[edit]

# Width and Depth Limits Commute in Residual Networks

*Proceedings of the 40th International Conference on Machine Learning*, PMLR 202:12700-12723, 2023.

#### Abstract

We show that taking the width and depth to infinity in a deep neural network with skip connections, when branches are scaled by $1/\sqrt{depth}$, 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.