Numpy搭建两层神经网络

在学习pytorch的时候，看见了标题的东西，就简单记录下自己的理解。
视频地址：here
时间：66:32
假定建立的神经网络当前没有偏置b，那么神经网络的结构构成如下：
$$\begin{array}{rl}& hidden=ReLU(x\ast w_1)\\ & \hat{y}=hidden\ast w_2\\ & loss=(\hat{y}-y{)}^2\end{array}$$
不妨稍加整理，其损失函数可以整理为：
$$\begin{array}{rl}loss& =(ReLU(x\ast w_1)\ast w_2-y{)}^2\\ & =(ReLU(X{W}_1)W_2-Y{)}^T(ReLU(X{W}_1)W_2-Y)\end{array}$$
那么，对其$w_1$求偏导，
$$\begin{array}{rl}\frac{\partial loss}{\partial w_1}& =\frac{\partial Z^{T}Z}{\partial w_1}\\ & =\frac{\partial Z^{T}Z}{\partial Z}\ast \frac{\partial Z}{\partial W_1}\\ & =2Z\ast \frac{\partial (ReLU(X{W}_1)W_2-Y)}{\partial W_1}\\ & =2(\hat{Y}-Y)\ast \frac{\partial (X{W}_1{W}_2-Y)}{\partial W_1}.whenX{W}_1⩾0\\ & =2X^T(\hat{Y}-Y)\ast W_2^T.whenX{W}_1⩾0\end{array}$$
同样的，有：
$$\begin{array}{rl}\frac{\partial loss}{\partial w_2}& =2W_1^T{X}^T(\hat{Y}-Y)\end{array}$$
这里，就简单的将$X{W}_1⩾0$，不考虑。也就是在计算梯度的时候，不考虑$ReLU$这个分段函数，简单的统一处理，也就是在梯度计算的时候，默认没有加入该激活函数，以简化问题。
但是，值得注意的是，在计算$(\hat{Y}-Y)$时候，还是添加上$ReLU$，经实践，可以加快收敛。

import numpy as np

# N is batch size; D_in is input dimension;
# H is hidden dimension; D_out is output dimension.
N, D_in, H, D_out = 64, 1000, 100, 10

# Create random input and output data
x = np.random.randn(N, D_in)
y = np.random.randn(N, D_out)

# Randomly initialize weights
w1 = np.random.randn(D_in, H)
w2 = np.random.randn(H, D_out)

learning_rate = 1e-6
for t in range(500):
    # Forward pass: compute predicted y
    h = x.dot(w1)
    h_relu = np.maximum(h, 0)
    y_pred = h_relu.dot(w2)

    # Compute and print loss
    loss = np.square(y_pred - y).sum()
    print(t, loss)

    # Backprop to compute gradients of w1 and w2 with respect to loss
    
    # loss = (y_pred - y) ** 2
    grad_y_pred = 2.0 * (y_pred - y)
    # 
#     grad_w2 = h_relu.T.dot(grad_y_pred)
#     grad_h_relu = grad_y_pred.dot(w2.T)
#     grad_h = grad_h_relu.copy()
#     grad_h[h < 0] = 0
#     grad_w1 = x.T.dot(grad_h) 
    grad_w1 = 2 * x.T.dot(y_pred - y).dot(w2.T)
    grad_w2 = 2 * w1.T.dot(x.T).dot(y_pred - y)
        
    # Update weights
    w1 -= learning_rate * grad_w1
    w2 -= learning_rate * grad_w2

【注】，改代码来自上面链接中视频中的代码。本文修改的部分仅仅是梯度计算的这里。仅是作为简单的理解和记录。