On the convergence of gradient-like flows with noisy gradient input

TL;DR

This paper investigates the long-term behavior of gradient-like flows with noisy inputs in convex optimization, revealing conditions for convergence, concentration, and robustness of modified methods.

Contribution

It provides a comprehensive analysis of convergence and concentration properties of mirror descent schemes under stochastic disturbances, including a noise-robust variant.

Findings

Dynamics converge to the solution set with vanishing noise.

Persistent noise causes concentration around interior solutions.

A rectified variant converges regardless of noise magnitude.

Abstract

In view of solving convex optimization problems with noisy gradient input, we analyze the asymptotic behavior of gradient-like flows under stochastic disturbances. Specifically, we focus on the widely studied class of mirror descent schemes for convex programs with compact feasible regions, and we examine the dynamics' convergence and concentration properties in the presence of noise. In the vanishing noise limit, we show that the dynamics converge to the solution set of the underlying problem (a.s.). Otherwise, when the noise is persistent, we show that the dynamics are concentrated around interior solutions in the long run, and they converge to boundary solutions that are sufficiently "sharp". Finally, we show that a suitably rectified variant of the method converges irrespective of the magnitude of the noise (or the structure of the underlying convex program), and we derive an explicit estimate for its rate of convergence.