Demystifying ResNet

TL;DR

This paper provides a theoretical explanation for why residual networks with shortcut of length two are easier to train and perform better, supported by extensive experiments comparing initialization methods and network depths.

Contribution

It offers a theoretical analysis of the role of shortcut depth in ResNet training difficulty and demonstrates the benefits of depth-two shortcuts through experiments.

Findings

Shortcut 2 leads to depth-invariant Hessian condition number.

Shortcut 1 behaves like no shortcut, with exploding condition number.

Small weight initialization with shortcut 2 improves training outcomes.

Abstract

The Residual Network (ResNet), proposed in He et al. (2015), utilized shortcut connections to significantly reduce the difficulty of training, which resulted in great performance boosts in terms of both training and generalization error. It was empirically observed in He et al. (2015) that stacking more layers of residual blocks with shortcut 2 results in smaller training error, while it is not true for shortcut of length 1 or 3. We provide a theoretical explanation for the uniqueness of shortcut 2. We show that with or without nonlinearities, by adding shortcuts that have depth two, the condition number of the Hessian of the loss function at the zero initial point is depth-invariant, which makes training very deep models no more difficult than shallow ones. Shortcuts of higher depth result in an extremely flat (high-order) stationary point initially, from which the optimization algorithm is hard to escape. The shortcut 1, however, is essentially equivalent to no shortcuts, which has a condition number exploding to infinity as the number of layers grows. We further argue that as the number of layers tends to infinity, it suffices to only look at the loss function at the zero initial point. Extensive experiments are provided accompanying our theoretical results. We show that initializing the network to small weights with shortcut 2 achieves significantly better results than random Gaussian (Xavier) initialization, orthogonal initialization, and shortcuts of deeper depth, from various perspectives ranging from final loss, learning dynamics and stability, to the behavior of the Hessian along the learning process.