BN层在实际中应用广泛。
上一次总结了使得训练变得简单的方法，比如SGD+momentum RMSProp Adam，BN是另外的方法。
cell 1 依旧是初始化设置
cell 2 读取cifar-10数据
cell 3 BN的前传
1 # Check the training-time forward pass by checking means and variances 2 # of features both before and after batch normalization 3 4 # Simulate the forward pass for a two-layer network 5 N, D1, D2, D3 = 200, 50, 60, 3 6 X = np.random.randn(N, D1) 7 W1 = np.random.randn(D1, D2) 8 W2 = np.random.randn(D2, D3) 9 a = np.maximum(0, X.dot(W1)).dot(W2) 10 11 print 'Before batch normalization:' 12 print ' means: ', a.mean(axis=0) 13 print ' stds: ', a.std(axis=0) 14 15 # Means should be close to zero and stds close to one 16 print 'After batch normalization (gamma=1, beta=0)' 17 a_norm, _ = batchnorm_forward(a, np.ones(D3), np.zeros(D3), {'mode': 'train'}) 18 print ' mean: ', a_norm.mean(axis=0) 19 print ' std: ', a_norm.std(axis=0) 20 21 # Now means should be close to beta and stds close to gamma 22 gamma = np.asarray([1.0, 2.0, 3.0]) 23 beta = np.asarray([11.0, 12.0, 13.0]) 24 a_norm, _ = batchnorm_forward(a, gamma, beta, {'mode': 'train'}) 25 print 'After batch normalization (nontrivial gamma, beta)' 26 print ' means: ', a_norm.mean(axis=0) 27 print ' stds: ', a_norm.std(axis=0)
相应的核心代码：
1 buf_mean = np.mean(x, axis=0) 2 buf_var = np.var(x, axis=0) 3 x_hat = x - buf_mean 4 x_hat = x_hat / (np.sqrt(buf_var + eps)) 5 6 out = gamma * x_hat + beta 7 #running_mean = momentum * running_mean + (1 - momentum) * sample_mean 8 #running_var = momentum * running_var + (1 - momentum) * sample_var 9 running_mean = momentum * running_mean + (1- momentum) * buf_mean 10 running_var = momentum * running_var + (1 - momentum) * buf_var
running_mean running_var 是在test时使用的，test时不再另外计算均值和方差。
test 时的前传核心代码：
1 x_hat = x - running_mean 2 x_hat = x_hat / (np.sqrt(running_var + eps)) 3 out = gamma * x_hat + beta
cell 5 BN反向传播
通过反向传播，计算beta gamma等参数。
核心代码：
1 dx_hat = dout * cache['gamma'] 2 dgamma = np.sum(dout * cache['x_hat'], axis=0) 3 dbeta = np.sum(dout, axis=0) 4 #x_hat = x - buf_mean 5 #x_hat = x_hat / (np.sqrt(buf_var + eps)) 6 t1 = cache['x'] - cache['mean'] 7 t2 = (-0.5)*((cache['var'] + cache['eps'])**(-1.5)) 8 t1 = t1 * t2 9 d_var = np.sum(dx_hat * t1, axis=0) 10 11 tmean1 = (-1)*((cache['var'] + cache['eps'])**(-0.5)) 12 d_mean = np.sum(dx_hat * tmean1, axis=0) 13 14 tmean1 = (-1)*tmean1 15 tx1 = dx_hat * tmean1 16 tx2 = d_mean * (1.0 / float(N)) 17 tx3 = d_var * (2 * (cache['x'] - cache['mean']) / N) 18 dx = tx1 + tx2 + tx3
cell 9 BN与其他层结合
形成的结构： {affine - [batch norm] - relu - [dropout]} x (L - 1) - affine - softmax
原理依旧。
之后是对cell 9 的模型，对cifar-10数据训练。
值得注意的是：
使用BN后，正则项与dropout层的需求降低。可以使用较高的学习率加快模型收敛。
附：通关CS231n企鹅群：578975100 validation：DL-CS231n