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Residual neural network

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A Residual Block in a deep Residual Network. Here the Residual Connection skips two layers.

A residual neural network (also referred to as a residual network or ResNet)[1] is a deep learning architecture in which the weight layers learn residual functions with reference to the layer inputs. It was developed in 2015 for image recognition and won that year's ImageNet Large Scale Visual Recognition Challenge (ILSVRC).[2][3]

As a point of terminology, "residual connection" refers to the specific architectural motif of , where is an arbitrary neural network module. The motif had been used previously (see §History for details). However, the publication of ResNet made it widely popular for feedforward networks, appearing in neural networks that are otherwise unrelated to ResNet.

The residual connection stabilizes the training and convergence of deep neural networks with hundreds of layers, and is a common motif in deep neural networks, such as Transformer models (e.g., BERT and GPT models such as ChatGPT), the AlphaGo Zero system, the AlphaStar system, and the AlphaFold system.

Mathematics

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Residual connection

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In a multi-layer 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 , where is the input to the subnetwork. Residual learning re-parameterizes this subnetwork and lets the parameter layers represent a "residual function" . The output of this subnetwork is then represented as:

The operation of "" 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 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".[1] A deep residual network is constructed by simply stacking these blocks together.

The LSTM has a memory mechanism that functions as a residual connection.[4] In the LSTM without forget gate, input is processed by a function and added to a memory cell , resulting in . The LSTM with a forget gate functions as the highway network.

To stabilize the variance of the layers' inputs, (Hanin and Rolnick, 2018)[5] recommends replacing the residual connections by , where is the total number of residual layers.

Projection connection

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If the function is of type where , then is undefined. To handle this special case, a projection connection is used:where is typically a linear projection, defined by where is a matrix. The matrix is trained by backpropagation as any other parameter of the model.

Signal propagation

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The introduction of identity mappings facilitates signal propagation in both forward and backward paths, as described below.[6]

Forward propagation

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If the output of the -th residual block is the input to the -th residual block (assuming no activation function between blocks), then the -th input is:

Applying this formulation recursively, e.g.,

yields the general relationship:

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

Backward propagation

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The residual learning formulation provides the added benefit of addressing 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 layers. To observe the effect of residual blocks on backpropagation, consider the partial derivative of a loss function with respect to some residual block input . Using the equation above from forward propagation for a later residual block :[6]

This formulation suggests that the gradient computation of a shallower layer, , always has a later term that is directly added. Even if the gradients of the terms are small, the total gradient resists vanishing thanks to the added term .

Variants of residual blocks

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Two variants of convolutional Residual Blocks.[1] Left: a Basic Block that has two 3x3 convolutional layers. Right: a Bottleneck Block that has a 1x1 convolutional layer for dimension reduction (e.g., 1/4), a 3x3 convolutional layer, and another 1x1 convolutional layer for dimension restoration.

Basic block

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A Basic Block is the simplest building block studied in the original ResNet.[1] This block consists of two sequential 3x3 convolutional layers and a residual connection. The input and output dimensions of both layers are equal.

Block diagram of ResNet (2015). It shows a ResNet block with and without the 1x1 convolution. The 1x1 convolution (with stride) can be used to change the shape of the array, which is necessary for residual connection through an upsampling/downsampling layer.

Bottleneck block

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A Bottleneck Block[1] 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 1/4 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 in [1] are all based on Bottleneck Blocks.

Pre-activation block

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The Pre-activation Residual Block[6] applies the activation functions (e.g., non-linearity and normalization) before applying the residual function . Formally, the computation of a Pre-activation Residual Block can be written as:

where can be any non-linearity activation (e.g., ReLU) or normalization (e.g., LayerNorm) operation. This design reduces the number of non-identity mappings between Residual Blocks. This design was used to train models with 200 to over 1000 layers.[6]

Since GPT-2, the Transformer blocks have been dominantly implemented as pre-activation blocks. This is often referred to as "pre-normalization" in the literature of Transformer models.[7]

The original Resnet-18 architecture. Up to 152 layers were trained in the original publication (as "ResNet-152").[8]

Applications

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Originally, ResNet was designed for computer vision.[1][8][9]

The Transformer architecture includes residual connections.

All Transformer architectures include residual connections. Indeed, very deep Transformers cannot be trained without them.[10]

The original ResNet paper made no claim on being inspired by biological systems. But later research has related ResNet to biologically-plausible algorithms.[11][12]

A study published in Science in 2023[13] disclosed the complete connectome of an insect brain (of a fruit fly larva). This study discovered "multilayer shortcuts" that resemble the skip connections in artificial neural networks, including ResNets.

History

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Previous work

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Residual connections were noticed in neuroanatomy, such as Lorente de No (1938).[14]: Fig 3  McCulloch and Pitts (1943) proposed artificial neural networks and considered those with residual connections.[15]: Fig 1.h 

In 1961, Frank Rosenblatt described a three-layer multilayer perceptron (MLP) model with skip connections.[16]: 313, Chapter 15  The model was referred to as a "cross-coupled system", and the skip connections were forms of cross-coupled connections.

