Maximal Initial Learning Rates in Deep ReLU Networks

12/14/2022
by   Gaurav Iyer, et al.
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Training a neural network requires choosing a suitable learning rate, involving 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 η^∗ - 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 η^∗, we observe that in constant-width fully-connected ReLU networks, η^∗ demonstrates different behavior to the maximum learning rate later in training. Specifically, we find that η^∗ is well predicted as a power of (depth×width), provided that (i) the width of the network is sufficiently large compared to the depth, and (ii) the input layer of the network is trained at a relatively small learning rate. We further analyze the relationship between η^∗ and the sharpness λ_1 of the network at initialization, indicating that they are closely though not inversely related. We formally prove bounds for λ_1 in terms of (depth×width) that align with our empirical results.

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