Width and Depth Limits Commute in Residual Networks

Soufiane Hayou, Greg Yang
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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v202-hayou23a, title = {Width and Depth Limits Commute in Residual Networks}, author = {Hayou, Soufiane and Yang, Greg}, booktitle = {Proceedings of the 40th International Conference on Machine Learning}, pages = {12700--12723}, year = {2023}, editor = {Krause, Andreas and Brunskill, Emma and Cho, Kyunghyun and Engelhardt, Barbara and Sabato, Sivan and Scarlett, Jonathan}, volume = {202}, series = {Proceedings of Machine Learning Research}, month = {23--29 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v202/hayou23a/hayou23a.pdf}, url = {https://proceedings.mlr.press/v202/hayou23a.html}, 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.} }
Endnote
%0 Conference Paper %T Width and Depth Limits Commute in Residual Networks %A Soufiane Hayou %A Greg Yang %B Proceedings of the 40th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2023 %E Andreas Krause %E Emma Brunskill %E Kyunghyun Cho %E Barbara Engelhardt %E Sivan Sabato %E Jonathan Scarlett %F pmlr-v202-hayou23a %I PMLR %P 12700--12723 %U https://proceedings.mlr.press/v202/hayou23a.html %V 202 %X 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.
APA
Hayou, S. & Yang, G.. (2023). Width and Depth Limits Commute in Residual Networks. Proceedings of the 40th International Conference on Machine Learning, in Proceedings of Machine Learning Research 202:12700-12723 Available from https://proceedings.mlr.press/v202/hayou23a.html.

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