Batch Normalization

This is a PyTorch implementation of Batch Normalization from paper Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift.

Internal Covariate Shift

The paper defines Internal Covariate Shift as the change in the distribution of network activations due to the change in network parameters during training. For example, let's say there are two layers $l_{1}$ and $l_{2}$ . During the beginning of the training $l_{1}$ outputs (inputs to $l_{2}$ ) could be in distribution $N (0.5, 1)$ . Then, after some training steps, it could move to $N (0.6, 1.5)$ . This is internal covariate shift.

Internal covariate shift will adversely affect training speed because the later layers ( $l_{2}$ in the above example) have to adapt to this shifted distribution.

By stabilizing the distribution, batch normalization minimizes the internal covariate shift.

Normalization

It is known that whitening improves training speed and convergence. Whitening is linearly transforming inputs to have zero mean, unit variance, and be uncorrelated.

Normalizing outside gradient computation doesn't work

Normalizing outside the gradient computation using pre-computed (detached) means and variances doesn't work. For instance. (ignoring variance), let $\overset{x}{^} = x - E [x]$ where $x = u + b$ and $b$ is a trained bias and $E [x]$ is an outside gradient computation (pre-computed constant).

Note that $\overset{x}{^}$ has no effect on $b$ . Therefore, $b$ will increase or decrease based $\frac{\partial L}{\partial x}$ , and keep on growing indefinitely in each training update. The paper notes that similar explosions happen with variances.

Batch Normalization

Whitening is computationally expensive because you need to de-correlate and the gradients must flow through the full whitening calculation.

The paper introduces a simplified version which they call Batch Normalization. First simplification is that it normalizes each feature independently to have zero mean and unit variance: $\overset{x}{^}^{(k)} = \frac{x ^{(k)} - E [ x ^{(k)} ]}{Va r [ x ^{(k)} ]}$ where $x = (x^{(1)} ... x^{(d)})$ is the $d$ -dimensional input.

The second simplification is to use estimates of mean $E [x^{(k)}]$ and variance $Va r [x^{(k)}]$ from the mini-batch for normalization; instead of calculating the mean and variance across the whole dataset.

Normalizing each feature to zero mean and unit variance could affect what the layer can represent. As an example paper illustrates that, if the inputs to a sigmoid are normalized most of it will be within $[- 1, 1]$ range where the sigmoid is linear. To overcome this each feature is scaled and shifted by two trained parameters $γ^{(k)}$ and $β^{(k)}$ . $y^{(k)} = γ^{(k)} \overset{x}{^}^{(k)} + β^{(k)}$ where $y^{(k)}$ is the output of the batch normalization layer.

Note that when applying batch normalization after a linear transform like $W u + b$ the bias parameter $b$ gets cancelled due to normalization. So you can and should omit bias parameter in linear transforms right before the batch normalization.

Batch normalization also makes the back propagation invariant to the scale of the weights and empirically it improves generalization, so it has regularization effects too.

Inference

We need to know $E [x^{(k)}]$ and $Va r [x^{(k)}]$ in order to perform the normalization. So during inference, you either need to go through the whole (or part of) dataset and find the mean and variance, or you can use an estimate calculated during training. The usual practice is to calculate an exponential moving average of mean and variance during the training phase and use that for inference.

Here's the training code and a notebook for training a CNN classifier that uses batch normalization for MNIST dataset.

97import torch 98from torch import nn

Batch Normalization Layer

Batch normalization layer $B N$ normalizes the input $X$ as follows:

When input $X \in R^{B \times C \times H \times W}$ is a batch of image representations, where $B$ is the batch size, $C$ is the number of channels, $H$ is the height and $W$ is the width. $γ \in R^{C}$ and $β \in R^{C}$ . $B N (X) = γ \frac{X - B , H , W E [ X ]}{B , H , W Va r [ X ] + ϵ} + β$

When input $X \in R^{B \times C}$ is a batch of embeddings, where $B$ is the batch size and $C$ is the number of features. $γ \in R^{C}$ and $β \in R^{C}$ . $B N (X) = γ \frac{X - B E [ X ]}{B Va r [ X ] + ϵ} + β$

