A Parallelizable Approach for Characterizing NE in Zero-Sum Games After a Linear Number of Iterations of Gradient Descent

TL;DR

This paper introduces a novel parallelizable method based on Hamiltonian dynamics that characterizes Nash equilibria in zero-sum games within a linear number of gradient descent iterations, outperforming standard approaches.

Contribution

It presents the first finite-iteration characterization of NE in online optimization using Hamiltonian dynamics, allowing for parallelization and arbitrary learning rates.

Findings

Method outperforms standard algorithms in experiments.

Characterizes NE in a finite linear number of iterations.

Works with arbitrary learning rates.

Abstract

We study online optimization methods for zero-sum games, a fundamental problem in adversarial learning in machine learning, economics, and many other domains. Traditional methods approximate Nash equilibria (NE) using either regret-based methods (time-average convergence) or contraction-map-based methods (last-iterate convergence). We propose a new method based on Hamiltonian dynamics in physics and prove that it can characterize the set of NE in a finite (linear) number of iterations of alternating gradient descent in the unbounded setting, modulo degeneracy, a first in online optimization. Unlike standard methods for computing NE, our proposed approach can be parallelized and works with arbitrary learning rates, both firsts in algorithmic game theory. Experimentally, we support our results by showing our approach drastically outperforms standard methods.