梯度下降算法：可能只会找到局部最优点
随机梯度下降算法：随机梯度下降更容易找到全局最优点，类似于快速排序选基点，随机选取能够提高找到全局最优点的概率
梯度下降算法实现代码：
import numpy as np
import matplotlib.pyplot as plt
from matplotlib import cm
from mpl_toolkits.mplot3d import Axes3D
x_data = [1.0, 2.0, 3.0]
y_data = [2.0, 4.0, 6.0]
w = 1.0
# 定义线性模型
def forward(x):
return x * w
# 定义所有样本的平均平方误差
def cost(xs, ys):
cost = 0
for x, y in zip(xs, ys):
y_pred = forward(x)
cost += (y_pred - y) ** 2
return cost / len(xs)
# 定义梯度函数
def gradient(xs, ys):
grad = 0
for x, y in zip(xs, ys):
grad += 2 * x * (x * w - y)
return grad / len(xs)
epoch_list = []
loss_list = []
print('Predict (before training)', 4, forward(4))
for epoch in range(100):
cost_val = cost(x_data, y_data)
grad_val = gradient(x_data, y_data)
w -= 0.01 * grad_val # 0.01是学习率
print('Epoch:', epoch, 'w=', w, 'loss=', cost_val)
epoch_list.append(epoch)
loss_list.append(cost_val)
print('Predict (after training)', 4, forward(4))
plt.plot(epoch_list, loss_list)
plt.ylabel('loss')
plt.xlabel('epoch')
plt.show()
实现结果：
Predict (before training) 4 4.0
Epoch: 0 w= 1.0933333333333333 loss= 4.666666666666667
Epoch: 1 w= 1.1779555555555554 loss= 3.8362074074074086
Epoch: 2 w= 1.2546797037037036 loss= 3.1535329869958857
Epoch: 3 w= 1.3242429313580246 loss= 2.592344272332262
Epoch: 4 w= 1.3873135910979424 loss= 2.1310222071581117
Epoch: 5 w= 1.4444976559288012 loss= 1.7517949663820642
Epoch: 6 w= 1.4963445413754464 loss= 1.440053319920117
Epoch: 7 w= 1.5433523841804047 loss= 1.1837878313441108
Epoch: 8 w= 1.5859728283235668 loss= 0.9731262101573632
Epoch: 9 w= 1.6246153643467005 loss= 0.7999529948031382
Epoch: 10 w= 1.659651263674342 loss= 0.6575969151946154
Epoch: 11 w= 1.6914171457314033 loss= 0.5405738908195378
Epoch: 12 w= 1.7202182121298057 loss= 0.44437576375991855
Epoch: 13 w= 1.7463311789976905 loss= 0.365296627844598
Epoch: 14 w= 1.7700069356245727 loss= 0.3002900634939416
Epoch: 15 w= 1.7914729549662791 loss= 0.2468517784170642
Epoch: 16 w= 1.8109354791694263 loss= 0.2029231330489788
Epoch: 17 w= 1.8285815011136133 loss= 0.16681183417217407
Epoch: 18 w= 1.8445805610096762 loss= 0.1371267415488235
Epoch: 19 w= 1.8590863753154396 loss= 0.11272427607497944
Epoch: 20 w= 1.872238313619332 loss= 0.09266436490145864
Epoch: 21 w= 1.8841627376815275 loss= 0.07617422636521683
Epoch: 22 w= 1.8949742154979183 loss= 0.06261859959338009
Epoch: 23 w= 1.904776622051446 loss= 0.051475271914629306
Epoch: 24 w= 1.9136641373266443 loss= 0.04231496130368814
Epoch: 25 w= 1.9217221511761575 loss= 0.03478477885657844
Epoch: 26 w= 1.9290280837330496 loss= 0.02859463421027894
Epoch: 27 w= 1.9356521292512983 loss= 0.023506060193480772
Epoch: 28 w= 1.9416579305211772 loss= 0.01932302619282764
Epoch: 29 w= 1.9471031903392007 loss= 0.015884386331668398
Epoch: 30 w= 1.952040225907542 loss= 0.01305767153735723
Epoch: 31 w= 1.9565164714895047 loss= 0.010733986344664803
