1. 简介

Batch Norm（Batch Normalization）是以进行时学习的 mini-batch 为单位，按 mini-batch 进行正规化（即就是进行使数据分布的均值为 0、方差为 1）。通过将这个处理插入到激活函数的前面（或者后面），可以减小数据分布的偏向。

2. 实现

考虑 mini-batch 的 $m$ 个输入样本数据 $\{x_1,x_2,\cdots,x_m\}$：

Input: Values of $x$ over a mini-batch: $\mathcal{B} = \{x_1,\cdots,x_m\}$
    Parameters to be learned: $\gamma,\beta$

Output: $\{y_i = \mathrm{BN}_{\gamma,\beta}(x_i)\}$

$$\begin{array}{c} \mu_B \leftarrow \frac{1}{m} \sum_{i=1}^{m} x_i \\ \sigma_B^2 \leftarrow \frac{1}{m} \sum_{i=1}^m (x_i - \mu_B)^2 \\ \hat{x_i} \leftarrow \frac{x_i - \mu_B}{\sqrt{\sigma_B^2+\varepsilon}} \\ y_i \leftarrow \gamma \hat{x_i} + \beta \equiv \mathrm{BN}_{\gamma,\beta}(x_i) \end{array}$$

其中，$\varepsilon$ 是一个微小值（比如 $10^{-7}$），是为了防止出现 $\sigma_B = 0$ 导致除零的情况；最后一项是对 $\hat{x_i}$ 进行缩放和平移。

• 使用 Batch Norm 后，神经网络的学习速度更快，并且，对权重初始值变得健壮（「对初始值健壮」表示不那么依赖初始值）。

3. Python 代码

# Batch-Norm 层
class BatchNorm:
    def __init__(self, gamma, beta, momentum=0.9, running_mean=None, running_var=None):
        self.gamma = gamma
        self.beta = beta 
        self.momentum = momentum 
        self.input_shape = None 
        
        # 测试时使用的平均值和方差
        self.running_mean = running_mean 
        self.running_var = running_var 

        # backward 时使用的中间数据
        self.batch_size = None 
        self.xc = None 
        self.xn = None 
        self.std = None 
        self.dgamma = None 
        self.dbeta = None 

    def forward(self, x, train_flg=True):
        self.input_shape = x.shape 
        if x.ndim != 2:
            N, C, H, W = x.shape
            x = x.reshape(N, -1)
        out = self.__forward(x, train_flg)
        return out.reshape(*self.input_shape)
    
    def __forward(self, x, train_flg):
        if self.running_mean is None:
            N, D = x.shape 
            self.running_mean = np.zeros(D)
            self.running_var = np.zeros(D)
        
        if train_flg:
            mu = x.mean(axis = 0)
            xc = x - mu 
            var = np.mean(xc**2, axis = 0)
            std = np.sqrt(var + 10e-7)
            xn = xc / std 

            self.batch_size = x.shape[0]
            self.xc = xc 
            self.xn = xn 
            self.std = std 
            self.running_mean = self.momentum * self.running_mean + (1 - self.momentum) * mu 
            self.running_var = self.momentum * self.running_var + (1 - self.momentum) * var 
        else:
            xc = x - self.running_mean
            xn = xc / np.sqrt(self.running_var + 10e-7)
        
        out = self.gamma * xn + self.beta 
        return out 

    def backward(self, dout):
        if dout.ndim != 2:
            N, C, H, W = dout.shape
            dout = dout.reshape(N, -1)
        dx = self.__backward(dout)
        dx = dx.reshape(*self.input_shape)
        return dx 
    
    def __backward(self, dout):
        dbeta = dout.sum(axis = 0)
        dgamma = np.sum(self.xn * dout, axis = 0)
        dxn = self.gamma * dout 
        dxc = dxn / self.std 
        dstd = -np.sum((dxn * self.xc) / (self.std * self.std), axis = 0)
        dvar = 0.5 * dstd / self.std 
        dxc += (2.0 / self.batch_size) * self.xc * dvar 
        dmu = np.sum(dxc, axis = 0)
        dx = dxc + dmu / self.batch_size

        self.dgamma = dgamma
        self.dbeta = dbeta
        
        return dx