Second-Order Algorithms for Finding Local Nash Equilibria in Zero-Sum Games

Kushagra Gupta, Xinjie Liu, Ross Emerson Allen, ufuk topcu, David Fridovich-Keil

Keywords: game theory, nonconvex-nonconcave optimization, dynamical systems, Nash equilibrium

TL;DR: We develop second-order algorithms for finding local Nash equilibrium in smooth, possibly nonconvex-nonconcave zero-sum games.

Abstract: Zero-sum games arise in a wide variety of problems, including robust optimization and adversarial learning. However, algorithms deployed for finding a local Nash equilibrium in these games often converge to non-Nash stationary points. This highlights a key challenge: for any algorithm, the stability properties of its underlying dynamical system can cause non-Nash points to be potential attractors. To overcome this challenge, algorithms must account for subtleties involving the curvatures of players' costs. To this end, we leverage dynamical system theory and develop a second-order algorithm for finding a local Nash equilibrium in the smooth, possibly nonconvex-nonconcave, zero-sum game setting. First, we prove that this novel method guarantees convergence to only local Nash equilibria with a local $\textit{linear}$ convergence rate. We then interpret a version of this method as a modified Gauss-Newton algorithm with local $\textit{superlinear}$ convergence to the neighborhood of a point that satisfies first-order local Nash equilibrium conditions. In comparison, current related state-of-the-art methods do not offer convergence rate guarantees. Furthermore, we show that this approach naturally generalizes to settings with convex and potentially coupled constraints while retaining earlier guarantees of convergence to only local (generalized) Nash equilibria.

Primary Area: optimization

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Submission Number: 8854