PyTorch梯度下降反向传播
前言：
反向传播的目的是计算成本函数C对网络中任意w或b的偏导数。一旦我们有了这些偏导数，我们将通过一些常数 α的乘积和该数量相对于成本函数的偏导数来更新网络中的权重和偏差。这是流行的梯度下降算法。而偏导数给出了最大上升的方向。因此，关于反向传播算法，我们继续查看下文。
我们向相反的方向迈出了一小步——最大下降的方向，也就是将我们带到成本函数的局部最小值的方向
如题：
意思是利用这个二次模型来预测数据，减小损失函数(MSE)的值。
代码如下：
import torch
import matplotlib.pyplot as plt
import os
os.environ["KMP_DUPLICATE_LIB_OK"] = "TRUE"
# 数据集
x_data = [1.0,2.0,3.0]
y_data = [2.0,4.0,6.0]
# 权重参数初始值均为1
w = torch.tensor([1.0,1.0,1.0])
w.requires_grad = True # 需要计算梯度
# 前向传播
def forward(x):
return w[0]*(x**2)+w[1]*x+w[2]
# 计算损失
def loss(x,y):
y_pred = forward(x)
return (y_pred-y) ** 2
# 训练模块
print('predict (before tranining) ',4, forward(4).item())
epoch_list = []
w_list = []
loss_list = []
for epoch in range(1000):
for x,y in zip(x_data,y_data):
l = loss(x,y)
l.backward() # 后向传播
print('\tgrad: ',x,y,w.grad.data)
w.data = w.data - 0.01 * w.grad.data # 梯度下降
w.grad.data.zero_() # 梯度清零操作
print('progress: ',epoch,l.item())
epoch_list.append(epoch)
w_list.append(w.data)
loss_list.append(l.item())
print('predict (after tranining) ',4, forward(4).item())
# 绘图
plt.plot(epoch_list,loss_list,'b')
plt.xlabel('Epoch')
plt.ylabel('Loss')
plt.grid()
plt.show()
结果如下：
predict (before tranining) 4 21.0
grad: 1.0 2.0 tensor([2., 2., 2.])
grad: 2.0 4.0 tensor([22.8800, 11.4400, 5.7200])
grad: 3.0 6.0 tensor([77.0472, 25.6824, 8.5608])
progress: 0 18.321826934814453
grad: 1.0 2.0 tensor([-1.1466, -1.1466, -1.1466])
grad: 2.0 4.0 tensor([-15.5367, -7.7683, -3.8842])
grad: 3.0 6.0 tensor([-30.4322, -10.1441, -3.3814])
progress: 1 2.858394145965576
grad: 1.0 2.0 tensor([0.3451, 0.3451, 0.3451])
grad: 2.0 4.0 tensor([2.4273, 1.2137, 0.6068])
grad: 3.0 6.0 tensor([19.4499, 6.4833, 2.1611])
progress: 2 1.1675907373428345
grad: 1.0 2.0 tensor([-0.3224, -0.3224, -0.3224])
grad: 2.0 4.0 tensor([-5.8458, -2.9229, -1.4614])
grad: 3.0 6.0 tensor([-3.8829, -1.2943, -0.4314])
progress: 3 0.04653334245085716
grad: 1.0 2.0 tensor([0.0137, 0.0137, 0.0137])
grad: 2.0 4.0 tensor([-1.9141, -0.9570, -0.4785])
grad: 3.0 6.0 tensor([6.8557, 2.2852, 0.7617])
progress: 4 0.14506366848945618
grad: 1.0 2.0 tensor([-0.1182, -0.1182, -0.1182])
grad: 2.0 4.0 tensor([-3.6644, -1.8322, -0.9161])
grad: 3.0 6.0 tensor([1.7455, 0.5818, 0.1939])
progress: 5 0.009403289295732975
grad: 1.0 2.0 tensor([-0.0333, -0.0333, -0.0333])
grad: 2.0 4.0 tensor([-2.7739, -1.3869, -0.6935])
grad: 3.0 6.0 tensor([4.0140, 1.3380, 0.4460])
progress: 6 0.04972923547029495
grad: 1.0 2.0 tensor([-0.0501, -0.0501, -0.0501])
grad: 2.0 4.0 tensor([-3.1150, -1.5575, -0.7788])
