Maximal Initial Learning Rates in Deep ReLU Networks

Gaurav Iyer, Boris Hanin, David Rolnick
Proceedings of the 40th International Conference on Machine Learning, PMLR 202:14500-14530, 2023.

Abstract

Training a neural network requires choosing a suitable learning rate, which involves a trade-off between speed and effectiveness of convergence. While there has been considerable theoretical and empirical analysis of how large the learning rate can be, most prior work focuses only on late-stage training. In this work, we introduce the maximal initial learning rate $\eta^{\ast}$ - the largest learning rate at which a randomly initialized neural network can successfully begin training and achieve (at least) a given threshold accuracy. Using a simple approach to estimate $\eta^{\ast}$, we observe that in constant-width fully-connected ReLU networks, $\eta^{\ast}$ behaves differently from the maximum learning rate later in training. Specifically, we find that $\eta^{\ast}$ is well predicted as a power of depth $\times$ width, provided that (i) the width of the network is sufficiently large compared to the depth, and (ii) the input layer is trained at a relatively small learning rate. We further analyze the relationship between $\eta^{\ast}$ and the sharpness $\lambda_{1}$ of the network at initialization, indicating they are closely though not inversely related. We formally prove bounds for $\lambda_{1}$ in terms of depth $\times$ width that align with our empirical results.

Cite this Paper

BibTeX
@InProceedings{pmlr-v202-iyer23a, title = {Maximal Initial Learning Rates in Deep {R}e{LU} Networks}, author = {Iyer, Gaurav and Hanin, Boris and Rolnick, David}, booktitle = {Proceedings of the 40th International Conference on Machine Learning}, pages = {14500--14530}, 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/iyer23a/iyer23a.pdf}, url = {https://proceedings.mlr.press/v202/iyer23a.html}, abstract = {Training a neural network requires choosing a suitable learning rate, which involves a trade-off between speed and effectiveness of convergence. While there has been considerable theoretical and empirical analysis of how large the learning rate can be, most prior work focuses only on late-stage training. In this work, we introduce the maximal initial learning rate $\eta^{\ast}$ - the largest learning rate at which a randomly initialized neural network can successfully begin training and achieve (at least) a given threshold accuracy. Using a simple approach to estimate $\eta^{\ast}$, we observe that in constant-width fully-connected ReLU networks, $\eta^{\ast}$ behaves differently from the maximum learning rate later in training. Specifically, we find that $\eta^{\ast}$ is well predicted as a power of depth $\times$ width, provided that (i) the width of the network is sufficiently large compared to the depth, and (ii) the input layer is trained at a relatively small learning rate. We further analyze the relationship between $\eta^{\ast}$ and the sharpness $\lambda_{1}$ of the network at initialization, indicating they are closely though not inversely related. We formally prove bounds for $\lambda_{1}$ in terms of depth $\times$ width that align with our empirical results.} }
Endnote
%0 Conference Paper %T Maximal Initial Learning Rates in Deep ReLU Networks %A Gaurav Iyer %A Boris Hanin %A David Rolnick %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-iyer23a %I PMLR %P 14500--14530 %U https://proceedings.mlr.press/v202/iyer23a.html %V 202 %X Training a neural network requires choosing a suitable learning rate, which involves a trade-off between speed and effectiveness of convergence. While there has been considerable theoretical and empirical analysis of how large the learning rate can be, most prior work focuses only on late-stage training. In this work, we introduce the maximal initial learning rate $\eta^{\ast}$ - the largest learning rate at which a randomly initialized neural network can successfully begin training and achieve (at least) a given threshold accuracy. Using a simple approach to estimate $\eta^{\ast}$, we observe that in constant-width fully-connected ReLU networks, $\eta^{\ast}$ behaves differently from the maximum learning rate later in training. Specifically, we find that $\eta^{\ast}$ is well predicted as a power of depth $\times$ width, provided that (i) the width of the network is sufficiently large compared to the depth, and (ii) the input layer is trained at a relatively small learning rate. We further analyze the relationship between $\eta^{\ast}$ and the sharpness $\lambda_{1}$ of the network at initialization, indicating they are closely though not inversely related. We formally prove bounds for $\lambda_{1}$ in terms of depth $\times$ width that align with our empirical results.
APA
Iyer, G., Hanin, B. & Rolnick, D.. (2023). Maximal Initial Learning Rates in Deep ReLU Networks. Proceedings of the 40th International Conference on Machine Learning, in Proceedings of Machine Learning Research 202:14500-14530 Available from https://proceedings.mlr.press/v202/iyer23a.html.

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