Learning threshold neurons via the "edge of stability"
This work addresses a gap in theoretical understanding of non-convex training dynamics for neural networks, which is incremental but provides insights into generalization benefits.
The paper tackles the problem of understanding neural network training with large learning rates, known as the 'edge of stability', by analyzing gradient descent in simplified two-layer models, and it proves the existence of this phenomenon and a phase transition where threshold neurons fail to learn below a certain step size.
Existing analyses of neural network training often operate under the unrealistic assumption of an extremely small learning rate. This lies in stark contrast to practical wisdom and empirical studies, such as the work of J. Cohen et al. (ICLR 2021), which exhibit startling new phenomena (the "edge of stability" or "unstable convergence") and potential benefits for generalization in the large learning rate regime. Despite a flurry of recent works on this topic, however, the latter effect is still poorly understood. In this paper, we take a step towards understanding genuinely non-convex training dynamics with large learning rates by performing a detailed analysis of gradient descent for simplified models of two-layer neural networks. For these models, we provably establish the edge of stability phenomenon and discover a sharp phase transition for the step size below which the neural network fails to learn "threshold-like" neurons (i.e., neurons with a non-zero first-layer bias). This elucidates one possible mechanism by which the edge of stability can in fact lead to better generalization, as threshold neurons are basic building blocks with useful inductive bias for many tasks.