The Cross-Entropy Loss Function for the Softmax Function

作者：凯鲁嘎吉 - 博客园 http://www.cnblogs.com/kailugaji/

本文介绍含有softmax函数的交叉熵损失函数的求导过程，并介绍一种交叉熵损失的等价形式，从而解决因log(0)而出现数值为NaN的问题。

1. softmax函数求导

2. 交叉熵损失求导(含softmax)

3. 交叉熵损失函数(含softmax函数)的等价形式

4. Python验证${p_i}\left( {{a_i} - \sum\limits_{j = 1}^K {{p_{ij}}{a_{ij}}} } \right) = 0$

 1 # -*- coding: utf-8 -*-
 2 # Author：凯鲁嘎吉 Coral Gajic
 3 # https://www.cnblogs.com/kailugaji/
 4 # 验证E(X-EX)=0
 5 import numpy as np
 6 a = np.random.rand(5)
 7 p = np.random.rand(5)
 8 p = p / p.sum(axis=0, keepdims=True)
 9 b = (np.dot(p, a)).sum()
10 print('a: ', a)
11 print('p: ', p)
12 print('b: ', b)
13 print('结果: ', (p * (a - b)).sum())

结果：

D:\ProgramData\Anaconda3\python.exe "D:/Python code/2023.3 exercise/向量间的距离度量/test.py"
a:  [0.90457897 0.08975555 0.6955031  0.74161145 0.50095907]
p:  [0.02797057 0.09509987 0.27454503 0.273575   0.32880953]
b:  0.5923907183986392
结果:  5.204170427930421e-17

Process finished with exit code 0

5. Python验证$L_1$与$L_2$等价

 1 # -*- coding: utf-8 -*-
 2 # Author：凯鲁嘎吉 Coral Gajic
 3 # https://www.cnblogs.com/kailugaji/
 4 # Softmax classification with cross-entropy
 5 import torch
 6 import numpy as np
 7 import matplotlib.pyplot as plt
 8 plt.rc('font',family='Times New Roman')
 9 
10 def sinkhorn(scores, eps = 5.0, n_iter = 3):
11     def remove_infs(x):
12         mm = x[torch.isfinite(x)].max().item()
13         x[torch.isinf(x)] = mm
14         x[x==0] = 1e-38
15         return x
16     scores = torch.tensor(scores)
17     n, m = scores.shape
18     scores1 = scores.view(n*m)
19     Q = torch.softmax(-scores1/eps, dim=0)
20     Q = remove_infs(Q).view(n,m).T
21     r, c = torch.ones(n), torch.ones(m) * (n / m)
22     for _ in range(n_iter):
23         u = (c/torch.sum(Q, dim=1))
24         Q *= remove_infs(u).unsqueeze(1)
25         v = (r/torch.sum(Q,dim=0))
26         Q *= remove_infs(v).unsqueeze(0)
27     bsum = torch.sum(Q, dim=0, keepdim=True)
28     Q = Q / remove_infs(bsum)
29     P = Q.T
30     assert torch.isnan(P.sum())==False
31     return P
32 
33 n = 128
34 m = 64
35 loss_1 = []
36 loss_2 = []
37 for _ in range(20):
38     a = torch.rand(n, m)
39     a = a ** 4
40     a = a / a.sum(dim = -1, keepdim = True)
41     P = torch.softmax(a, dim=1)
42     b = np.random.rand(n, m)
43     b = b ** 1.5
44     Q = sinkhorn(b, 0.5, 10)
45     # 方法1：
46     loss_11 = -(Q * torch.log(P)).sum(-1).mean()
47     loss_1.append(loss_11)
48     # 方法2：
49     loss_22 = ((P - Q) * a).sum(-1).mean()
50     loss_2.append(loss_22)
51 
52 loss_1, index = torch.sort(torch.tensor(loss_1), 0)
53 loss_2 = np.array(loss_2)[index]
54 print('方法1--损失：\n', np.array(loss_1))
55 print('方法2--损失：\n', loss_2)
56 grad_1 = np.gradient(np.array(loss_1))
57 grad_2 = np.gradient(np.array(loss_2))
58 print('方法1--梯度：\n', np.array(grad_1))
59 print('方法2--梯度：\n', np.array(grad_2))
60 plt.subplots(1, 2, figsize=(16, 7))
61 plt.subplot(1, 2, 1)
62 plt.plot(loss_1, loss_2, color = 'red', ls = '-')
63 plt.xlabel('Loss 1')
64 plt.ylabel('Loss 2')
65 plt.subplot(1, 2, 2)
66 plt.scatter(grad_1*1E6, grad_2*1E6, color = 'blue')
67 plt.xlabel('Gradient 1')
68 plt.ylabel('Gradient 2')
69 plt.savefig('softmax cross-entropy loss.png', bbox_inches='tight', dpi=500)
70 plt.show()

结果：

D:\ProgramData\Anaconda3\python.exe "D:/Python code/2023.3 exercise/向量间的距离度量/softmax_cross_entropy_loss_test.py"
方法1--损失：
 [4.15883989 4.15890663 4.15894403 4.15897117 4.15902341 4.15904347
 4.1590823  4.1590913  4.15910622 4.15913114 4.15913474 4.1591434
 4.15914856 4.15916808 4.15916826 4.15917904 4.15918445 4.15918608
 4.15925961 4.15926385]
方法2--损失：
 [0.00017298 0.00024753 0.00028277 0.00030974 0.00036783 0.0003804
 0.00041808 0.00043415 0.00044729 0.00047444 0.00047943 0.00048301
 0.00049451 0.00050864 0.00051069 0.0005161  0.00053111 0.00052533
 0.00060765 0.0006024 ]
方法1--梯度：
 [6.67441917e-05 5.20709542e-05 3.22701985e-05 3.96937794e-05
 3.61504422e-05 2.94408723e-05 2.39134622e-05 1.19608872e-05
 1.99222036e-05 1.42617498e-05 6.12662792e-06 6.90728703e-06
 1.23429982e-05 9.85375545e-06 5.47883732e-06 8.09470613e-06
 3.52000565e-06 3.75770611e-05 3.88866185e-05 4.24487449e-06]
方法2--梯度：
 [ 7.45563606e-05  5.48997609e-05  3.11016626e-05  4.25261140e-05
  3.53301332e-05  2.51239797e-05  2.68772106e-05  1.46074152e-05
  2.01447210e-05  1.60698704e-05  4.28303591e-06  7.54056029e-06
  1.28157065e-05  8.08914041e-06  3.73246714e-06  1.02123578e-05
  4.61507539e-06  3.82697805e-05  3.85335993e-05 -5.25437451e-06]

Process finished with exit code 0

可以看到，虽然理论上可以证明$L_1$与$L_2$的梯度相等，但实际应用时还是有点偏颇，当数据量与数据维度足够大时，$L_1$与$L_2$的数值呈现线性相关。

6. 参考文献

[1] Softmax classification with cross-entropy (2/2)

[2] Liu Q, Zhou Q, Yang R, et al. Robust Representation Learning by Clustering with Bisimulation Metrics for Visual Reinforcement Learning with Distractions[C]. AAAI, 2023.

[3] Python小练习：Sinkhorn-Knopp算法