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# Maximal Initial Learning Rates in Deep ReLU Networks

*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.