grad: 3.0 6.0 tensor([2.8534, 0.9511, 0.3170])
progress: 7 0.025129113346338272
grad: 1.0 2.0 tensor([-0.0205, -0.0205, -0.0205])
grad: 2.0 4.0 tensor([-2.8858, -1.4429, -0.7215])
grad: 3.0 6.0 tensor([3.2924, 1.0975, 0.3658])
progress: 8 0.03345605731010437
grad: 1.0 2.0 tensor([-0.0134, -0.0134, -0.0134])
grad: 2.0 4.0 tensor([-2.9247, -1.4623, -0.7312])
grad: 3.0 6.0 tensor([2.9909, 0.9970, 0.3323])
progress: 9 0.027609655633568764
grad: 1.0 2.0 tensor([0.0033, 0.0033, 0.0033])
grad: 2.0 4.0 tensor([-2.8414, -1.4207, -0.7103])
grad: 3.0 6.0 tensor([3.0377, 1.0126, 0.3375])
progress: 10 0.02848036028444767
grad: 1.0 2.0 tensor([0.0148, 0.0148, 0.0148])
grad: 2.0 4.0 tensor([-2.8174, -1.4087, -0.7043])
grad: 3.0 6.0 tensor([2.9260, 0.9753, 0.3251])
progress: 11 0.02642466314136982
grad: 1.0 2.0 tensor([0.0280, 0.0280, 0.0280])
grad: 2.0 4.0 tensor([-2.7682, -1.3841, -0.6920])
grad: 3.0 6.0 tensor([2.8915, 0.9638, 0.3213])
progress: 12 0.025804826989769936
grad: 1.0 2.0 tensor([0.0397, 0.0397, 0.0397])
grad: 2.0 4.0 tensor([-2.7330, -1.3665, -0.6832])
grad: 3.0 6.0 tensor([2.8243, 0.9414, 0.3138])
progress: 13 0.02462013065814972
grad: 1.0 2.0 tensor([0.0514, 0.0514, 0.0514])
grad: 2.0 4.0 tensor([-2.6934, -1.3467, -0.6734])
grad: 3.0 6.0 tensor([2.7756, 0.9252, 0.3084])
progress: 14 0.023777369409799576
grad: 1.0 2.0 tensor([0.0624, 0.0624, 0.0624])
grad: 2.0 4.0 tensor([-2.6580, -1.3290, -0.6645])
grad: 3.0 6.0 tensor([2.7213, 0.9071, 0.3024])
progress: 15 0.0228563379496336
grad: 1.0 2.0 tensor([0.0731, 0.0731, 0.0731])
grad: 2.0 4.0 tensor([-2.6227, -1.3113, -0.6557])
grad: 3.0 6.0 tensor([2.6725, 0.8908, 0.2969])
progress: 16 0.022044027224183083
grad: 1.0 2.0 tensor([0.0833, 0.0833, 0.0833])
grad: 2.0 4.0 tensor([-2.5893, -1.2946, -0.6473])
grad: 3.0 6.0 tensor([2.6240, 0.8747, 0.2916])
progress: 17 0.02125072106719017
grad: 1.0 2.0 tensor([0.0931, 0.0931, 0.0931])
grad: 2.0 4.0 tensor([-2.5568, -1.2784, -0.6392])
grad: 3.0 6.0 tensor([2.5780, 0.8593, 0.2864])
progress: 18 0.020513182505965233
grad: 1.0 2.0 tensor([0.1025, 0.1025, 0.1025])
grad: 2.0 4.0 tensor([-2.5258, -1.2629, -0.6314])
grad: 3.0 6.0 tensor([2.5335, 0.8445, 0.2815])
progress: 19 0.019810274243354797
grad: 1.0 2.0 tensor([0.1116, 0.1116, 0.1116])
grad: 2.0 4.0 tensor([-2.4958, -1.2479, -0.6239])
grad: 3.0 6.0 tensor([2.4908, 0.8303, 0.2768])
progress: 20 0.019148115068674088
grad: 1.0 2.0 tensor([0.1203, 0.1203, 0.1203])
grad: 2.0 4.0 tensor([-2.4669, -1.2335, -0.6167])
grad: 3.0 6.0 tensor([2.4496, 0.8165, 0.2722])
