Equilibrium Invariance, Proximality, and Surrogation: Moreau-Smoothed Best-Response Pathways in Stochastic Nonsmooth Games

TL;DR

This paper introduces Moreau-smoothed best-response schemes for nonsmooth, nonconvex stochastic games, providing convergence guarantees and complexity bounds for equilibrium computation in challenging settings.

Contribution

It develops novel Moreau-smoothed BR algorithms for nonsmooth, nonconvex games, extending equilibrium analysis and convergence guarantees beyond smooth cases.

Findings

Linear and sublinear convergence rates established.

Equilibrium invariance under smoothing demonstrated.

Numerical experiments show promising results.

Abstract

Best-response (BR) schemes represent an important avenue for learning equilibria in noncooperative games. However, extant rate guarantees for BR schemes generally necessitate stringent smoothness requirements on player objectives and the availability of suitably defined eigenvalue bounds, significantly limiting the reach of such schemes, and few schemes if any exist for the efficient resolution of a broad class of nonsmooth and nonconvex games with expectation-valued objectives. This motivates our study of Moreau-smoothed BR schemes that allow for nonsmooth objectives. First, we consider a class of nonsmooth and strongly convex games (but potentially non-monotone) under uncertainty. By presenting an equilibrium invariance claim, we present synchronous and asynchronous schemes, equipped with linear and sublinear rate guarantees and associated complexity statements. Second, faced by weakly convex player objectives, we incorporate surrogation into the Moreau-smoothed best-response and show that the resulting smoothed quasi-Nash equilibrium (QNE) constitutes an $\mathcal{O}(\eta)$-QNE of the original weakly convex game, where $\eta > 0$ denotes the Moreau smoothing parameter. In this setting, we again present synchronous and asynchronous BR schemes, equipped with linear and sublinear rates and analogous complexity bounds. Preliminary numerics on a range of such games appear promising.