Epoch: 32 w= 1.9605749341504843 loss= 0.008823813841374291
Epoch: 33 w= 1.9642546069631057 loss= 0.007253567147113681
Epoch: 34 w= 1.9675908436465492 loss= 0.005962754575689583
Epoch: 35 w= 1.970615698239538 loss= 0.004901649272531298
Epoch: 36 w= 1.9733582330705144 loss= 0.004029373553099482
Epoch: 37 w= 1.975844797983933 loss= 0.0033123241439168096
Epoch: 38 w= 1.9780992835054327 loss= 0.0027228776607060357
Epoch: 39 w= 1.980143350378259 loss= 0.002238326453885249
Epoch: 40 w= 1.9819966376762883 loss= 0.001840003826269386
Epoch: 41 w= 1.983676951493168 loss= 0.0015125649231412608
Epoch: 42 w= 1.9852004360204722 loss= 0.0012433955919298103
Epoch: 43 w= 1.9865817286585614 loss= 0.0010221264385926248
Epoch: 44 w= 1.987834100650429 loss= 0.0008402333603648631
Epoch: 45 w= 1.9889695845897222 loss= 0.0006907091659248264
Epoch: 46 w= 1.9899990900280147 loss= 0.0005677936325753796
Epoch: 47 w= 1.9909325082920666 loss= 0.0004667516012495216
Epoch: 48 w= 1.9917788075181404 loss= 0.000383690560742734
Epoch: 49 w= 1.9925461188164473 loss= 0.00031541069384432885
Epoch: 50 w= 1.9932418143935788 loss= 0.0002592816085930997
Epoch: 51 w= 1.9938725783835114 loss= 0.0002131410058905752
Epoch: 52 w= 1.994444471067717 loss= 0.00017521137977565514
Epoch: 53 w= 1.9949629871013967 loss= 0.0001440315413480261
Epoch: 54 w= 1.9954331083052663 loss= 0.0001184003283899171
Epoch: 55 w= 1.9958593515301082 loss= 9.733033217332803e-05
Epoch: 56 w= 1.9962458120539648 loss= 8.000985883901657e-05
Epoch: 57 w= 1.9965962029289281 loss= 6.57716599593935e-05
Epoch: 58 w= 1.9969138906555615 loss= 5.406722767150764e-05
Epoch: 59 w= 1.997201927527709 loss= 4.444566413387458e-05
Epoch: 60 w= 1.9974630809584561 loss= 3.65363112808981e-05
Epoch: 61 w= 1.9976998600690001 loss= 3.0034471708953996e-05
Epoch: 62 w= 1.9979145397958935 loss= 2.4689670610172655e-05
Epoch: 63 w= 1.9981091827482769 loss= 2.0296006560253656e-05
Epoch: 64 w= 1.9982856590251044 loss= 1.6684219437262796e-05
Epoch: 65 w= 1.9984456641827613 loss= 1.3715169898293847e-05
Epoch: 66 w= 1.9985907355257035 loss= 1.1274479219506377e-05
Epoch: 67 w= 1.9987222668766378 loss= 9.268123006398985e-06
Epoch: 68 w= 1.9988415219681517 loss= 7.61880902783969e-06
Epoch: 69 w= 1.9989496465844576 loss= 6.262999634617916e-06
Epoch: 70 w= 1.9990476795699081 loss= 5.1484640551938914e-06
Epoch: 71 w= 1.9991365628100501 loss= 4.232266273994499e-06
Epoch: 72 w= 1.999217150281112 loss= 3.479110977946351e-06
Epoch: 73 w= 1.999290216254875 loss= 2.859983851026929e-06
Epoch: 74 w= 1.9993564627377531 loss= 2.3510338359374262e-06
Epoch: 75 w= 1.9994165262155628 loss= 1.932654303533636e-06
Epoch: 76 w= 1.999470983768777 loss= 1.5887277332523938e-06
Epoch: 77 w= 1.9995203586170245 loss= 1.3060048068548734e-06
Epoch: 78 w= 1.9995651251461022 loss= 1.0735939958924364e-06
Epoch: 79 w= 1.9996057134657994 loss= 8.825419799121559e-07
