Width and Depth Limits Commute in Residual Networks
This provides theoretical justification for using infinite-width approximations in residual networks, which is incremental but useful for researchers in deep learning theory and Bayesian methods.
The paper demonstrates that in deep residual networks with specific scaling, the covariance structure remains consistent regardless of the order in which width and depth limits are taken, explaining the practical relevance of infinite-width-then-depth approximations for networks with comparable depth and width, and shows that pre-activations become Gaussian, aiding Bayesian deep learning applications.
We show that taking the width and depth to infinity in a deep neural network with skip connections, when branches are scaled by $1/\sqrt{depth}$ (the only nontrivial scaling), result in the same covariance structure no matter how that limit is taken. This explains why the standard infinite-width-then-depth approach provides practical insights even for networks with depth of the same order as width. We also demonstrate that the pre-activations, in this case, have Gaussian distributions which has direct applications in Bayesian deep learning. We conduct extensive simulations that show an excellent match with our theoretical findings.