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 $\mathcal{N}(0.5, 1)$. Then, after some training steps, it could move to $\mathcal{N}(0.6, 1.5)$. 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.


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 where $x = u + b$ and $b$ is a trained bias and $\mathbb{E}[x]$ is an outside gradient computation (pre-computed constant).

Note that $\hat{x}$ has no effect on $b$. Therefore, $b$ will increase or decrease based $\frac{\partial{\mathcal{L}}}{\partial x}$, 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: where $x = (x^{(1)} … x^{(d)})$ is the $d$-dimensional input.

The second simplification is to use estimates of mean $\mathbb{E}[x^{(k)}]$ and variance $Var[x^{(k)}]$ 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 $\gamma^{(k)}$ and $\beta^{(k)}$. where $y^{(k)}$ is the output of the batch normalization layer.

Note that when applying batch normalization after a linear transform like $Wu + b$ 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.


We need to know $\mathbb{E}[x^{(k)}]$ and $Var[x^{(k)}]$ 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.

Open In Colab View Run

98import torch
99from torch import nn
101from labml_helpers.module import Module

Batch Normalization Layer

Batch normalization layer $\text{BN}$ normalizes the input $X$ as follows:

When input $X \in \mathbb{R}^{B \times C \times H \times W}$ 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. $\gamma \in \mathbb{R}^{C}$ and $\beta \in \mathbb{R}^{C}$.

When input $X \in \mathbb{R}^{B \times C}$ is a batch of embeddings, where $B$ is the batch size and $C$ is the number of features. $\gamma \in \mathbb{R}^{C}$ and $\beta \in \mathbb{R}^{C}$.

When input $X \in \mathbb{R}^{B \times C \times L}$ 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. $\gamma \in \mathbb{R}^{C}$ and $\beta \in \mathbb{R}^{C}$.

104class BatchNorm(Module):
  • channels is the number of features in the input
  • eps is $\epsilon$, used in $\sqrt{Var[x^{(k)}] + \epsilon}$ 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.

132    def __init__(self, channels: int, *,
133                 eps: float = 1e-5, momentum: float = 0.1,
134                 affine: bool = True, track_running_stats: bool = True):
144        super().__init__()
146        self.channels = channels
148        self.eps = eps
149        self.momentum = momentum
150        self.affine = affine
151        self.track_running_stats = track_running_stats

Create parameters for $\gamma$ and $\beta$ for scale and shift

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

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

158        if self.track_running_stats:
159            self.register_buffer('exp_mean', torch.zeros(channels))
160            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]

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

Keep the original shape

170        x_shape = x.shape

Get the batch size

172        batch_size = x_shape[0]

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

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

Reshape into [batch_size, channels, n]

177        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

181        if or not self.track_running_stats:

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

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

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

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

Variance for each feature $Var[x^{(k)}] = \mathbb{E}[(x^{(k)})^2] - \mathbb{E}[x^{(k)}]^2$

189            var = mean_x2 - mean ** 2

Update exponential moving averages

192            if and self.track_running_stats:
193                self.exp_mean = (1 - self.momentum) * self.exp_mean + self.momentum * mean
194                self.exp_var = (1 - self.momentum) * self.exp_var + self.momentum * var

Use exponential moving averages as estimates

196        else:
197            mean = self.exp_mean
198            var = self.exp_var


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

Scale and shift

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

Reshape to original and return

207        return x_norm.view(x_shape)

Simple test

210def _test():
214    from labml.logger import inspect
216    x = torch.zeros([2, 3, 2, 4])
217    inspect(x.shape)
218    bn = BatchNorm(3)
220    x = bn(x)
221    inspect(x.shape)
222    inspect(bn.exp_var.shape)
226if __name__ == '__main__':
227    _test()