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

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).

Appeared in ICML 2019

References in corpus (4)

• First-order Methods Almost Always Avoid Saddle Points
• How To Make the Gradients Small Stochastically: Even Faster Convex and Nonconvex SGD
• Katyusha X: Practical Momentum Method for Stochastic Sum-of-Nonconvex Optimization
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Cited by in corpus (1)

• Global Convergence of Multi-Agent Policy Gradient in Markov Potential Games