Epoch: 80 w= 1.9996425135423248 loss= 7.254887315754342e-07
Epoch: 81 w= 1.999675878945041 loss= 5.963839812987369e-07
Epoch: 82 w= 1.999706130243504 loss= 4.902541385825727e-07
Epoch: 83 w= 1.9997335580874436 loss= 4.0301069098738336e-07
Epoch: 84 w= 1.9997584259992822 loss= 3.312926995781724e-07
Epoch: 85 w= 1.9997809729060159 loss= 2.723373231729343e-07
Epoch: 86 w= 1.9998014154347876 loss= 2.2387338352920307e-07
Epoch: 87 w= 1.9998199499942075 loss= 1.8403387118941732e-07
Epoch: 88 w= 1.9998367546614149 loss= 1.5128402140063082e-07
Epoch: 89 w= 1.9998519908930161 loss= 1.2436218932547864e-07
Epoch: 90 w= 1.9998658050763347 loss= 1.0223124683409346e-07
Epoch: 91 w= 1.9998783299358769 loss= 8.403862850836479e-08
Epoch: 92 w= 1.9998896858085284 loss= 6.908348768398496e-08
Epoch: 93 w= 1.9998999817997325 loss= 5.678969725349543e-08
Epoch: 94 w= 1.9999093168317574 loss= 4.66836551287917e-08
Epoch: 95 w= 1.9999177805941268 loss= 3.8376039345125727e-08
Epoch: 96 w= 1.9999254544053418 loss= 3.154680994333735e-08
Epoch: 97 w= 1.9999324119941766 loss= 2.593287985380858e-08
Epoch: 98 w= 1.9999387202080534 loss= 2.131797981222471e-08
Epoch: 99 w= 1.9999444396553017 loss= 1.752432687141379e-08
Predict (after training) 4 7.999777758621207
实现效果图：
随机梯度和普通的区别
- cost是所有样本算出来的损失，loss是随机的一个样本损失，引入了一个随机噪声，随机噪声可能会把我们向前推动，那么将来我们在更新的过程中，就有可能跨越鞍点，从而向最优值前进，而普通梯度下降可能只会停留在局部最优点停滞不前
- 随机梯度下降只需要求出一个样本均值 ，而普通梯度下降是计算所有数据
- 普通的是需要把计算所得的均值进行相加，随机梯度下降是对每一个样本来求梯度，然后进行更新
- 随机梯度算法更新的次数大于普通梯度算法
随机梯度代码：
import numpy as np
import matplotlib.pyplot as plt
from matplotlib import cm
from mpl_toolkits.mplot3d import Axes3D
x_data = [1.0, 2.0, 3.0]
y_data = [2.0, 4.0, 6.0]
w = 1.0
def forward(x):
return x * w
def loss(x, y):
y_pred = forward(x)
return (y_pred - y) ** 2
def gradient(x, y):
return 2 * x * (x * w - y)
epoch_list = []
loss_list = []
print('Predict (before training)', 4, forward(4))
for epoch in range(100):
for x, y in zip(x_data, y_data):
grad = gradient(x, y)
w -= 0.01 * grad
print("\tgrad:", x, y, grad)
l = loss(x, y)
print("progress", epoch, "w=", w, "loss=", l)
print("\n")
epoch_list.append(epoch)
loss_list.append(l)
print('Predict (after training)', 4, forward(4))
plt.plot(epoch_list, loss_list)
plt.ylabel('loss')
plt.xlabel('epoch')
plt.show()
训练结果：
Predict (before training) 4 4.0
grad: 1.0 2.0 -2.0
grad: 2.0 4.0 -7.84
grad: 3.0 6.0 -16.2288
progress 0 w= 1.260688 loss= 4.919240100095999
grad: 1.0 2.0 -1.478624
grad: 2.0 4.0 -5.796206079999999
grad: 3.0 6.0 -11.998146585599997
progress 1 w= 1.453417766656 loss= 2.688769240265834
grad: 1.0 2.0 -1.093164466688
grad: 2.0 4.0 -4.285204709416961
grad: 3.0 6.0 -8.87037374849311
progress 2 w= 1.5959051959019805 loss= 1.4696334962911515
grad: 1.0 2.0 -0.8081896081960389
grad: 2.0 4.0 -3.1681032641284723
grad: 3.0 6.0 -6.557973756745939
progress 3 w= 1.701247862192685 loss= 0.8032755585999681
