# Layer Normalization

This is a PyTorch implementation of Layer Normalization.

### Limitations of Batch Normalization

• You need to maintain running means.
• Tricky for RNNs. Do you need different normalizations for each step?
• Doesn’t work with small batch sizes; large NLP models are usually trained with small batch sizes.
• Need to compute means and variances across devices in distributed training.

## Layer Normalization

Layer normalization is a simpler normalization method that works on a wider range of settings. Layer normalization transforms the inputs to have zero mean and unit variance across the features. Note that batch normalization fixes the zero mean and unit variance for each element. Layer normalization does it for each batch across all elements.

Layer normalization is generally used for NLP tasks.

We have used layer normalization in most of the transformer implementations.

35from typing import Union, List
36
37import torch
38from torch import nn, Size
39
40from labml_helpers.module import Module

## Layer Normalization

Layer normalization $\text{LN}$ normalizes the input $X$ as follows:

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 $X \in \mathbb{R}^{L \times B \times C}$ is a batch of a sequence of embeddings, where $B$ is the batch size, $C$ is the number of channels, $L$ is the length of the sequence. $\gamma \in \mathbb{R}^{C}$ and $\beta \in \mathbb{R}^{C}$.

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. This is not a widely used scenario. $\gamma \in \mathbb{R}^{C \times H \times W}$ and $\beta \in \mathbb{R}^{C \times H \times W}$.

43class LayerNorm(Module):
• normalized_shape $S$ is the shape of the elements (except the batch). The input should then be $X \in \mathbb{R}^{* \times S[0] \times S[1] \times … \times S[n]}$
• eps is $\epsilon$, used in $\sqrt{Var[X] + \epsilon}$ for numerical stability
• elementwise_affine is whether to scale and shift the normalized value

We’ve tried to use the same names for arguments as PyTorch LayerNorm implementation.

72    def __init__(self, normalized_shape: Union[int, List[int], Size], *,
73                 eps: float = 1e-5,
74                 elementwise_affine: bool = True):
84        super().__init__()
85
86        self.normalized_shape = normalized_shape
87        self.eps = eps
88        self.elementwise_affine = elementwise_affine

Create parameters for $\gamma$ and $\beta$ for gain and bias

90        if self.elementwise_affine:
91            self.gain = nn.Parameter(torch.ones(normalized_shape))
92            self.bias = nn.Parameter(torch.zeros(normalized_shape))

x is a tensor of shape [*, S[0], S[1], ..., S[n]]. * could be any number of dimensions. For example, in an NLP task this will be [seq_len, batch_size, features]

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

Sanity check to make sure the shapes match

102        assert self.normalized_shape == x.shape[-len(self.normalized_shape):]

The dimensions to calculate the mean and variance on

105        dims = [-(i + 1) for i in range(len(self.normalized_shape))]

Calculate the mean of all elements; i.e. the means for each element $\mathbb{E}[X]$

109        mean = x.mean(dim=dims, keepdims=True)

Calculate the squared mean of all elements; i.e. the means for each element $\mathbb{E}[X^2]$

112        mean_x2 = (x ** 2).mean(dim=dims, keepdims=True)

Variance of all element $Var[X] = \mathbb{E}[X^2] - \mathbb{E}[X]^2$

114        var = mean_x2 - mean ** 2

Normalize

117        x_norm = (x - mean) / torch.sqrt(var + self.eps)

Scale and shift

119        if self.elementwise_affine:
120            x_norm = self.gain * x_norm + self.bias
123        return x_norm

Simple test

126def _test():
130    from labml.logger import inspect
131
132    x = torch.zeros([2, 3, 2, 4])
133    inspect(x.shape)
134    ln = LayerNorm(x.shape[2:])
135
136    x = ln(x)
137    inspect(x.shape)
138    inspect(ln.gain.shape)
142if __name__ == '__main__':
143    _test()