softmax 反向传播 代码 python 实现

概念

反向传播求导

可以看到，softmax 计算了多个神经元的输入，在反向传播求导时，需要考虑对不同神经元的参数求导。
分两种情况考虑：

• 当求导的参数位于分子时
• 当求导的参数位于分母时
  $$p_i=softmax(z_i)=\frac{e^{z_i}}{{\sum }_{j=1}^N{e}^{z_j}}=\frac{e^{z_1}}{e^{z_1}+e^{z_2}+e^{z_3}}$$
  当求导的参数位于分子时：

$$\begin{array}{c}\frac{\partial (\frac{e^{z_i}}{{\sum }_{j=1}^N{e}^{z_j}})}{\partial (e^{z_i})}=\frac{\partial (\frac{e^{z_1}}{e^{z_1}+e^{z_2}+e^{z_3}})}{\partial (e^{z_i})}\\ =\frac{e^{z_1}(e^{z_1}+e^{z_2}+e^{z_3})-e^{z_1}\cdot e^{z_1}}{(e^{z_1}+e^{z_2}+e^{z_3}{)}^2}\\ =p_i-p_i^2\\ =p_i(1-p_i)\end{array}$$
当求导的参数位于分母时($e^{z_2}$ or $e^{z_3}$ 这两个是对称的，求导结果是一样的)：
$$\begin{array}{c}\frac{\partial (\frac{e^{z_i}}{{\sum }_{j=1}^N{e}^{z_j}})}{\partial (e^{z_j})}=\frac{\partial (\frac{e^{z_1}}{e^{z_1}+e^{z_2}+e^{z_3}})}{\partial (e^{z_i})}\\ =\frac{0-e^{z_1}\cdot e^{z_2}}{(e^{z_1}+e^{z_2}+e^{z_3}{)}^2}\\ =-p_j\cdot p_i\end{array}$$

代码

import torch
import math

def my_softmax(features):
    _sum = 0
    for i in features:
        _sum += math.e ** i
    return torch.Tensor([ math.e ** i / _sum for i in features ])

def my_softmax_grad(outputs):    
    n = len(outputs)
    grad = []
    for i in range(n):
        temp = []
        for j in range(n):
            if i == j:
                temp.append(outputs[i] * (1- outputs[i]))
            else:
                temp.append(-outputs[j] * outputs[i])
        grad.append(torch.Tensor(temp))
    return grad

if __name__ == '__main__':

    features = torch.randn(10)
    features.requires_grad_()

    torch_softmax = torch.nn.functional.softmax
    p1 = torch_softmax(features,dim=0)
    p2 = my_softmax(features)
    print(torch.allclose(p1,p2))
    
    n = len(p1)
    p2_grad = my_softmax_grad(p2)
    for i in range(n):
        p1_grad = torch.autograd.grad(p1[i],features, retain_graph=True)
        print(torch.allclose(p1_grad[0], p2_grad[i]))
        

相关资料

https://zhuanlan.zhihu.com/p/105722023

https://www.cnblogs.com/gzyatcnblogs/articles/15937870.html