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

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.

```
98import torch
99from torch import nn
100
101from labml_helpers.module import Module
```

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__()
145
146 self.channels = channels
147
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 self.training 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 self.training 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
```

Normalize

`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
215
216 x = torch.zeros([2, 3, 2, 4])
217 inspect(x.shape)
218 bn = BatchNorm(3)
219
220 x = bn(x)
221 inspect(x.shape)
222 inspect(bn.exp_var.shape)
```

```
226if __name__ == '__main__':
227 _test()
```