progress: 21 0.018520694226026535
grad: 1.0 2.0 tensor([0.1286, 0.1286, 0.1286])
grad: 2.0 4.0 tensor([-2.4392, -1.2196, -0.6098])
grad: 3.0 6.0 tensor([2.4101, 0.8034, 0.2678])
progress: 22 0.017927465960383415
grad: 1.0 2.0 tensor([0.1367, 0.1367, 0.1367])
grad: 2.0 4.0 tensor([-2.4124, -1.2062, -0.6031])
grad: 3.0 6.0 tensor([2.3720, 0.7907, 0.2636])
progress: 23 0.01736525259912014
grad: 1.0 2.0 tensor([0.1444, 0.1444, 0.1444])
grad: 2.0 4.0 tensor([-2.3867, -1.1933, -0.5967])
grad: 3.0 6.0 tensor([2.3354, 0.7785, 0.2595])
progress: 24 0.016833148896694183
grad: 1.0 2.0 tensor([0.1518, 0.1518, 0.1518])
grad: 2.0 4.0 tensor([-2.3619, -1.1810, -0.5905])
grad: 3.0 6.0 tensor([2.3001, 0.7667, 0.2556])
progress: 25 0.01632905937731266
grad: 1.0 2.0 tensor([0.1589, 0.1589, 0.1589])
grad: 2.0 4.0 tensor([-2.3380, -1.1690, -0.5845])
grad: 3.0 6.0 tensor([2.2662, 0.7554, 0.2518])
progress: 26 0.01585075818002224
grad: 1.0 2.0 tensor([0.1657, 0.1657, 0.1657])
grad: 2.0 4.0 tensor([-2.3151, -1.1575, -0.5788])
grad: 3.0 6.0 tensor([2.2336, 0.7445, 0.2482])
progress: 27 0.015397666022181511
grad: 1.0 2.0 tensor([0.1723, 0.1723, 0.1723])
grad: 2.0 4.0 tensor([-2.2929, -1.1465, -0.5732])
grad: 3.0 6.0 tensor([2.2022, 0.7341, 0.2447])
progress: 28 0.014967591501772404
grad: 1.0 2.0 tensor([0.1786, 0.1786, 0.1786])
grad: 2.0 4.0 tensor([-2.2716, -1.1358, -0.5679])
grad: 3.0 6.0 tensor([2.1719, 0.7240, 0.2413])
progress: 29 0.014559715054929256
grad: 1.0 2.0 tensor([0.1846, 0.1846, 0.1846])
grad: 2.0 4.0 tensor([-2.2511, -1.1255, -0.5628])
grad: 3.0 6.0 tensor([2.1429, 0.7143, 0.2381])
progress: 30 0.014172340743243694
grad: 1.0 2.0 tensor([0.1904, 0.1904, 0.1904])
grad: 2.0 4.0 tensor([-2.2313, -1.1157, -0.5578])
grad: 3.0 6.0 tensor([2.1149, 0.7050, 0.2350])
progress: 31 0.013804304413497448
grad: 1.0 2.0 tensor([0.1960, 0.1960, 0.1960])
grad: 2.0 4.0 tensor([-2.2123, -1.1061, -0.5531])
grad: 3.0 6.0 tensor([2.0879, 0.6960, 0.2320])
progress: 32 0.013455045409500599
grad: 1.0 2.0 tensor([0.2014, 0.2014, 0.2014])
grad: 2.0 4.0 tensor([-2.1939, -1.0970, -0.5485])
grad: 3.0 6.0 tensor([2.0620, 0.6873, 0.2291])
progress: 33 0.013122711330652237
grad: 1.0 2.0 tensor([0.2065, 0.2065, 0.2065])
grad: 2.0 4.0 tensor([-2.1763, -1.0881, -0.5441])
grad: 3.0 6.0 tensor([2.0370, 0.6790, 0.2263])
progress: 34 0.01280694268643856
grad: 1.0 2.0 tensor([0.2114, 0.2114, 0.2114])
grad: 2.0 4.0 tensor([-2.1592, -1.0796, -0.5398])
grad: 3.0 6.0 tensor([2.0130, 0.6710, 0.2237])
progress: 35 0.012506747618317604
grad: 1.0 2.0 tensor([0.2162, 0.2162, 0.2162])
