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小編給大家分享一下PyTorch中梯度下降及反向傳播的示例分析，希望大家閱讀完這篇文章之后都有所收獲，下面讓我們一起去探討吧！
線性模型
線性模型介紹
線性模型是很常見的機器學習模型，通常通過線性的公式來擬合訓練數據集。訓練集包括(x,y)，x為特征,y為目標。如下圖：
將真實值和預測值用于構建損失函數，訓練的目標是最小化這個函數，從而更新w。當損失函數達到最小時(理想上,實際情況可能會陷入局部最優)，此時的模型為最優模型，線性模型常見的的損失函數：
線性模型例子
下面通過一個例子可以觀察不同權重(w)對模型損失函數的影響。
#author:yuquanle
#data:2018.2.5
#Study of Linear Model
import numpy as np
import matplotlib.pyplot as plt
x_data = [1.0, 2.0, 3.0]
y_data = [2.0, 4.0, 6.0]
def forward(x):
return x * w
def loss(x, y):
y_pred = forward(x)
return (y_pred - y)*(y_pred - y)
w_list = []
mse_list = []
for w in np.arange(0.0, 4.1, 0.1):
print("w=", w)
l_sum = 0
for x_val, y_val in zip(x_data, y_data):
# error
l = loss(x_val, y_val)
l_sum += l
print("MSE=", l_sum/3)
w_list.append(w)
mse_list.append(l_sum/3)
plt.plot(w_list, mse_list)
plt.ylabel("Loss")
plt.xlabel("w")
plt.show()
輸出結果：
w= 0.0
MSE= 18.6666666667
w= 0.1
MSE= 16.8466666667
w= 0.2
MSE= 15.12
w= 0.3
MSE= 13.4866666667
w= 0.4
MSE= 11.9466666667
w= 0.5
MSE= 10.5
w= 0.6
MSE= 9.14666666667調整w，loss變化圖：
可以發現當w=2時，loss最小。但是現實中最常見的情況是，我們知道數據集，定義好損失函數之后(loss)，我們并不會從0到n去設置w的值，然后求loss，最后選取使得loss最小的w作為最佳模型的參數。更常見的做法是，首先隨機初始化w的值，然后根據loss函數定義對w求梯度，然后通過w的梯度來更新w的值，這就是經典的梯度下降法思想。
梯度下降法
梯度的本意是一個向量，表示某一函數在該點處的方向導數沿著該方向取得最大值，即函數在該點處沿著該方向（此梯度的方向）變化最快，變化率最大（為該梯度的模）。
梯度下降是迭代法的一種,可以用于求解最小二乘問題(線性和非線性都可以)。在求解機器學習算法的模型參數，即無約束優化問題時，梯度下降（Gradient Descent）是最常采用的方法之一，另一種常用的方法是最小二乘法。在求解損失函數的最小值時，可以通過梯度下降法來一步步的迭代求解，得到最小化的損失函數和模型參數值。即每次更新參數w減去其梯度(通常會乘以學習率)。
#author:yuquanle
#data:2018.2.5
#Study of SGD
x_data = [1.0, 2.0, 3.0]
y_data = [2.0, 4.0, 6.0]
# any random value
w = 1.0
# forward pass
def forward(x):
return x * w
def loss(x, y):
y_pred = forward(x)
return (y_pred - y)*(y_pred - y)
# compute gradient (loss對w求導)
def gradient(x, y):
return 2*x*(x*w - y)
# Before training
print("predict (before training)", 4, forward(4))
# Training loop
for epoch in range(20):
for x, y in zip(x_data, y_data):
grad = gradient(x, y)
w = w - 0.01 * grad
print("\t grad: ",x, y, grad)
l = loss(x, y)
print("progress:", epoch, l)
# After training
print("predict (after training)", 4, forward(4))
輸出結果：
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 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 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 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 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 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 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 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 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 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 0.021418922423117836
predict (after training) 4 7.804863933862125反向傳播
但是在定義好模型之后，使用pytorch框架不需要我們手動的求導，我們可以通過反向傳播將梯度往回傳播。通常有二個過程，forward和backward:
#author:yuquanle
#data:2018.2.6
#Study of BackPagation
import torch
from torch import nn
from torch.autograd import Variable
x_data = [1.0, 2.0, 3.0]
y_data = [2.0, 4.0, 6.0]
# Any random value
w = Variable(torch.Tensor([1.0]), requires_grad=True)
# forward pass
def forward(x):
return x*w
# Before training
print("predict (before training)", 4, forward(4))
def loss(x, y):
y_pred = forward(x)
return (y_pred-y)*(y_pred-y)
# Training: forward, backward and update weight
# Training loop
for epoch in range(10):
for x, y in zip(x_data, y_data):
l = loss(x, y)
l.backward()
print("\t grad:", x, y, w.grad.data[0])
w.data = w.data - 0.01 * w.grad.data
# Manually zero the gradients after running the backward pass and update w
w.grad.data.zero_()
print("progress:", epoch, l.data[0])
# After training
print("predict (after training)", 4, forward(4))
輸出結果：
predict (before training) 4 Variable containing:
4
[torch.FloatTensor of size 1]
grad: 1.0 2.0 -2.0
grad: 2.0 4.0 -7.840000152587891
grad: 3.0 6.0 -16.228801727294922
progress: 0 7.315943717956543
grad: 1.0 2.0 -1.478623867034912
grad: 2.0 4.0 -5.796205520629883
grad: 3.0 6.0 -11.998146057128906
progress: 1 3.9987640380859375
grad: 1.0 2.0 -1.0931644439697266
grad: 2.0 4.0 -4.285204887390137
grad: 3.0 6.0 -8.870372772216797
progress: 2 2.1856532096862793
grad: 1.0 2.0 -0.8081896305084229
grad: 2.0 4.0 -3.1681032180786133
grad: 3.0 6.0 -6.557973861694336
progress: 3 1.1946394443511963
grad: 1.0 2.0 -0.5975041389465332
grad: 2.0 4.0 -2.3422164916992188
grad: 3.0 6.0 -4.848389625549316
progress: 4 0.6529689431190491
grad: 1.0 2.0 -0.4417421817779541
grad: 2.0 4.0 -1.7316293716430664
grad: 3.0 6.0 -3.58447265625
progress: 5 0.35690122842788696
grad: 1.0 2.0 -0.3265852928161621
grad: 2.0 4.0 -1.2802143096923828
grad: 3.0 6.0 -2.650045394897461
progress: 6 0.195076122879982
grad: 1.0 2.0 -0.24144840240478516
grad: 2.0 4.0 -0.9464778900146484
grad: 3.0 6.0 -1.9592113494873047
progress: 7 0.10662525147199631
grad: 1.0 2.0 -0.17850565910339355
grad: 2.0 4.0 -0.699742317199707
grad: 3.0 6.0 -1.4484672546386719
progress: 8 0.0582793727517128
grad: 1.0 2.0 -0.1319713592529297
grad: 2.0 4.0 -0.5173273086547852
grad: 3.0 6.0 -1.070866584777832
progress: 9 0.03185431286692619
predict (after training) 4 Variable containing:
7.8049
[torch.FloatTensor of size 1]
Process finished with exit code 0看完了這篇文章，相信你對“PyTorch中梯度下降及反向傳播的示例分析”有了一定的了解，如果想了解更多相關知識，歡迎關注億速云行業資訊頻道，感謝各位的閱讀！
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