神经网络的训练就是梯度反向传播的过程，也是面试的时候手撕的重要考点之一！
下面我搭建了两层神经网络，使用sigmoid激活函数，具体的公式推导就忽略了，但是要注意的是，记住公式是最为关键的。

import numpy as np
np.random.seed(17)
def sigmoid(x):
    return 1 / (1 + np.exp(-x))

def sigmoid_grad(x):
    return (1.0 - sigmoid(x)) * sigmoid(x)

def softmax(x):
    if x.ndim == 2:
        x = x.T
        x = x - np.max(x, axis=0)
        y = np.exp(x) / np.sum(np.exp(x), axis=0)
        return y.T
    x = x - np.max(x)  # 溢出对策
    return np.exp(x) / np.sum(np.exp(x))


# 搭建两层全连接神经网络，使用sigmoid激活函数，完成10分类任务
class TwoLayerNet:
    def __init__(self, input_size, hidden_size, output_size, weight_init_std=0.01):
        # 初始化权重
        self.params = {}
        self.params['W1'] = weight_init_std * np.random.randn(input_size, hidden_size)
        self.params['b1'] = np.zeros(hidden_size)
        self.params['W2'] = weight_init_std * np.random.randn(hidden_size, output_size)
        self.params['b2'] = np.zeros(output_size)

    def predict(self, x):
        W1, W2 = self.params['W1'], self.params['W2']
        b1, b2 = self.params['b1'], self.params['b2']

        a1 = np.dot(x, W1) + b1
        z1 = sigmoid(a1)
        a2 = np.dot(z1, W2) + b2
        y = softmax(a2)

        return y

    # x:输入数据, t:监督数据
    def loss(self, x, t):
        y = self.predict(x)
        LOSS = (1/y.shape[0])*np.sum(-t * np.log(y) - (1-t) * np.log(1-y))
        return LOSS 

    def accuracy(self, x, t):
        y = self.predict(x)
        y = np.argmax(y, axis=1)
        t = np.argmax(t, axis=1)

        accuracy = np.sum(y == t) / float(x.shape[0])
        return accuracy

    # x:输入数据, t:监督数据
    def gradient(self, x, t):
        W1, W2 = self.params['W1'], self.params['W2']
        b1, b2 = self.params['b1'], self.params['b2']
        grads = {}

        batch_num = x.shape[0]

        # forward
        a1 = np.dot(x, W1) + b1
        z1 = sigmoid(a1)
        a2 = np.dot(z1, W2) + b2
        y = softmax(a2)

        # backward
        dy = (y - t) / batch_num   # 输出层反向误差
        grads['W2'] = np.dot(z1.T, dy)
        grads['b2'] = np.sum(dy, axis=0)

        da1 = np.dot(dy, W2.T)   # 隐层反向传播误差
        dz1 = sigmoid_grad(a1) * da1
        grads['W1'] = np.dot(x.T, dz1)
        grads['b1'] = np.sum(dz1, axis=0)

        return grads


# 数据集：随机的初始的数据集，输入为10分类，数据集大小是3万
x_train = np.random.randn(30000, 784)
t_train = np.random.randint(0, 2, size=(x_train.shape[0], 10))
train_size = x_train.shape[0]
batch_size = 512

# 任务1：梯度检查
 # 梯度检查
net = TwoLayerNet(input_size=784, hidden_size=80, output_size=10)
grad = net.gradient(x_train, t_train)
print("-------梯度检查---------")
print(grad["W1"].shape)
print(grad["b1"].shape)
print(grad["W2"].shape)
print(grad["b2"].shape)


# 任务2：模型训练
if __name__ == "__main__":
    net = TwoLayerNet(input_size=784, hidden_size=80, output_size=10)
    batch_size = 512
    learning_rate = 0.01
    iters = 1000  # 适当设定循环的次数
    loss_history = []
    iter_per_epoch = max(train_size / batch_size, 1)

    for i in range(iters):
        batch_mask = np.random.choice(train_size, batch_size)
        x_batch = x_train[batch_mask]
        t_batch = t_train[batch_mask]
        grad = net.gradient(x_batch, t_batch)
        # 更新参数
        for key in ('W1', 'b1', 'W2', 'b2'):
            net.params[key] -= learning_rate * grad[key]
        loss = net.loss(x_batch, t_batch)
        if (i+1) % 100 == 0:
            loss_history.append(round(loss, 4))
            print("iterrs:%d, loss:%.4f" % (i+1, loss))
print(loss_history)

参考文献：

斋藤康毅：深度学习入门_ 基于Python的理论与实现