grad: 2.0 4.0 tensor([-2.1428, -1.0714, -0.5357])
grad: 3.0 6.0 tensor([1.9899, 0.6633, 0.2211])
progress: 36 0.012220758944749832
grad: 1.0 2.0 tensor([0.2207, 0.2207, 0.2207])
grad: 2.0 4.0 tensor([-2.1270, -1.0635, -0.5317])
grad: 3.0 6.0 tensor([1.9676, 0.6559, 0.2186])
progress: 37 0.01194891706109047
grad: 1.0 2.0 tensor([0.2251, 0.2251, 0.2251])
grad: 2.0 4.0 tensor([-2.1118, -1.0559, -0.5279])
grad: 3.0 6.0 tensor([1.9462, 0.6487, 0.2162])
progress: 38 0.011689926497638226
grad: 1.0 2.0 tensor([0.2292, 0.2292, 0.2292])
grad: 2.0 4.0 tensor([-2.0971, -1.0485, -0.5243])
grad: 3.0 6.0 tensor([1.9255, 0.6418, 0.2139])
progress: 39 0.01144315768033266
grad: 1.0 2.0 tensor([0.2333, 0.2333, 0.2333])
grad: 2.0 4.0 tensor([-2.0829, -1.0414, -0.5207])
grad: 3.0 6.0 tensor([1.9057, 0.6352, 0.2117])
progress: 40 0.011208509095013142
grad: 1.0 2.0 tensor([0.2371, 0.2371, 0.2371])
grad: 2.0 4.0 tensor([-2.0693, -1.0346, -0.5173])
grad: 3.0 6.0 tensor([1.8865, 0.6288, 0.2096])
progress: 41 0.0109840864315629
grad: 1.0 2.0 tensor([0.2408, 0.2408, 0.2408])
grad: 2.0 4.0 tensor([-2.0561, -1.0280, -0.5140])
grad: 3.0 6.0 tensor([1.8681, 0.6227, 0.2076])
progress: 42 0.010770938359200954
grad: 1.0 2.0 tensor([0.2444, 0.2444, 0.2444])
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grad: 3.0 6.0 tensor([1.8503, 0.6168, 0.2056])
progress: 43 0.010566935874521732
grad: 1.0 2.0 tensor([0.2478, 0.2478, 0.2478])
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grad: 1.0 2.0 tensor([0.2510, 0.2510, 0.2510])
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progress: 52 0.00910158734768629
grad: 1.0 2.0 tensor([0.2728, 0.2728, 0.2728])
grad: 2.0 4.0 tensor([-1.9387, -0.9694, -0.4847])
grad: 3.0 6.0 tensor([1.7050, 0.5683, 0.1894])
progress: 53 0.00897257961332798
grad: 1.0 2.0 tensor([0.2750, 0.2750, 0.2750])
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grad: 2.0 4.0 tensor([-1.8034, -0.9017, -0.4508])
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progress: 75 0.007156429346650839
grad: 1.0 2.0 tensor([0.3058, 0.3058, 0.3058])
grad: 2.0 4.0 tensor([-1.7992, -0.8996, -0.4498])
grad: 3.0 6.0 tensor([1.5173, 0.5058, 0.1686])
progress: 76 0.007105532102286816
grad: 1.0 2.0 tensor([0.3066, 0.3066, 0.3066])
grad: 2.0 4.0 tensor([-1.7952, -0.8976, -0.4488])
grad: 3.0 6.0 tensor([1.5121, 0.5040, 0.1680])
progress: 77 0.00705681974068284
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grad: 2.0 4.0 tensor([-1.7913, -0.8956, -0.4478])
grad: 3.0 6.0 tensor([1.5070, 0.5023, 0.1674])