grad: 1.0 2.0 -0.59750427561463
grad: 2.0 4.0 -2.3422167604093502
grad: 3.0 6.0 -4.848388694047353
progress 4 w= 1.7791289594933983 loss= 0.43905614881022015
grad: 1.0 2.0 -0.44174208101320334
grad: 2.0 4.0 -1.7316289575717576
grad: 3.0 6.0 -3.584471942173538
progress 5 w= 1.836707389300983 loss= 0.2399802903801062
grad: 1.0 2.0 -0.3265852213980338
grad: 2.0 4.0 -1.2802140678802925
grad: 3.0 6.0 -2.650043120512205
progress 6 w= 1.8792758133988885 loss= 0.1311689630744999
grad: 1.0 2.0 -0.241448373202223
grad: 2.0 4.0 -0.946477622952715
grad: 3.0 6.0 -1.9592086795121197
progress 7 w= 1.910747160155559 loss= 0.07169462478267678
grad: 1.0 2.0 -0.17850567968888198
grad: 2.0 4.0 -0.6997422643804168
grad: 3.0 6.0 -1.4484664872674653
progress 8 w= 1.9340143044689266 loss= 0.03918700813247573
grad: 1.0 2.0 -0.13197139106214673
grad: 2.0 4.0 -0.5173278529636143
grad: 3.0 6.0 -1.0708686556346834
progress 9 w= 1.9512159834655312 loss= 0.021418922423117836
grad: 1.0 2.0 -0.09756803306893769
grad: 2.0 4.0 -0.38246668963023644
grad: 3.0 6.0 -0.7917060475345892
progress 10 w= 1.9639333911678687 loss= 0.01170720245384975
grad: 1.0 2.0 -0.07213321766426262
grad: 2.0 4.0 -0.2827622132439096
grad: 3.0 6.0 -0.5853177814148953
progress 11 w= 1.9733355232910992 loss= 0.006398948863435593
grad: 1.0 2.0 -0.05332895341780164
grad: 2.0 4.0 -0.2090494973977819
grad: 3.0 6.0 -0.4327324596134101
progress 12 w= 1.9802866323953892 loss= 0.003497551760830656
grad: 1.0 2.0 -0.039426735209221686
grad: 2.0 4.0 -0.15455280202014876
grad: 3.0 6.0 -0.3199243001817109
progress 13 w= 1.9854256707695 loss= 0.001911699652671057
grad: 1.0 2.0 -0.02914865846100012
grad: 2.0 4.0 -0.11426274116712065
grad: 3.0 6.0 -0.2365238742159388
progress 14 w= 1.9892250235079405 loss= 0.0010449010656399273
grad: 1.0 2.0 -0.021549952984118992
grad: 2.0 4.0 -0.08447581569774698
grad: 3.0 6.0 -0.17486493849433593
progress 15 w= 1.9920339305797026 loss= 0.0005711243580809696
grad: 1.0 2.0 -0.015932138840594856
grad: 2.0 4.0 -0.062453984255132156
grad: 3.0 6.0 -0.12927974740812687
progress 16 w= 1.994110589284741 loss= 0.0003121664271570621
grad: 1.0 2.0 -0.011778821430517894
grad: 2.0 4.0 -0.046172980007630926
grad: 3.0 6.0 -0.09557806861579543
progress 17 w= 1.9956458879852805 loss= 0.0001706246229305199
grad: 1.0 2.0 -0.008708224029438938
grad: 2.0 4.0 -0.03413623819540135
grad: 3.0 6.0 -0.07066201306448505
progress 18 w= 1.9967809527381737 loss= 9.326038746484765e-05
grad: 1.0 2.0 -0.006438094523652627
grad: 2.0 4.0 -0.02523733053271826
grad: 3.0 6.0 -0.052241274202728505
progress 19 w= 1.9976201197307648 loss= 5.097447086306101e-05
grad: 1.0 2.0 -0.004759760538470381
grad: 2.0 4.0 -0.01865826131080439
grad: 3.0 6.0 -0.03862260091336722
progress 20 w= 1.998240525958391 loss= 2.7861740127856012e-05
grad: 1.0 2.0 -0.0035189480832178432
grad: 2.0 4.0 -0.01379427648621423
