On exponential convergence of SGD in non-convex over-parametrized learning

TL;DR

This paper extends theoretical understanding of stochastic gradient descent (SGD) convergence, showing exponential rates in over-parametrized, non-convex settings satisfying the Polyak-Lojasiewicz condition, aligning theory with empirical observations.

Contribution

It generalizes previous convex results to a broader class of non-convex functions satisfying the PL condition, relevant for modern over-parametrized models.

Findings

SGD exhibits exponential convergence under the PL condition in non-convex over-parametrized models.

The PL condition is applicable to many neural network classes.

Theoretical results align with practical efficiency of SGD in deep learning.

Abstract

Large over-parametrized models learned via stochastic gradient descent (SGD) methods have become a key element in modern machine learning. Although SGD methods are very effective in practice, most theoretical analyses of SGD suggest slower convergence than what is empirically observed. In our recent work [8] we analyzed how interpolation, common in modern over-parametrized learning, results in exponential convergence of SGD with constant step size for convex loss functions. In this note, we extend those results to a much broader non-convex function class satisfying the Polyak-Lojasiewicz (PL) condition. A number of important non-convex problems in machine learning, including some classes of neural networks, have been recently shown to satisfy the PL condition. We argue that the PL condition provides a relevant and attractive setting for many machine learning problems, particularly in the over-parametrized regime.