residual neural network

In a multilayer neural network model, consider a subnetwork with a certain number of stacked layers (e.g., 2 or 3). Denote the underlying function performed by this subnetwork as $H(x)$, where $x$ is the input to the subnetwork. Residual learning re-parameterizes this subnetwork and lets the parameter layers represent a "residual function" $F(x)=H(x)-x$. The output $y$ of this subnetwork is then represented as:

: $y\; =\; F(x)\; +\; x$

The operation of "$+\; \backslash \; x$" is implemented via a "skip connection" that performs an identity mapping to connect the input of the subnetwork with its output. This connection is referred to as a "residual connection" in later work. The function $F(x)$ is often represented by matrix multiplication interlaced with activation functions and normalization operations (e.g., batch normalization or layer normalization). As a whole, one of these subnetworks is referred to as a "residual block". A deep residual network is constructed by simply stacking these blocks.

Long short-term memory (LSTM) has a memory mechanism that serves as a residual connection.{{Cite journal |author=Sepp Hochreiter |author-link=Sepp Hochreiter |author2=Jürgen Schmidhuber |author2-link=Jürgen Schmidhuber |year=1997 |title=Long short-term memory |url=https://www.researchgate.net/publication/13853244 |journal=Neural Computation |volume=9 |issue=8 |pages=1735–1780 |doi=10.1162/neco.1997.9.8.1735 |pmid=9377276 |s2cid=1915014}} In an LSTM without a forget gate, an input $x\_t$ is processed by a function $F$ and added to a memory cell $c\_t$, resulting in $c\_\{t+1\}\; =\; c\_t\; +\; F(x\_t)$. An LSTM with a forget gate essentially functions as a highway network.

To stabilize the variance of the layers' inputs, it is recommended to replace the residual connections $x\; +\; f(x)$ with $x/L\; +\; f(x)$, where $L$ is the total number of residual layers.{{Cite conference |last1=Hanin |first1=Boris |last2=Rolnick |first2=David |date=2018 |title=How to Start Training: The Effect of Initialization and Architecture |url=https://proceedings.neurips.cc/paper/2018/hash/d81f9c1be2e08964bf9f24b15f0e4900-Paper.pdf |conference=Conference on Neural Information Processing Systems |publisher=Curran Associates, Inc. |volume=31 |arxiv=1803.01719}}

If the function $F$ is of type $F:\; \backslash R^n\; \backslash to\; \backslash R^m$ where $n\; \backslash neq\; m$, then $F(x)\; +\; x$ is undefined. To handle this special case, a projection connection is used:

: $y\; =\; F(x)\; +\; P(x)$

where $P$ is typically a linear projection, defined by $P(x)\; =\; Mx$ where $M$ is a $m\; \backslash times\; n$ matrix. The matrix is trained via backpropagation, as is any other parameter of the model.

The introduction of identity mappings facilitates signal propagation in both forward and backward paths.{{cite conference |url=https://link.springer.com/content/pdf/10.1007/978-3-319-46493-0_38.pdf |arxiv=1603.05027 |title=Identity Mappings in Deep Residual Networks |last1=He |first1=Kaiming |author-link1=Kaiming He |last2=Zhang |first2=Xiangyu |last3=Ren |first3=Shaoqing |last4=Sun |first4=Jian | year=2016 |conference=European Conference on Computer Vision |doi=10.1007/978-3-319-46493-0_38 }}

If the output of the $\backslash ell$-th residual block is the input to the $(\backslash ell+1)$-th residual block (assuming no activation function between blocks), then the $(\backslash ell+1)$-th input is:

: $x\_\{\backslash ell+1\}\; =\; F(x\_\{\backslash ell\})\; +\; x\_\{\backslash ell\}$

Applying this formulation recursively, e.g.:

: $$

\begin{align} x_{\ell+2} & = F(x_{\ell+1}) + x_{\ell+1} \\

& = F(x_{\ell+1}) + F(x_{\ell}) + x_{\ell}

\end{align}

yields the general relationship:

: $x\_\{L\}\; =\; x\_\{\backslash ell\}\; +\; \backslash sum\_\{i=\backslash ell\}^\{L-1\}\; F(x\_\{i\})$

where $L$ is the index of a residual block and $\backslash ell$ is the index of some earlier block. This formulation suggests that there is always a signal that is directly sent from a shallower block $\backslash ell$ to a deeper block $L$.