When input $X \in R^{B \times C \times L}$ is a batch of a sequence embeddings, where $B$ is the batch size, $C$ is the number of features, and $L$ is the length of the sequence. $γ \in R^{C}$ and $β \in R^{C}$ . $B N (X) = γ \frac{X - B , L E [ X ]}{B , L Va r [ X ] + ϵ} + β$

102class BatchNorm(nn.Module):

channels is the number of features in the input
eps is $ϵ$ , used in $Va r [x^{(k)}] + ϵ$ for numerical stability
momentum is the momentum in taking the exponential moving average
affine is whether to scale and shift the normalized value
track_running_stats is whether to calculate the moving averages or mean and variance

We've tried to use the same names for arguments as PyTorch BatchNorm implementation.

130 def __init__(self, channels: int, *, 131 eps: float = 1e-5, momentum: float = 0.1, 132 affine: bool = True, track_running_stats: bool = True):

142 super().__init__() 143 144 self.channels = channels 145 146 self.eps = eps 147 self.momentum = momentum 148 self.affine = affine 149 self.track_running_stats = track_running_stats

Create parameters for $γ$ and $β$ for scale and shift

151 if self.affine: 152 self.scale = nn.Parameter(torch.ones(channels)) 153 self.shift = nn.Parameter(torch.zeros(channels))

Create buffers to store exponential moving averages of mean $E [x^{(k)}]$ and variance $Va r [x^{(k)}]$

156 if self.track_running_stats: 157 self.register_buffer('exp_mean', torch.zeros(channels)) 158 self.register_buffer('exp_var', torch.ones(channels))

x is a tensor of shape [batch_size, channels, *] . * denotes any number of (possibly 0) dimensions. For example, in an image (2D) convolution this will be [batch_size, channels, height, width]

160 def forward(self, x: torch.Tensor):

Keep the original shape

168 x_shape = x.shape

Get the batch size

170 batch_size = x_shape[0]

Sanity check to make sure the number of features is the same

172 assert self.channels == x.shape[1]

Reshape into [batch_size, channels, n]

175 x = x.view(batch_size, self.channels, -1)

We will calculate the mini-batch mean and variance if we are in training mode or if we have not tracked exponential moving averages

179 if self.training or not self.track_running_stats:

Calculate the mean across first and last dimension; i.e. the means for each feature $E [x^{(k)}]$

182 mean = x.mean(dim=[0, 2])

Calculate the squared mean across first and last dimension; i.e. the means for each feature $E [(x^{(k)})^{2}]$

185 mean_x2 = (x ** 2).mean(dim=[0, 2])

Variance for each feature $Va r [x^{(k)}] = E [(x^{(k)})^{2}] - E [x^{(k)}]^{2}$

187 var = mean_x2 - mean ** 2

Update exponential moving averages

190 if self.training and self.track_running_stats: 191 self.exp_mean = (1 - self.momentum) * self.exp_mean + self.momentum * mean 192 self.exp_var = (1 - self.momentum) * self.exp_var + self.momentum * var

Use exponential moving averages as estimates

194 else: 195 mean = self.exp_mean 196 var = self.exp_var

Normalize $\overset{x}{^}^{(k)} = \frac{x ^{(k)} - E [ x ^{(k)} ]}{Va r [ x ^{(k)} ] + ϵ}$

199 x_norm = (x - mean.view(1, -1, 1)) / torch.sqrt(var + self.eps).view(1, -1, 1)

Scale and shift $y^{(k)} = γ^{(k)} \overset{x}{^}^{(k)} + β^{(k)}$

201 if self.affine: 202 x_norm = self.scale.view(1, -1, 1) * x_norm + self.shift.view(1, -1, 1)

Reshape to original and return

205 return x_norm.view(x_shape)

Simple test

208def _test():

212 from labml.logger import inspect 213 214 x = torch.zeros([2, 3, 2, 4]) 215 inspect(x.shape) 216 bn = BatchNorm(3) 217 218 x = bn(x) 219 inspect(x.shape) 220 inspect(bn.exp_var.shape)

224if __name__ == '__main__': 225 _test()