Multiplicative Weights Update as a Distributed Constrained Optimization Algorithm: Convergence to Second-order Stationary Points Almost Always

TL;DR

This paper demonstrates that the Multiplicative Weights Update algorithm converges to second-order stationary points in constrained non-concave maximization problems, extending understanding beyond unconstrained cases.

Contribution

It provides the first convergence analysis of MWU for constrained non-concave maximization, showing almost sure convergence to second-order critical points.

Findings

MWU converges to second-order stationary points with small stepsizes.

Convergence holds almost always under certain conditions.

The analysis combines dynamical systems techniques and Baum Eagon inequality.

Abstract

Non-concave maximization has been the subject of much recent study in the optimization and machine learning communities, specifically in deep learning. Recent papers Ge et al, Lee et al (and references therein) indicate that first order methods work well and avoid saddle points. Results as in Lee et al, however, are limited to the \textit{unconstrained} case or for cases where the critical points are in the interior of the feasibility set, which fail to capture some of the most interesting applications. In this paper we focus on \textit{constrained} non-concave maximization. We analyze a variant of a well-established algorithm in machine learning called Multiplicative Weights Update (MWU) for the maximization problem $\max_{\mathbf{x} \in D} P(\mathbf{x})$, where $P$ is non-concave, twice continuously differentiable and $D$ is a product of simplices. We show that MWU converges almost always for small enough stepsizes to critical points that satisfy the second order KKT conditions. We combine techniques from dynamical systems as well as taking advantage of a recent connection between Baum Eagon inequality and MWU (Palaiopanos et al).