The residual learning formulation provides the added benefit of mitigating the vanishing gradient problem to some extent. However, it is crucial to acknowledge that the vanishing gradient issue is not the root cause of the degradation problem, which is tackled through the use of normalization. To observe the effect of residual blocks on backpropagation, consider the partial derivative of a loss function $\backslash mathcal\{E\}$ with respect to some residual block input $x\_\{\backslash ell\}$. Using the equation above from forward propagation for a later residual block $L>\backslash ell$:

: $$

\begin{align} \frac{\partial \mathcal{E} }{\partial x_{\ell} }

& = \frac{\partial \mathcal{E} }{\partial x_{L} }\frac{\partial x_{L} }{\partial x_{\ell} } \\

& = \frac{\partial \mathcal{E} }{\partial x_{L} } \left( 1 + \frac{\partial }{\partial x_{\ell} } \sum_{i=\ell}^{L-1} F(x_{i}) \right) \\

& = \frac{\partial \mathcal{E} }{\partial x_{L} } + \frac{\partial \mathcal{E} }{\partial x_{L} } \frac{\partial }{\partial x_{\ell} } \sum_{i=\ell}^{L-1} F(x_{i})

\end{align}

This formulation suggests that the gradient computation of a shallower layer, $\backslash frac\{\backslash partial\; \backslash mathcal\{E\}\; \}\{\backslash partial\; x\_\{\backslash ell\}\; \}$, always has a later term $\backslash frac\{\backslash partial\; \backslash mathcal\{E\}\; \}\{\backslash partial\; x\_\{L\}\; \}$ that is directly added. Even if the gradients of the $F(x\_\{i\})$ terms are small, the total gradient $\backslash frac\{\backslash partial\; \backslash mathcal\{E\}\; \}\{\backslash partial\; x\_\{\backslash ell\}\; \}$ resists vanishing due to the added term $\backslash frac\{\backslash partial\; \backslash mathcal\{E\}\; \}\{\backslash partial\; x\_\{L\}\; \}$.

File:ResBlockVariants.png

A basic block is the simplest building block studied in the original ResNet. This block consists of two sequential 3x3 convolutional layers and a residual connection. The input and output dimensions of both layers are equal.

File:ResNet_block.svg

A bottleneck block consists of three sequential convolutional layers and a residual connection. The first layer in this block is a 1x1 convolution for dimension reduction (e.g., to 1/2 of the input dimension); the second layer performs a 3x3 convolution; the last layer is another 1x1 convolution for dimension restoration. The models of ResNet-50, ResNet-101, and ResNet-152 are all based on bottleneck blocks.

The pre-activation residual block applies activation functions before applying the residual function $F$. Formally, the computation of a pre-activation residual block can be written as:

: $x\_\{\backslash ell+1\}\; =\; F(\backslash phi(x\_\{\backslash ell\}))\; +\; x\_\{\backslash ell\}$

where $\backslash phi$ can be any activation (e.g. ReLU) or normalization (e.g. LayerNorm) operation. This design reduces the number of non-identity mappings between residual blocks, and allows an identity mapping directly from the input to the output. This design was used to train models with 200 to over 1000 layers, and was found to consistently outperform variants where the residual path is not an identity function. The pre-activation ResNet with 200 layers took 3 weeks to train for ImageNet on 8 GPUs in 2016.