grad: 3.0 6.0 -0.028554152326460525
progress 21 w= 1.99869919972735 loss= 1.5228732143933469e-05
grad: 1.0 2.0 -0.002601600545300009
grad: 2.0 4.0 -0.01019827413757568
grad: 3.0 6.0 -0.021110427464781978
progress 22 w= 1.9990383027488265 loss= 8.323754426231206e-06
grad: 1.0 2.0 -0.001923394502346909
grad: 2.0 4.0 -0.007539706449199102
grad: 3.0 6.0 -0.01560719234984198
progress 23 w= 1.9992890056818404 loss= 4.549616284094891e-06
grad: 1.0 2.0 -0.0014219886363191492
grad: 2.0 4.0 -0.005574195454370212
grad: 3.0 6.0 -0.011538584590544687
progress 24 w= 1.999474353368653 loss= 2.486739429417538e-06
grad: 1.0 2.0 -0.0010512932626940419
grad: 2.0 4.0 -0.004121069589761106
grad: 3.0 6.0 -0.008530614050808794
progress 25 w= 1.9996113831376856 loss= 1.3592075910762856e-06
grad: 1.0 2.0 -0.0007772337246287897
grad: 2.0 4.0 -0.0030467562005451754
grad: 3.0 6.0 -0.006306785335127074
progress 26 w= 1.9997126908902887 loss= 7.429187207079447e-07
grad: 1.0 2.0 -0.0005746182194226179
grad: 2.0 4.0 -0.002252503420136165
grad: 3.0 6.0 -0.00466268207967957
progress 27 w= 1.9997875889274812 loss= 4.060661735575354e-07
grad: 1.0 2.0 -0.0004248221450375844
grad: 2.0 4.0 -0.0016653028085471533
grad: 3.0 6.0 -0.0034471768136938863
progress 28 w= 1.9998429619451539 loss= 2.2194855602869353e-07
grad: 1.0 2.0 -0.00031407610969225175
grad: 2.0 4.0 -0.0012311783499932005
grad: 3.0 6.0 -0.0025485391844828342
progress 29 w= 1.9998838998815958 loss= 1.213131374411496e-07
grad: 1.0 2.0 -0.00023220023680847746
grad: 2.0 4.0 -0.0009102249282886277
grad: 3.0 6.0 -0.0018841656015560204
progress 30 w= 1.9999141657892625 loss= 6.630760559646474e-08
grad: 1.0 2.0 -0.00017166842147497974
grad: 2.0 4.0 -0.0006729402121816719
grad: 3.0 6.0 -0.0013929862392156878
progress 31 w= 1.9999365417379913 loss= 3.624255915449335e-08
grad: 1.0 2.0 -0.0001269165240174175
grad: 2.0 4.0 -0.0004975127741477792
grad: 3.0 6.0 -0.0010298514424817995
progress 32 w= 1.9999530845453979 loss= 1.9809538924707548e-08
grad: 1.0 2.0 -9.383090920422887e-05
grad: 2.0 4.0 -0.00036781716408107457
grad: 3.0 6.0 -0.0007613815296476645
progress 33 w= 1.9999653148414271 loss= 1.0827542027017377e-08
grad: 1.0 2.0 -6.937031714571162e-05
grad: 2.0 4.0 -0.0002719316432120422
grad: 3.0 6.0 -0.0005628985014531906
progress 34 w= 1.999974356846045 loss= 5.9181421028034105e-09
grad: 1.0 2.0 -5.1286307909848006e-05
grad: 2.0 4.0 -0.00020104232700646207
grad: 3.0 6.0 -0.0004161576169003922
progress 35 w= 1.9999810417085633 loss= 3.2347513278475087e-09
grad: 1.0 2.0 -3.7916582873442906e-05
grad: 2.0 4.0 -0.0001486330048638962
grad: 3.0 6.0 -0.0003076703200690645
progress 36 w= 1.9999859839076413 loss= 1.7680576050779005e-09
grad: 1.0 2.0 -2.8032184717474706e-05
grad: 2.0 4.0 -0.0001098861640933535
grad: 3.0 6.0 -0.00022746435967313516
progress 37 w= 1.9999896377347262 loss= 9.6638887447731e-10
