Stochastic Gradient Descent on Nonconvex Functions with General Noise Models

TL;DR

This paper proves that stochastic gradient descent (SGD) either diverges or converges to a stationary point for a broad class of nonconvex functions and noise models, extending theoretical guarantees of SGD's behavior.

Contribution

It establishes convergence properties of SGD under very general nonconvex functions and noise models, relaxing many previous restrictive assumptions.

Findings

SGD iterates diverge or converge to stationary points with probability one.

Gradient norms at SGD iterates tend to zero in probability and expectation under certain conditions.

Broader applicability of SGD with rigorous convergence guarantees for nonconvex problems.

Abstract

Stochastic Gradient Descent (SGD) is a widely deployed optimization procedure throughout data-driven and simulation-driven disciplines, which has drawn a substantial interest in understanding its global behavior across a broad class of nonconvex problems and noise models. Recent analyses of SGD have made noteworthy progress in this direction, and these analyses have innovated important and insightful new strategies for understanding SGD. However, these analyses often have imposed certain restrictions (e.g., convexity, global Lipschitz continuity, uniform Holder continuity, expected smoothness, etc.) that leave room for innovation. In this work, we address this gap by proving that, for a rather general class of nonconvex functions and noise models, SGD's iterates either diverge to infinity or converge to a stationary point with probability one. By further restricting to globally Holder continuous functions and the expected smoothness noise model, we prove that -- regardless of whether the iterates diverge or remain finite -- the norm of the gradient function evaluated at SGD's iterates converges to zero with probability one and in expectation. As a result of our work, we broaden the scope of nonconvex problems and noise models to which SGD can be applied with rigorous guarantees of its global behavior.