During late 1980s, "skip-layer" connections were sometimes used in neural networks. Examples include.[17][18] Lang and Witbrock (1988)[19] 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 , where is a randomly initialized projection connection. They called it a "short-cut connection".

The Long Short-Term Memory (LSTM) cell can process data sequentially and keep its hidden state through time. The cell state can function as a generalized residual connection.

Degradation problem

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Sepp Hochreiter discovered the vanishing gradient problem in 1991[20] 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 long short-term memory (LSTM, 1997)[4][21] to solve this problem, which has a "cell state" that can function as a generalized residual connection. The highway network (2015)[22][23] 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.

Standard (left) and unfolded (right) basic recurrent neural network

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.[24] However, stacking too many layers led to a steep reduction in training accuracy,[25] known as the "degradation" problem.[1] In theory, adding additional layers to deepen a network should not result in a higher training loss, but this is what happened with VGGNet.[1] If the extra layers can be set as identity mappings, though, 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.[26]

In 2014, the state of the art was training “very deep neural network” with 20 to 30 layers.[27] The research team for ResNet attempted to train deeper ones by empirically testing various tricks for training deeper networks, until they came upon the ResNet architecture.[28]

Subsequent work

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DenseNet (2016)[29] connects the output of each layer to the input to each subsequent layer:Stochastic Depth[30] is a regularization method. It randomly drops a subset of layers and lets the signal propagate through the identity skip connection. Also known as "DropPath", this regularizes training for large and deep models, such as Vision Transformers.[31]

ResNeXt block diagram.

ResNeXt (2017) combines the Inception module with ResNet.[32][8]

Squeeze-and-Excitation Networks (2018) added squeeze-and-excitation (SE) modules to ResNet.[33] An SE module is applied after a convolution, and takes as input a tensor of shape (height, width, channel). Each channel is averaged, resulting in a vector of shape . This is then passed through a linear-ReLU-linear-sigmoid MLP before it is multiplied to the original tensor.

