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

Zhuoyu Xiao, Uday V. Shanbhag

TL;DR

This work presents synchronous and asynchronous BR schemes, equipped with linear and sublinear rates and analogous complexity bounds, and incorporates surrogation into the Moreau-smoothed best-response and shows 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.

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 $\mathcal{O}(η)$-QNE of the original weakly convex game, where $η> 0$ 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.

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

TL;DR

This work presents synchronous and asynchronous BR schemes, equipped with linear and sublinear rates and analogous complexity bounds, and incorporates surrogation into the Moreau-smoothed best-response and shows that the resulting smoothed quasi-Nash equilibrium (QNE) constitutes an -QNE of the original weakly convex game, where denotes the Moreau smoothing parameter.

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 -QNE of the original weakly convex game, where 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.
Paper Structure (27 sections, 27 theorems, 118 equations, 2 figures, 3 tables, 4 algorithms)

This paper contains 27 sections, 27 theorems, 118 equations, 2 figures, 3 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Related work
  3. Outline and contributions
  4. Preliminaries
  5. Surrogation method
  6. Moreau envelopes in convex and weakly convex regimes
  7. Quasi-Nash equilibrium under weak convexity
  8. Inexact BR schemes for stochastic strongly convex non-monotone games
  9. MS-SBR under strong convexity
  10. Convergence analysis
  11. Complexity analysis
  12. MS-ABR under strong convexity
  13. Convergence analysis
  14. Complexity analysis
  15. Inexact BR schemes for stochastic weakly convex games
  16. ...and 12 more sections

Key Result

Proposition 2.1

Consider a closed convex proper function $f: \mathbb{R}^{n}\to \bar{\mathbb{R}}$ and its Moreau envelope $f^{\eta}(x)$ with $\eta > 0$. Then the following hold: (i) $x^{*}$ minimizes $f$ over $\mathbb{R}^{n}$ if and only if $x^{*}$ minimizes $f^{\eta}$ over $\mathbb{R}^{n}$; (ii) $f^{\eta}$ is $\tfr

Figures (2)

  • Figure 1: MS-SBR and MS-ABR for stochastic strongly convex games.
  • Figure 2: MS-SSBR and MS-SABR for stochastic weakly convex games.

Theorems & Definitions (62)

  • Definition 2.1
  • Proposition 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3: renaud-leclaire-papadakis-2025
  • Definition 2.2: QNE under B-stationarity
  • Proposition 2.4: QNE under weak convexity
  • proof
  • Theorem 2.1: Existence of QNE
  • proof
  • ...and 52 more