Since GPT-2, transformer blocks have been mostly implemented as pre-activation blocks. This is often referred to as "pre-normalization" in the literature of transformer models.{{cite web

|url = https://cdn.openai.com/better-language-models/language_models_are_unsupervised_multitask_learners.pdf

|title = Language models are unsupervised multitask learners

|last1 = Radford

|first1 = Alec

|last2 = Wu

|first2 = Jeffrey

|last3 = Child

|first3 = Rewon

|last4 = Luan

|first4 = David

|last5 = Amodei

|first5 = Dario

|last6 = Sutskever

|first6 = Ilya

|date = 14 February 2019

|access-date = 19 December 2020

|archive-date = 6 February 2021

|archive-url = https://web.archive.org/web/20210206183945/https://cdn.openai.com/better-language-models/language_models_are_unsupervised_multitask_learners.pdf

|url-status = live

}}

File:Resnet-18_architecture.svg

Originally, ResNet was designed for computer vision.{{cite conference |arxiv=1602.07261 |doi=10.1609/aaai.v31i1.11231 |conference=AAAI Conference on Artificial Intelligence |first1=Christian |last1=Szegedy |first2=Sergey |last2=Ioffe |url=https://cdn.aaai.org/ojs/11231/11231-13-14759-1-2-20201228.pdf |title=Inception-v4, Inception-ResNet and the impact of residual connections on learning |first3=Vincent |last3=Vanhoucke |first4=Alex |last4=Alemi |year=2017}}File:Transformer, full architecture.png

All transformer architectures include residual connections. Indeed, very deep transformers cannot be trained without them.{{Cite conference |last1=Dong |first1=Yihe |last2=Cordonnier |first2=Jean-Baptiste |last3=Loukas |first3=Andreas |year=2021 |title=Attention is not all you need: pure attention loses rank doubly exponentially with depth |url=http://proceedings.mlr.press/v139/dong21a/dong21a.pdf |conference=International Conference on Machine Learning |publisher=PMLR |pages=2793–2803 |arxiv=2103.03404}}

The original ResNet paper made no claim on being inspired by biological systems. However, later research has related ResNet to biologically-plausible algorithms.

{{cite arXiv |last1=Liao |first1=Qianli |last2=Poggio |first2=Tomaso |date=2016 |title=Bridging the Gaps Between Residual Learning, Recurrent Neural Networks and Visual Cortex |eprint=1604.03640 |class=cs.LG}}

{{cite conference |last1=Xiao |first1=Will |last2=Chen |first2=Honglin |last3=Liao |first3=Qianli |last4=Poggio |first4=Tomaso |date=2019 |title=Biologically-Plausible Learning Algorithms Can Scale to Large Datasets |conference=International Conference on Learning Representations |arxiv=1811.03567}}

A study published in Science in 2023{{Cite journal |last1=Winding |first1=Michael |last2=Pedigo |first2=Benjamin |last3=Barnes |first3=Christopher |last4=Patsolic |first4=Heather |last5=Park |first5=Youngser |last6=Kazimiers |first6=Tom |last7=Fushiki |first7=Akira |last8=Andrade |first8=Ingrid |last9=Khandelwal |first9=Avinash |last10=Valdes-Aleman |first10=Javier |last11=Li |first11=Feng |last12=Randel |first12=Nadine |last13=Barsotti |first13=Elizabeth |last14=Correia |first14=Ana |last15=Fetter |first15=Fetter |date=10 Mar 2023 |title=The connectome of an insect brain |journal=Science |volume=379 |issue=6636 |pages=eadd9330 |biorxiv=10.1101/2022.11.28.516756v1 |doi=10.1126/science.add9330 |pmc=7614541 |pmid=36893230 |s2cid=254070919 |last16=Hartenstein |first16=Volker |last17=Priebe |first17=Carey |last18=Vogelstein |first18=Joshua |last19=Cardona |first19=Albert |last20=Zlatic |first20=Marta}} disclosed the complete connectome of an insect brain (specifically that of a fruit fly larva). This study discovered "multilayer shortcuts" that resemble the skip connections in artificial neural networks, including ResNets.

