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

2025-07-15

AI Summary

Online optimization of Nash equilibria (NE) in zero-sum games faces a fundamental trade-off: existing methods either achieve time-average convergence (regret minimization) or last-iterate convergence (under contraction assumptions), but struggle to simultaneously ensure computational efficiency and parallelizability. Method: We propose an alternating gradient descent framework grounded in Hamiltonian dynamics modeling, operating under non-contractive, unbounded settings. Contribution/Results: Our method exactly characterizes the NE set in linear iterations—achieving the first linear-iteration exact characterization of NE—enabling fully time-parallel execution without reliance on problem-specific learning rates. We establish theoretical guarantees of no-regret behavior and global convergence. Empirical evaluations demonstrate significant improvements in both convergence speed and stability over state-of-the-art approaches. This work introduces a novel paradigm for online equilibrium computation in algorithmic game theory.

Technology Category

• Game Theory and Economic Paradigms: Equilibrium
• Multiagent Systems: Mechanism Design
• Machine Learning: Online Learning & Bandits

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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.

Problem

• Characterizing Nash equilibria in zero-sum games efficiently
• Proposing a parallelizable method for NE computation
• Enabling arbitrary learning rates in gradient descent

Innovation

• Uses Hamiltonian dynamics for NE characterization
• Parallelizable with arbitrary learning rates
• Finite linear iterations for NE convergence