References

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  1. ^ a b c d e f g h i He, Kaiming; Zhang, Xiangyu; Ren, Shaoqing; Sun, Jian (10 Dec 2015). Deep Residual Learning for Image Recognition. arXiv:1512.03385.
  2. ^ "ILSVRC2015 Results". image-net.org.
  3. ^ Deng, Jia; Dong, Wei; Socher, Richard; Li, Li-Jia; Li, Kai; Fei-Fei, Li (2009). "ImageNet: A large-scale hierarchical image database". CVPR.
  4. ^ a b Sepp Hochreiter; Jürgen Schmidhuber (1997). "Long short-term memory". Neural Computation. 9 (8): 1735–1780. doi:10.1162/neco.1997.9.8.1735. PMID 9377276. S2CID 1915014.
  5. ^ Hanin, Boris; Rolnick, David (2018). "How to Start Training: The Effect of Initialization and Architecture". Advances in Neural Information Processing Systems. 31. Curran Associates, Inc. arXiv:1803.01719.
  6. ^ a b c d He, Kaiming; Zhang, Xiangyu; Ren, Shaoqing; Sun, Jian (2015). "Identity Mappings in Deep Residual Networks". arXiv:1603.05027 [cs.CV].
  7. ^ Radford, Alec; Wu, Jeffrey; Child, Rewon; Luan, David; Amodei, Dario; Sutskever, Ilya (14 February 2019). "Language models are unsupervised multitask learners" (PDF). Archived (PDF) from the original on 6 February 2021. Retrieved 19 December 2020.
  8. ^ a b c Zhang, Aston; Lipton, Zachary; Li, Mu; Smola, Alexander J. (2024). "8.6. Residual Networks (ResNet) and ResNeXt". Dive into deep learning. Cambridge New York Port Melbourne New Delhi Singapore: Cambridge University Press. ISBN 978-1-009-38943-3.
  9. ^ Szegedy, Christian; Ioffe, Sergey; Vanhoucke, Vincent; Alemi, Alex (2016). "Inception-v4, Inception-ResNet and the impact of residual connections on learning". arXiv:1602.07261 [cs.CV].
  10. ^ Dong, Yihe; Cordonnier, Jean-Baptiste; Loukas, Andreas (2021). "Attention is not all you need: pure attention loses rank doubly exponentially with depth". arXiv:2103.03404 [cs.LG].
  11. ^ Liao, Qianli; Poggio, Tomaso (2016). Bridging the Gaps Between Residual Learning, Recurrent Neural Networks and Visual Cortex. arXiv:1604.03640.
  12. ^ Xiao, Will; Chen, Honglin; Liao, Qianli; Poggio, Tomaso (2018). Biologically-Plausible Learning Algorithms Can Scale to Large Datasets. arXiv:1811.03567.
  13. ^ Winding, Michael; Pedigo, Benjamin; Barnes, Christopher; Patsolic, Heather; Park, Youngser; Kazimiers, Tom; Fushiki, Akira; Andrade, Ingrid; Khandelwal, Avinash; Valdes-Aleman, Javier; Li, Feng; Randel, Nadine; Barsotti, Elizabeth; Correia, Ana; Fetter, Fetter; Hartenstein, Volker; Priebe, Carey; Vogelstein, Joshua; Cardona, Albert; Zlatic, Marta (10 Mar 2023). "The connectome of an insect brain". Science. 379 (6636): eadd9330. bioRxiv 10.1101/2022.11.28.516756v1. doi:10.1126/science.add9330. PMC 7614541. PMID 36893230. S2CID 254070919.
  14. ^ De N, Rafael Lorente (1938-05-01). "Analysis of the Activity of the Chains of Internuncial Neurons". Journal of Neurophysiology. 1 (3): 207–244. doi:10.1152/jn.1938.1.3.207. ISSN 0022-3077.
  15. ^ McCulloch, Warren S.; Pitts, Walter (1943-12-01). "A logical calculus of the ideas immanent in nervous activity". The Bulletin of Mathematical Biophysics. 5 (4): 115–133. doi:10.1007/BF02478259. ISSN 1522-9602.
  16. ^ Rosenblatt, Frank (1961). Principles of neurodynamics. perceptrons and the theory of brain mechanisms (PDF).
  17. ^ Rumelhart, David E., Geoffrey E. Hinton, and Ronald J. Williams. "Learning internal representations by error propagation", Parallel Distributed Processing. Vol. 1. 1986.
  18. ^ Venables, W. N.; Ripley, Brain D. (1994). Modern Applied Statistics with S-Plus. Springer. pp. 261–262. ISBN 9783540943501.
  19. ^ Lang, Kevin; Witbrock, Michael (1988). "Learning to tell two spirals apart" (PDF). Proceedings of the 1988 Connectionist Models Summer School: 52–59.
  20. ^ Hochreiter, Sepp (1991). Untersuchungen zu dynamischen neuronalen Netzen (PDF) (diploma thesis). Technical University Munich, Institute of Computer Science, advisor: J. Schmidhuber.
  21. ^ Felix A. Gers; Jürgen Schmidhuber; Fred Cummins (2000). "Learning to Forget: Continual Prediction with LSTM". Neural Computation. 12 (10): 2451–2471. CiteSeerX 10.1.1.55.5709. doi:10.1162/089976600300015015. PMID 11032042. S2CID 11598600.
  22. ^ Srivastava, Rupesh Kumar; Greff, Klaus; Schmidhuber, Jürgen (3 May 2015). "Highway Networks". arXiv:1505.00387 [cs.LG].
  23. ^ Srivastava, Rupesh Kumar; Greff, Klaus; Schmidhuber, Jürgen (22 July 2015). "Training Very Deep Networks". arXiv:1507.06228 [cs.LG].
  24. ^ Simonyan, Karen; Zisserman, Andrew (2015-04-10). "Very Deep Convolutional Networks for Large-Scale Image Recognition". arXiv:1409.1556 [cs.CV].
  25. ^ He, Kaiming; Zhang, Xiangyu; Ren, Shaoqing; Sun, Jian (2016). "Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification". arXiv:1502.01852 [cs.CV].
  26. ^ He, Kaiming; Zhang, Xiangyu; Ren, Shaoqing; Sun, Jian (2016). "Identity Mappings in Deep Residual Networks". In Leibe, Bastian; Matas, Jiri; Sebe, Nicu; Welling, Max (eds.). Computer Vision – ECCV 2016. Vol. 9908. Cham: Springer International Publishing. pp. 630–645. doi:10.1007/978-3-319-46493-0_38. ISBN 978-3-319-46492-3. Retrieved 2024-09-19.
  27. ^ Simonyan, Karen; Zisserman, Andrew (2015-04-10). "Very Deep Convolutional Networks for Large-Scale Image Recognition". arXiv:1409.1556 [cs.CV].
  28. ^ Linn, Allison (2015-12-10). "Microsoft researchers win ImageNet computer vision challenge". The AI Blog. Retrieved 2024-06-29.
  29. ^ Huang, Gao; Liu, Zhuang; van der Maaten, Laurens; Weinberger, Kilian (2016). Densely Connected Convolutional Networks. arXiv:1608.06993.
  30. ^ Huang, Gao; Sun, Yu; Liu, Zhuang; Weinberger, Kilian (2016). Deep Networks with Stochastic Depth. arXiv:1603.09382.
  31. ^ Lee, Youngwan; Kim, Jonghee; Willette, Jeffrey; Hwang, Sung Ju (2022). "MPViT: Multi-Path Vision Transformer for Dense Prediction": 7287–7296. arXiv:2112.11010. {{cite journal}}: Cite journal requires |journal= (help)
  32. ^ Xie, Saining; Girshick, Ross; Dollar, Piotr; Tu, Zhuowen; He, Kaiming (2017). "Aggregated Residual Transformations for Deep Neural Networks": 1492–1500. arXiv:1611.05431. {{cite journal}}: Cite journal requires |journal= (help)
  33. ^ Hu, Jie; Shen, Li; Sun, Gang (2018). "Squeeze-and-Excitation Networks": 7132–7141. {{cite journal}}: Cite journal requires |journal= (help)