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<math>\gamma</math> and <math>\beta</math> allow the network to learn to undo the normalization, if this is beneficial.<ref name=":1">{{Cite book |last1=Goodfellow |first1=Ian |title=Deep learning |last2=Bengio |first2=Yoshua |last3=Courville |first3=Aaron |date=2016 |publisher=The MIT Press |isbn=978-0-262-03561-3 |series=Adaptive computation and machine learning |___location=Cambridge, Massachusetts |chapter=8.7.1. Batch Normalization}}</ref> BatchNorm can be interpreted as removing the purely linear transformations, so that its layers focus solely on modelling the nonlinear aspects of data, which may be beneficial, as a neural network can always be augmented with a linear transformation layer on top.<ref>{{Cite journal |last1=Desjardins |first1=Guillaume |last2=Simonyan |first2=Karen |last3=Pascanu |first3=Razvan |last4=kavukcuoglu |first4=koray |date=2015 |title=Natural Neural Networks |url=https://proceedings.neurips.cc/paper_files/paper/2015/hash/2de5d16682c3c35007e4e92982f1a2ba-Abstract.html |journal=Advances in Neural Information Processing Systems |publisher=Curran Associates, Inc. |volume=28}}</ref><ref name=":1" />
It is claimed in the original publication that BatchNorm works by reducing internal covariance shift, though the claim has both supporters<ref>{{Cite journal |last1=Xu |first1=Jingjing |last2=Sun |first2=Xu |last3=Zhang |first3=Zhiyuan |last4=Zhao |first4=Guangxiang |last5=Lin |first5=Junyang |date=2019 |title=Understanding and Improving Layer Normalization |url=https://proceedings.neurips.cc/paper/2019/hash/2f4fe03d77724a7217006e5d16728874-Abstract.html |journal=Advances in Neural Information Processing Systems |publisher=Curran Associates, Inc. |volume=32 |arxiv=1911.07013}}</ref><ref>{{Cite journal |last1=Awais |first1=Muhammad |last2=Bin Iqbal |first2=Md. Tauhid |last3=Bae |first3=Sung-Ho |date=November 2021 |title=Revisiting Internal Covariate Shift for Batch Normalization
=== Special cases ===
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* <math>b^{(l)}_c</math> is the bias term for the <math>c</math>-th channel of the <math>l</math>-th layer.
In order to preserve the translational invariance, BatchNorm treats all outputs from the same kernel in the same batch as more data in a batch. That is, it is applied once per ''kernel'' <math>c</math> (equivalently, once per channel <math>c</math>), not per ''activation'' <math>x^{(l+1)}_{h, w, c}</math>:
<math display="block">
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return y
</syntaxhighlight>For multilayered [[Recurrent neural network|recurrent neural networks]] (RNN), BatchNorm is usually applied only for the ''input-to-hidden'' part, not the ''hidden-to-hidden'' part.<ref name=":4">{{Cite book |last1=Laurent |first1=Cesar |last2=Pereyra |first2=Gabriel |last3=Brakel |first3=Philemon |last4=Zhang |first4=Ying |last5=Bengio |first5=Yoshua |chapter=Batch normalized recurrent neural networks |date=March 2016 |title=2016 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP) |publisher=IEEE |pages=2657–2661 |doi=10.1109/ICASSP.2016.7472159 |arxiv=1510.01378 |isbn=978-1-4799-9988-0}}</ref> Let the hidden state of the <math>l</math>-th layer at time <math>t</math> be <math>h_t^{(l)}</math>. The standard RNN, without normalization, satisfies<math display="block">h^{(l)}_t = \phi(W^{(l)} h_t^{l-1} + U^{(l)} h_{t-1}^{l} + b^{(l)}) </math>where <math>W^{(l)}, U^{(l)}, b^{(l)}</math> are weights and biases, and <math>\phi</math> is the activation function. Applying BatchNorm, this becomes<math display="block">h^{(l)}_t = \phi(\mathrm{BN}(W^{(l)} h_t^{l-1}) + U^{(l)} h_{t-1}^{l}) </math>There are two possible ways to define what a "batch" is in BatchNorm for RNNs: ''frame-wise'' and ''sequence-wise''. Concretely, consider applying an RNN to process a batch of sentences. Let <math>h_{b, t}^{(l)}</math> be the hidden state of the <math>l</math>-th layer for the <math>t</math>-th token of the <math>b</math>-th input sentence. Then frame-wise BatchNorm means normalizing over <math>b</math>:<math display="block">
\begin{aligned}
\mu_t^{(l)} &= \frac{1}{B} \sum_{b=1}^B h_{i,t}^{(l)} \\