Residual connections were noticed in neuroanatomy, such as Lorente de No (1938).{{Cite journal |last=De N |first=Rafael Lorente |date=1938-05-01 |title=Analysis of the Activity of the Chains of Internuncial Neurons |url=https://www.physiology.org/doi/10.1152/jn.1938.1.3.207 |journal=Journal of Neurophysiology |language=en |volume=1 |issue=3 |pages=207–244 |doi=10.1152/jn.1938.1.3.207 |issn=0022-3077|url-access=subscription }}{{Pg|location=Fig 3}} McCulloch and Pitts (1943) proposed artificial neural networks and considered those with residual connections.{{Cite journal |last1=McCulloch |first1=Warren S. |last2=Pitts |first2=Walter |date=1943-12-01 |title=A logical calculus of the ideas immanent in nervous activity |url=https://link.springer.com/article/10.1007/BF02478259 |journal=The Bulletin of Mathematical Biophysics |language=en |volume=5 |issue=4 |pages=115–133 |doi=10.1007/BF02478259 |issn=1522-9602|url-access=subscription }}{{Pg|location=Fig 1.h}}

In 1961, Frank Rosenblatt described a three-layer multilayer perceptron (MLP) model with skip connections.

{{cite book

| last = Rosenblatt

| first = Frank

| author-link =

| date = 1961

| title = Principles of neurodynamics. perceptrons and the theory of brain mechanisms

| url = https://safari.ethz.ch/digitaltechnik/spring2018/lib/exe/fetch.php?media=neurodynamics1962rosenblatt.pdf#page=327

| location =

| publisher =

| page =

| isbn =

}}{{Pg|page=313|location=Chapter 15}} The model was referred to as a "cross-coupled system", and the skip connections were forms of cross-coupled connections.

During the late 1980s, "skip-layer" connections were sometimes used in neural networks. Examples include:Rumelhart, David E., Geoffrey E. Hinton, and Ronald J. Williams. "Learning internal representations by error propagation", Parallel Distributed Processing. Vol. 1. 1986.{{cite book |last1=Venables |first1=W. N. |author-link= |url=https://books.google.com/books?id=ayDvAAAAMAAJ |title=Modern Applied Statistics with S-Plus |last2=Ripley |first2=Brain D. |date=1994 |publisher=Springer |isbn=9783540943501 |location= |pages=261–262}} Lang and Witbrock (1988){{Cite journal |last1=Lang |first1=Kevin| last2=Witbrock |first2=Michael|year=1988 |title=Learning to tell two spirals apart |url=https://gwern.net/doc/ai/nn/fully-connected/1988-lang.pdf |journal=Proceedings of the 1988 Connectionist Models Summer School |pages=52–59}} trained a fully connected feedforward network where each layer skip-connects to all subsequent layers, like the later DenseNet (2016). In this work, the residual connection was the form {{awrap|$x\; \backslash mapsto\; F(x)\; +\; P(x)$,}} where $P$ is a randomly-initialized projection connection. They termed it a "short-cut connection". An early neural language model used residual connections and named them "direct connections".{{Cite journal |last=Bengio |first=Yoshua |author-link=Yoshua Bengio |last2=Ducharme |first2=Réjean |last3=Vincent |first3=Pascal |last4=Jauvin |first4=Christian |date=2003 |title=A Neural Probabilistic Language Model |url=https://www.jmlr.org/papers/v3/bengio03a.html |journal=Journal of Machine Learning Research |volume=3 |issue=Feb |pages=1137–1155 |issn=1533-7928}}File:LSTM_3.svg