grad: 1.0 2.0 -2.0724530547688857e-05
grad: 2.0 4.0 -8.124015974608767e-05
grad: 3.0 6.0 -0.00016816713067413502
progress 38 w= 1.999992339052936 loss= 5.282109892545845e-10
grad: 1.0 2.0 -1.5321894128117464e-05
grad: 2.0 4.0 -6.006182498197177e-05
grad: 3.0 6.0 -0.00012432797771566584
progress 39 w= 1.9999943361699042 loss= 2.887107421958329e-10
grad: 1.0 2.0 -1.1327660191629008e-05
grad: 2.0 4.0 -4.4404427951505454e-05
grad: 3.0 6.0 -9.191716585732479e-05
progress 40 w= 1.9999958126624442 loss= 1.5780416225633037e-10
grad: 1.0 2.0 -8.37467511161094e-06
grad: 2.0 4.0 -3.282872643772805e-05
grad: 3.0 6.0 -6.795546372551087e-05
progress 41 w= 1.999996904251097 loss= 8.625295142578772e-11
grad: 1.0 2.0 -6.191497806007362e-06
grad: 2.0 4.0 -2.4270671399762023e-05
grad: 3.0 6.0 -5.0240289795056015e-05
progress 42 w= 1.999997711275687 loss= 4.71443308235547e-11
grad: 1.0 2.0 -4.5774486259198e-06
grad: 2.0 4.0 -1.794359861406747e-05
grad: 3.0 6.0 -3.714324913239864e-05
progress 43 w= 1.9999983079186507 loss= 2.5768253628059826e-11
grad: 1.0 2.0 -3.3841626985164908e-06
grad: 2.0 4.0 -1.326591777761621e-05
grad: 3.0 6.0 -2.7460449796734565e-05
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progress 72 w= 1.999999999734279 loss= 6.354692062078993e-19
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progress 81 w= 1.999999999982466 loss= 2.7669155644059242e-21
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progress 85 w= 1.9999999999947617 loss= 2.469467919185614e-22
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progress 89 w= 1.9999999999984353 loss= 2.2033851437431755e-23
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progress 91 w= 1.9999999999991447 loss= 6.5840863393251405e-24
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progress 92 w= 1.9999999999993676 loss= 3.5991747246272455e-24
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progress 93 w= 1.9999999999995324 loss= 1.969312363793734e-24
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progress 94 w= 1.9999999999996543 loss= 1.0761829795642296e-24
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progress 95 w= 1.9999999999997444 loss= 5.875191475205477e-25
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progress 96 w= 1.999999999999811 loss= 3.2110109830478153e-25
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progress 97 w= 1.9999999999998603 loss= 1.757455879087579e-25
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progress 98 w= 1.9999999999998967 loss= 9.608404711682446e-26
grad: 1.0 2.0 -2.0650148258027912e-13
grad: 2.0 4.0 -8.100187187665142e-13
grad: 3.0 6.0 -1.6786572132332367e-12
progress 99 w= 1.9999999999999236 loss= 5.250973729513143e-26
效果展示：
显然两个图还是有区别的
| 周国平说，人生要有不较劲的智慧。常与同好争高下，不与傻瓜论短长。和错的人纠缠，只会把你拖入污浊的泥淖。不要与烂人烂事纠缠，因为你的好时光值得浪费在一切美好的事情上。 |
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