progress: 78 0.007009552326053381
grad: 1.0 2.0 tensor([0.3081, 0.3081, 0.3081])
grad: 2.0 4.0 tensor([-1.7875, -0.8937, -0.4469])
grad: 3.0 6.0 tensor([1.5021, 0.5007, 0.1669])
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grad: 1.0 2.0 tensor([0.3087, 0.3087, 0.3087])
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progress: 80 0.006920332089066505
grad: 1.0 2.0 tensor([0.3094, 0.3094, 0.3094])
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grad: 3.0 6.0 tensor([1.4884, 0.4961, 0.1654])
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grad: 1.0 2.0 tensor([0.3106, 0.3106, 0.3106])
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progress: 90 0.0065572685562074184
grad: 1.0 2.0 tensor([0.3143, 0.3143, 0.3143])
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grad: 3.0 6.0 tensor([1.4542, 0.4847, 0.1616])
progress: 91 0.0065271081402897835
grad: 1.0 2.0 tensor([0.3147, 0.3147, 0.3147])
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grad: 3.0 6.0 tensor([1.4510, 0.4837, 0.1612])
progress: 92 0.00649801641702652
grad: 1.0 2.0 tensor([0.3150, 0.3150, 0.3150])
grad: 2.0 4.0 tensor([-1.7441, -0.8720, -0.4360])
grad: 3.0 6.0 tensor([1.4478, 0.4826, 0.1609])
progress: 93 0.0064699104987084866
grad: 1.0 2.0 tensor([0.3153, 0.3153, 0.3153])
grad: 2.0 4.0 tensor([-1.7415, -0.8708, -0.4354])
grad: 3.0 6.0 tensor([1.4448, 0.4816, 0.1605])
progress: 94 0.006442630663514137
grad: 1.0 2.0 tensor([0.3156, 0.3156, 0.3156])
grad: 2.0 4.0 tensor([-1.7391, -0.8695, -0.4348])
grad: 3.0 6.0 tensor([1.4418, 0.4806, 0.1602])
progress: 95 0.006416172254830599
grad: 1.0 2.0 tensor([0.3159, 0.3159, 0.3159])
grad: 2.0 4.0 tensor([-1.7366, -0.8683, -0.4342])
grad: 3.0 6.0 tensor([1.4389, 0.4796, 0.1599])
progress: 96 0.006390606984496117
grad: 1.0 2.0 tensor([0.3161, 0.3161, 0.3161])
grad: 2.0 4.0 tensor([-1.7343, -0.8671, -0.4336])
grad: 3.0 6.0 tensor([1.4361, 0.4787, 0.1596])
progress: 97 0.0063657015562057495
grad: 1.0 2.0 tensor([0.3164, 0.3164, 0.3164])
grad: 2.0 4.0 tensor([-1.7320, -0.8660, -0.4330])
grad: 3.0 6.0 tensor([1.4334, 0.4778, 0.1593])
progress: 98 0.0063416799530386925
grad: 1.0 2.0 tensor([0.3166, 0.3166, 0.3166])
grad: 2.0 4.0 tensor([-1.7297, -0.8649, -0.4324])
grad: 3.0 6.0 tensor([1.4308, 0.4769, 0.1590])
progress: 99 0.00631808303296566
predict (after tranining) 4 8.544171333312988
损失值随着迭代次数的增加呈递减趋势，如下图所示：
可以看出：x=4时的预测值约为8.5，与真实值8有所差距，可通过提高迭代次数或者调整学习率、初始参数等方法来减小差距。
参考文献：
[1] https://www.bilibili.com/video/av93365242
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