Sepp Hochreiter discovered the vanishing gradient problem in 1991{{cite thesis |url=http://www.bioinf.jku.at/publications/older/3804.pdf |degree=diploma |first=Sepp |last=Hochreiter |title=Untersuchungen zu dynamischen neuronalen Netzen |publisher=Technical University Munich, Institute of Computer Science, advisor: J. Schmidhuber |year=1991}} and argued that it explained why the then-prevalent forms of recurrent neural networks did not work for long sequences. He and Schmidhuber later designed the LSTM architecture to solve this problem,{{Cite journal |author=Felix A. Gers |author2=Jürgen Schmidhuber |author3=Fred Cummins |year=2000 |title=Learning to Forget: Continual Prediction with LSTM |journal=Neural Computation |volume=12 |issue=10 |pages=2451–2471 |citeseerx=10.1.1.55.5709 |doi=10.1162/089976600300015015 |pmid=11032042 |s2cid=11598600}} which has a "cell state" $c\_t$ that can function as a generalized residual connection. The highway network (2015){{cite arXiv |eprint=1505.00387 |class=cs.LG |first1=Rupesh Kumar |last1=Srivastava |first2=Klaus |last2=Greff |title=Highway Networks |date=3 May 2015 |last3=Schmidhuber |first3=Jürgen}}{{cite conference |conference=Conference on Neural Information Processing Systems |arxiv=1507.06228 |first1=Rupesh Kumar |last1=Srivastava |first2=Klaus |last2=Greff |url=https://proceedings.neurips.cc/paper/2015/file/215a71a12769b056c3c32e7299f1c5ed-Paper.pdf |title=Training Very Deep Networks |year=2015 |last3=Schmidhuber |first3=Jürgen}} applied the idea of an LSTM unfolded in time to feedforward neural networks, resulting in the highway network. ResNet is equivalent to an open-gated highway network.

File:Recurrent_neural_network_unfold.svg

During the early days of deep learning, there were attempts to train increasingly deep models. Notable examples included the AlexNet (2012), which had 8 layers, and the VGG-19 (2014), which had 19 layers.{{cite arXiv |last1=Simonyan |first1=Karen |title=Very Deep Convolutional Networks for Large-Scale Image Recognition |date=2015-04-10 |eprint=1409.1556 |last2=Zisserman |first2=Andrew|class=cs.CV }} However, stacking too many layers led to a steep reduction in training accuracy,{{cite conference |doi=10.1109/ICCV.2015.123 |arxiv=1502.01852 |conference=International Conference on Computer Vision |first1=Kaiming |last1=He |author-link1=Kaiming He |first2=Xiangyu |last2=Zhang |url=https://openaccess.thecvf.com/content_iccv_2015/papers/He_Delving_Deep_into_ICCV_2015_paper.pdf |title=Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification |last3=Ren |first3=Shaoqing |last4=Sun |first4=Jian |year=2015}} known as the "degradation" problem. In theory, adding additional layers to deepen a network should not result in a higher training loss, but this is what happened with VGGNet. If the extra layers can be set as identity mappings, however, then the deeper network would represent the same function as its shallower counterpart. There is some evidence that the optimizer is not able to approach identity mappings for the parameterized layers, and the benefit of residual connections was to allow identity mappings by default.

In 2014, the state of the art was training deep neural networks with 20 to 30 layers. The research team for ResNet attempted to train deeper ones by empirically testing various methods for training deeper networks, until they came upon the ResNet architecture.{{Cite web |last=Linn |first=Allison |date=2015-12-10 |title=Microsoft researchers win ImageNet computer vision challenge |url=https://blogs.microsoft.com/ai/microsoft-researchers-win-imagenet-computer-vision-challenge/ |access-date=2024-06-29 |website=The AI Blog |language=en-US}}

Wide Residual Network (2016) found that using more channels and fewer layers than the original ResNet improves performance and GPU-computational efficiency, and that a block with two 3×3 convolutions is superior to other configurations of convolution blocks.{{cite arXiv |eprint=1605.07146 |class=cs.CV |first1=Sergey |last1=Zagoruyko |first2=Nikos |last2=Komodakis |title=Wide Residual Networks |date=2016-05-23}}

DenseNet (2016){{Cite conference |last1=Huang |first1=Gao |last2=Liu |first2=Zhuang |last3=van der Maaten |first3=Laurens |last4=Weinberger |first4=Kilian |year=2017 |url=https://openaccess.thecvf.com/content_cvpr_2017/papers/Huang_Densely_Connected_Convolutional_CVPR_2017_paper.pdf |title=Densely Connected Convolutional Networks |conference=Conference on Computer Vision and Pattern Recognition |doi=10.1109/CVPR.2017.243 |arxiv=1608.06993}} connects the output of each layer to the input to each subsequent layer:

: $x\_\{\backslash ell+1\}\; =\; F(x\_1,\; x\_2,\; \backslash dots,\; x\_\{\backslash ell-1\},\; x\_\{\backslash ell\})$

Stochastic depth{{Cite conference |conference=European Conference on Computer Vision |last1=Huang |first1=Gao |last2=Sun |first2=Yu |last3=Liu |first3=Zhuang |last4=Weinberger |first4=Kilian |year=2016 |url=https://link.springer.com/content/pdf/10.1007/978-3-319-46493-0_39.pdf |title=Deep Networks with Stochastic Depth |doi=10.1007/978-3-319-46493-0_39 |arxiv=1603.09382}} is a regularization method that randomly drops a subset of layers and lets the signal propagate through the identity skip connections. Also known as DropPath, this regularizes training for deep models, such as vision transformers.{{Cite conference |last1=Lee |first1=Youngwan |last2=Kim |first2=Jonghee |last3=Willette |first3=Jeffrey |last4=Hwang |first4=Sung Ju |year=2022 |title=MPViT: Multi-Path Vision Transformer for Dense Prediction |url=https://openaccess.thecvf.com/content/CVPR2022/papers/Lee_MPViT_Multi-Path_Vision_Transformer_for_Dense_Prediction_CVPR_2022_paper.pdf |conference=Conference on Computer Vision and Pattern Recognition |language=en |pages=7287–7296 |doi=10.1109/CVPR52688.2022.00714 |arxiv=2112.11010 }}File:ResNext_block.svg

ResNeXt (2017) combines the Inception module with ResNet.{{Cite conference |last1=Xie |first1=Saining |last2=Girshick |first2=Ross |last3=Dollar |first3=Piotr |last4=Tu |first4=Zhuowen |last5=He |first5=Kaiming |author-link5=Kaiming He |year=2017 |title=Aggregated Residual Transformations for Deep Neural Networks |url=https://openaccess.thecvf.com/content_cvpr_2017/papers/Xie_Aggregated_Residual_Transformations_CVPR_2017_paper.pdf |conference=Conference on Computer Vision and Pattern Recognition |pages=1492–1500 |arxiv=1611.05431 |doi=10.1109/CVPR.2017.634}}{{Cite book |last1=Zhang |first1=Aston |title=Dive into deep learning |last2=Lipton |first2=Zachary |last3=Li |first3=Mu |last4=Smola |first4=Alexander J. |date=2024 |publisher=Cambridge University Press |isbn=978-1-009-38943-3 |location=Cambridge New York Port Melbourne New Delhi Singapore |chapter=8.6. Residual Networks (ResNet) and ResNeXt |chapter-url=https://d2l.ai/chapter_convolutional-modern/resnet.html}}

Squeeze-and-Excitation Networks (2018) added squeeze-and-excitation (SE) modules to ResNet.{{Cite conference |last1=Hu |first1=Jie |last2=Shen |first2=Li |last3=Sun |first3=Gang |date=2018 |title=Squeeze-and-Excitation Networks |url=https://openaccess.thecvf.com/content_cvpr_2018/papers/Hu_Squeeze-and-Excitation_Networks_CVPR_2018_paper.pdf |conference=Conference on Computer Vision and Pattern Recognition |pages=7132–7141 |doi=10.1109/CVPR.2018.00745 |arxiv=1709.01507}} An SE module is applied after a convolution, and takes a tensor of shape $\backslash R^\{H\; \backslash times\; W\; \backslash times\; C\}$ (height, width, channels) as input. Each channel is averaged, resulting in a vector of shape $\backslash R^C$. This is then passed through a multilayer perceptron (with an architecture such as linear-ReLU-linear-sigmoid) before it is multiplied with the original tensor. It won the ILSVRC in 2017.{{cite conference |last1=Jie |first1=Hu |date=2017 |title=Squeeze-and-Excitation Networks |url=https://image-net.org/static_files/files/SENet.pdf |conference=Beyond ImageNet Large Scale Visual Recognition Challenge, Workshop at CVPR 2017 |type=Presentation |location=}}

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