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# Normalization in Deep Learning

### Layer Normalization

Ba, Kiro and Hinton 2016 introduced Layer Normalization. Let $$\mathbf{a} \in \mathbb{R}^D$$ be a $$D$$-dimensional vector of activations. Layer normalization first computes the mean and standard deviation across the $$D$$ dimensions:

$\mathbf{\mu} = \frac{1}{D} \sum_{d=1}^D \mathbf{a}_d$ $\sigma = \sqrt{\frac{1}{D} \sum_{d=1}^D (\mathbf{a}_d - \mu)^2 + \epsilon}$

where $$\epsilon > 0$$ is a small constant to avoid division by zero. Then, Layer normalization normalizes the activations:

$\frac{ \mathbf{a} - \mathbf{\mu}}{\sigma}$

One can optionally introduce learnable parameters $$\mathbf{\gamma}, \mathbf{\beta} \in \mathbb{R}^D$$ to scale and shift the normalized activations:

$\mathbf{\gamma} \odot \frac{ \mathbf{a} - \mathbf{\mu}}{\sigma} + \mathbf{\beta}$

where $$\odot$$ denotes elementwise multiplication.

### Root Mean Square Layer Normalization

Zhang and Sennrich NeurIPS 2019 introduced Root Mean Square (RMS) layer normalization. Rather than centering, RMS Layer Norm normalizes the activations by their root mean square. Let $$\mathbf{a} \in \mathbb{R}^D$$ be a vector of activations. RMS Layer Norm first calculates the root mean square:

$RMS(\mathbf{a}) = \sqrt{\frac{1}{D} \sum_d \mathbf{a}_d^2}$

RMS Layer Norm then normalizes the activations:

$\frac{\mathbf{a}}{RMS(\mathbf{a})}$

One can optionally introduce learnable parameters $$\mathbf{\gamma} \in \mathbb{R}^D$$ to scale the normalized activations:

$\mathbf{\gamma} \odot \frac{\mathbf{a}}{RMS(\mathbf{a})}$

This forces the vectors to lie on a $$\sqrt{D}$$-scaled hypersphere.