(\sigma_t^{(l)})^2 &= \frac{1}{B} \sum_{b=1}^B (h_t^{(l)} - \mu_t^{(l)})^2
\end{aligned}
</math>and sequence-wise means normalizing over <math>(b, t)</math>:<math display="block">
\begin{aligned}
\mu^{(l)} &= \frac{1}{BT} \sum_{b=1}^B\sum_{t=1}^T h_{i,t}^{(l)} \\
(\sigma^{(l)})^2 &= \frac{1}{BT} \sum_{b=1}^B\sum_{t=1}^T (h_t^{(l)} - \mu^{(l)})^2
\end{aligned}
</math>Frame-wise BatchNorm is suited for causal tasks such as next-character prediction, where future frames are unavailable, forcing normalization per frame. Sequence-wise BatchNorm is suited for tasks such as speech recognition, where the entire sequences are available, but with variable lengths. In a batch, the smaller sequences are padded with zeroes to match the size of the longest sequence of the batch. In such setups, frame-wise is not recommended, because the number of unpadded frames decreases along the time axis, leading to increasingly poorer statistics estimates.<ref name=":4" />
It is also possible to apply BatchNorm to [[Long short-term memory|LSTMs]].<ref>{{cite arXiv | eprint=1603.09025 | last1=Cooijmans | first1=Tim | last2=Ballas | first2=Nicolas | last3=Laurent | first3=César | last4=Gülçehre | first4=Çağlar | last5=Courville | first5=Aaron | title=Recurrent Batch Normalization | date=2016 | class=cs.LG }}</ref>
=== Improvements ===
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</math>
In [[recurrent neural network]]s<ref name=":2" /> and [[Transformer (deep learning architecture)|transformers]],<ref>{{cite arXiv |last1=Phuong |first1=Mary |title=Formal Algorithms for Transformers |date=2022-07-19 |eprint=2207.09238 |last2=Hutter |first2=Marcus|class=cs.LG }}</ref> LayerNorm is applied individually to each timestep. For example, if the hidden vector in an RNN at timestep <math>t</math> is <math>x^{(t)} \in \mathbb{R}^{D}
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=== Root mean square layer normalization ===
'''Root mean square layer normalization''' ('''RMSNorm'''):<ref>{{cite arXiv |last1=Zhang |first1=Biao |title=Root Mean Square Layer Normalization |date=2019-10-16 |eprint=1910.07467 |last2=Sennrich |first2=Rico|class=cs.LG }}</ref>
<math display="block">
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</math>
Essentially, it is LayerNorm where we enforce <math>\mu, \epsilon = 0</math>. It is also called '''L2 normalization'''. It is a special case of '''Lp normalization''', or '''power normalization''':<math display="block">
\hat{x_i} = \frac{x_i}{\left(\frac 1D \sum_{i=1}^D |x_i|^p \right)^{1/p}}, \quad y_i = \gamma \hat{x_i} + \beta
</math>where <math>p > 0</math> is a constant.
=== Adaptive ===
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By reassigning <math>W_i \leftarrow \frac{W_i}{\|W_i\|_s}</math> after each update of the discriminator, we can upper-bound <math>\|W_i\|_s \leq 1</math>, and thus upper-bound <math>\|D \|_L</math>.
The algorithm can be further accelerated by [[memoization]]: at step <math>t</math>, store <math>x^*_i(t)</math>. Then, at step <math>t+1</math>, use <math>x^*_i(t)</math> as the initial guess for the algorithm. Since <math>W_i(t+1)</math> is very close to <math>W_i(t)</math>, so is <math>x^*_i(t)</math> to <math>x^*_i(t+1)</math>, thus allowing rapid convergence.
== CNN-specific normalization ==
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Some normalization methods were designed for use in [[Transformer (deep learning architecture)|transformers]].
The original 2017 transformer used the "post-LN" configuration for its LayerNorms. It was difficult to train, and required careful [[Hyperparameter optimization|hyperparameter tuning]] and a "warm-up" in [[learning rate]], where it starts small and gradually increases. The pre-LN convention, proposed several times in 2018,<ref>{{
'''FixNorm'''<ref>{{
In '''nGPT''', many vectors are normalized to have unit L2 norm:<ref>{{
== Miscellaneous ==
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== Further reading ==
* {{Cite web |title=Normalization Layers |url=https://nn.labml.ai/normalization/index.html |access-date=2024-08-07 |website=labml.ai Deep Learning Paper Implementations |language=en}}
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