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:
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
代码语言：javascript复制# 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
