Norm-Preservation: Why Residual Networks Can Become Extremely Deep?

TL;DR

This paper analyzes why residual networks can be extremely deep by focusing on norm preservation through skip connections, providing theoretical insights and proposing a regularization method to enhance training stability and performance.

Contribution

It offers a theoretical explanation for the benefits of skip connections in residual networks and introduces Procrustes ResNets, a regularization technique to improve norm preservation and training of deep networks.

Findings

Skip connections facilitate gradient norm preservation and stable back-propagation.

Stacking more residual blocks enhances norm-preservation.

Regularizing singular values improves learning dynamics and classification accuracy.

Abstract

Augmenting neural networks with skip connections, as introduced in the so-called ResNet architecture, surprised the community by enabling the training of networks of more than 1,000 layers with significant performance gains. This paper deciphers ResNet by analyzing the effect of skip connections, and puts forward new theoretical results on the advantages of identity skip connections in neural networks. We prove that the skip connections in the residual blocks facilitate preserving the norm of the gradient, and lead to stable back-propagation, which is desirable from optimization perspective. We also show that, perhaps surprisingly, as more residual blocks are stacked, the norm-preservation of the network is enhanced. Our theoretical arguments are supported by extensive empirical evidence. Can we push for extra norm-preservation? We answer this question by proposing an efficient method to regularize the singular values of the convolution operator and making the ResNet's transition layers extra norm-preserving. Our numerical investigations demonstrate that the learning dynamics and the classification performance of ResNet can be improved by making it even more norm preserving. Our results and the introduced modification for ResNet, referred to as Procrustes ResNets, can be used as a guide for training deeper networks and can also inspire new deeper architectures.