On the Convergence of Adaptive Gradient Methods for Nonconvex Optimization

TL;DR

This paper provides a detailed convergence analysis of adaptive gradient methods like AMSGrad, RMSProp, and AdaGrad for nonconvex optimization, establishing new theoretical guarantees and convergence rates.

Contribution

It offers the first high-probability convergence bounds for these methods and improves understanding of their behavior in nonconvex settings.

Findings

Adaptive gradient methods converge to stationary points in expectation.

New dimension-dependent convergence rates are established.

High-probability bounds for AMSGrad, RMSProp, and AdaGrad are proven.

Abstract

Adaptive gradient methods are workhorses in deep learning. However, the convergence guarantees of adaptive gradient methods for nonconvex optimization have not been thoroughly studied. In this paper, we provide a fine-grained convergence analysis for a general class of adaptive gradient methods including AMSGrad, RMSProp and AdaGrad. For smooth nonconvex functions, we prove that adaptive gradient methods in expectation converge to a first-order stationary point. Our convergence rate is better than existing results for adaptive gradient methods in terms of dimension. In addition, we also prove high probability bounds on the convergence rates of AMSGrad, RMSProp as well as AdaGrad, which have not been established before. Our analyses shed light on better understanding the mechanism behind adaptive gradient methods in optimizing nonconvex objectives.