-4.782271863468782 , y=0.04740554143735653
迭代次数=210 , x=-4.7866264261994065 , y=0.04552828199643733
迭代次数=211 , x=-4.790893897675418 , y=0.04372536202937853
迭代次数=212 , x=-4.795076019721909 , y=0.04199383769301529
迭代次数=213 , x=-4.799174499327471 , y=0.040330881720372046
迭代次数=214 , x=-4.803191009340921 , y=0.03873377880424548
迭代次数=215 , x=-4.8071271891541025 , y=0.03719992116359734
迭代次数=216 , x=-4.81098464537102 , y=0.03572680428551903
迭代次数=217 , x=-4.8147649524636 , y=0.03431202283581244
迭代次数=218 , x=-4.818469653414327 , y=0.032953266731514406
迭代次数=219 , x=-4.822100260346041 , y=0.03164831736894655
迭代次数=220 , x=-4.825658255139119 , y=0.03039504400113639
迭代次数=221 , x=-4.829145090036337 , y=0.029191400258691366
迭代次数=222 , x=-4.8325621882356105 , y=0.028035420808447144
迭代次数=223 , x=-4.835910944470898 , y=0.026925218144432692
迭代次数=224 , x=-4.83919272558148 , y=0.025858979505913215
迭代次数=225 , x=-4.842408871069851 , y=0.024834963917478928
迭代次数=226 , x=-4.845560693648454 , y=0.023851499346346648
迭代次数=227 , x=-4.848649479775485 , y=0.022906979972231293
迭代次数=228 , x=-4.851676490179975 , y=0.021999863565331006
迭代次数=229 , x=-4.854642960376376 , y=0.021128668968143832
迭代次数=230 , x=-4.857550101168848 , y=0.020291973677005377
迭代次数=231 , x=-4.8603990991454715 , y=0.019488411519395888
迭代次数=232 , x=-4.863191117162562 , y=0.01871667042322786
迭代次数=233 , x=-4.86592729481931 , y=0.017975490274468108
迭代次数=234 , x=-4.868608748922925 , y=0.017263660859599066
迭代次数=235 , x=-4.871236573944466 , y=0.01658001988955901
迭代次数=236 , x=-4.873811842465576 , y=0.01592345110193253
迭代次数=237 , x=-4.876335605616265 , y=0.015292882438296003
迭代次数=238 , x=-4.8788088935039395 , y=0.014687284293739469
迭代次数=239 , x=-4.88123271563386 , y=0.014105667835707466
迭代次数=240 , x=-4.883608061321183 , y=0.0135470833894134
迭代次数=241 , x=-4.8859359000947595 , y=0.013010618887192679
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迭代次数=947 , x=-4.999999927094815 , y=5.315165999904393e-15
迭代次数=948 , x=-4.999999928552919 , y=5.104685403463387e-15
迭代次数=949 , x=-4.99999992998186 , y=4.902539891336765e-15
迭代次数=950 , x=-4.999999931382223 , y=4.708399353082313e-15
迭代次数=951 , x=-4.999999932754578 , y=4.521946736311216e-15
迭代次数=952 , x=-4.9999999340994865 , y=4.342877678330884e-15
迭代次数=953 , x=-4.999999935417497 , y=4.170899680969215e-15
迭代次数=954 , x=-4.999999936709147 , y=4.00573201762615e-15
迭代次数=955 , x=-4.999999937974964 , y=3.847105062781733e-15
迭代次数=956 , x=-4.999999939215465 , y=3.694759663424568e-15
迭代次数=957 , x=-4.999999940431156 , y=3.5484471849855757e-15
迭代次数=958 , x=-4.999999941622533 , y=3.4079286889040525e-15
迭代次数=959 , x=-4.999999942790082 , y=3.2729746823358834e-15
迭代次数=960 , x=-4.999999943934281 , y=3.143364866988692e-15
迭代次数=961 , x=-4.999999945055595 , y=3.0188875948317313e-15
迭代次数=962 , x=-4.9999999461544835 , y=2.8993396460763947e-15
迭代次数=963 , x=-4.999999947231394 , y=2.7845258185883794e-15
迭代次数=964 , x=-4.999999948286765 , y=2.6742586310795195e-15
迭代次数=965 , x=-4.9999999493210305 , y=2.568357946077282e-15
迭代次数=966 , x=-4.99999995033461 , y=2.4666509661192143e-15
迭代次数=967 , x=-4.999999951327918 , y=2.3689716016943313e-15
迭代次数=968 , x=-4.999999952301359 , y=2.275160314405063e-15
迭代次数=969 , x=-4.999999953255332 , y=2.1850639576511013e-15
迭代次数=970 , x=-4.999999954190225 , y=2.09853545259545e-15
迭代次数=971 , x=-4.9999999551064205 , y=2.015433482166417e-15
迭代次数=972 , x=-4.9999999560042925 , y=1.9356222803226718e-15
迭代次数=973 , x=-4.999999956884206 , y=1.858971657935043e-15
迭代次数=974 , x=-4.999999957746522 , y=1.785356407301427e-15
迭代次数=975 , x=-4.999999958591592 , y=1.714656274447702e-15
迭代次数=976 , x=-4.9999999594197595 , y=1.6467559191386671e-15
迭代次数=977 , x=-4.9999999602313645 , y=1.5815443691992526e-15
迭代次数=978 , x=-4.999999961026737 , y=1.5189152163327875e-15
迭代次数=979 , x=-4.9999999618062025 , y=1.4587661642675952e-15
迭代次数=980 , x=-4.999999962570079 , y=1.4009990148541529e-15
迭代次数=981 , x=-4.999999963318677 , y=1.3455194473500166e-15
迭代次数=982 , x=-4.999999964052304 , y=1.2922368746807184e-15
迭代次数=983 , x=-4.999999964771257 , y=1.2410643044559728e-15
迭代次数=984 , x=-4.9999999654758325 , y=1.1919181432809784e-15
迭代次数=985 , x=-4.999999966166316 , y=1.1447181932211491e-15
迭代次数=986 , x=-4.9999999668429895 , y=1.0993873433458025e-15
迭代次数=987 , x=-4.999999967506129 , y=1.0558516334096632e-15
迭代次数=988 , x=-4.999999968156007 , y=1.014039904201337e-15
迭代次数=989 , x=-4.999999968792887 , y=9.738839228862646e-16
迭代次数=990 , x=-4.999999969417029 , y=9.35318104328613e-16
迭代次数=991 , x=-4.999999970028688 , y=8.982795223043283e-16
迭代次数=992 , x=-4.999999970628115 , y=8.6270765009058e-16
迭代次数=993 , x=-4.999999971215552 , y=8.2854443021488e-16
迭代次数=994 , x=-4.999999971791241 , y=7.957340777936058e-16
迭代次数=995 , x=-4.9999999723554165 , y=7.642229975095171e-16
迭代次数=996 , x=-4.999999972908308 , y=7.339597514082964e-16
迭代次数=997 , x=-4.999999973450143 , y=7.048949292174407e-16
迭代次数=998 , x=-4.99999997398114 , y=6.769810733816692e-16
迭代次数=999 , x=-4.999999974501518 , y=6.501726065697697e-16
迭代次数=1000 , x=-4.999999975011487 , y=6.244257615841334e-16

plt.figure()
plt.scatter(range(max_iter),y_list)
plt.show()

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lect01_codes03_eg梯度下降

PythonML

purpose:梯度下降的实例

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