Exponential Convergence of Gradient Methods in Concave Network Zero-sum Games

TL;DR

This paper demonstrates that gradient-based methods, including optimistic variants, achieve exponential convergence when computing Nash equilibria in concave network zero-sum games, extending classical results to more complex multiplayer settings.

Contribution

It generalizes convergence results from two-player to multiplayer concave network zero-sum games, establishing exponential convergence of gradient methods under various conditions.

Findings

Gradient Ascent converges exponentially in linear payoff settings.

Optimistic Gradient Ascent achieves last iterate convergence in strongly concave, Lipschitz, and smooth cases.

Experimental results support theoretical convergence rates.

Abstract

Motivated by Generative Adversarial Networks, we study the computation of Nash equilibrium in concave network zero-sum games (NZSGs), a multiplayer generalization of two-player zero-sum games first proposed with linear payoffs. Extending previous results, we show that various game theoretic properties of convex-concave two-player zero-sum games are preserved in this generalization. We then generalize last iterate convergence results obtained previously in two-player zero-sum games. We analyze convergence rates when players update their strategies using Gradient Ascent, and its variant, Optimistic Gradient Ascent, showing last iterate convergence in three settings -- when the payoffs of players are linear, strongly concave and Lipschitz, and strongly concave and smooth. We provide experimental results that support these theoretical findings.