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PyTorch: 梯度下降及反向傳播的實(shí)例詳解

更新時(shí)間：2019年08月20日 11:29:47 作者：yuquanle

今天小編就為大家分享一篇PyTorch: 梯度下降及反向傳播的實(shí)例詳解，具有很好的參考價(jià)值，希望對(duì)大家有所幫助。一起跟隨小編過來看看吧

線性模型

線性模型介紹

線性模型是很常見的機(jī)器學(xué)習(xí)模型，通常通過線性的公式來擬合訓(xùn)練數(shù)據(jù)集。訓(xùn)練集包括(x,y)，x為特征,y為目標(biāo)。如下圖：

將真實(shí)值和預(yù)測(cè)值用于構(gòu)建損失函數(shù)，訓(xùn)練的目標(biāo)是最小化這個(gè)函數(shù)，從而更新w。當(dāng)損失函數(shù)達(dá)到最小時(shí)(理想上,實(shí)際情況可能會(huì)陷入局部最優(yōu))，此時(shí)的模型為最優(yōu)模型，線性模型常見的的損失函數(shù)：

線性模型例子

下面通過一個(gè)例子可以觀察不同權(quán)重(w)對(duì)模型損失函數(shù)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(jié)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(diào)整w，loss變化圖：

可以發(fā)現(xiàn)當(dāng)w=2時(shí)，loss最小。但是現(xiàn)實(shí)中最常見的情況是，我們知道數(shù)據(jù)集，定義好損失函數(shù)之后(loss)，我們并不會(huì)從0到n去設(shè)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(shù)。更常見的做法是，首先隨機(jī)初始化w的值，然后根據(jù)loss函數(shù)定義對(duì)w求梯度，然后通過w的梯度來更新w的值，這就是經(jīng)典的梯度下降法思想。

梯度下降法

梯度的本意是一個(gè)向量，表示某一函數(shù)在該點(diǎn)處的方向?qū)?shù)沿著該方向取得最大值，即函數(shù)在該點(diǎn)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(jī)器學(xué)習(xí)算法的模型參數(shù)，即無約束優(yōu)化問題時(shí)，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(shù)的最小值時(shí)，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(shù)和模型參數(shù)值。即每次更新參數(shù)w減去其梯度(通常會(huì)乘以學(xué)習(xí)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(duì)w求導(dǎo))
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(jié)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(dòng)的求導(dǎo)，我們可以通過反向傳播將梯度往回傳播。通常有二個(gè)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(jié)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

以上這篇PyTorch: 梯度下降及反向傳播的實(shí)例詳解就是小編分享給大家的全部?jī)?nèi)容了，希望能給大家一個(gè)參考，也希望大家多多支持腳本之家。

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