Last-Iterate Guarantees for Learning in Co-coercive Games

2026-04-21

AI Summary

This work addresses the lack of finite-time guarantees for the last-iterate convergence of stochastic gradient descent (SGD) in cocoercive games under non-vanishing noise. Focusing on a more realistic noise model—where the second moment of the stochastic gradient is affinely related to the squared norm of the current iterate—the paper establishes, for the first time, a finite-time convergence bound for the last iterate of SGD to the set of Nash equilibria. By leveraging the structure of cocoercive games and carefully analyzing the impact of persistent noise, the authors prove almost sure convergence of the algorithm and derive a last-iterate convergence rate of $O(\log t / t^{1/3})$. Additionally, they provide convergence guarantees in the time-averaged sense.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Game Theory and Economic Paradigms: Cooperative Game Theory
• Machine Learning: Online Learning & Bandits

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements

Abstract

We establish finite-time last-iterate guarantees for vanilla stochastic gradient descent in co-coercive games under noisy feedback. This is a broad class of games that is more general than strongly monotone games, allows for multiple Nash equilibria, and includes examples such as quadratic games with negative semidefinite interaction matrices and potential games with smooth concave potentials. Prior work in this setting has relied on relative noise models, where the noise vanishes as iterates approach equilibrium, an assumption that is often unrealistic in practice. We work instead under a substantially more general noise model in which the second moment of the noise is allowed to scale affinely with the squared norm of the iterates, an assumption natural in learning with unbounded action spaces. Under this model, we prove a last-iterate bound of order $O(\log(t)/t^{1/3})$, the first such bound for co-coercive games under non-vanishing noise. We additionally establish almost sure convergence of the iterates to the set of Nash equilibria and derive time-average convergence guarantees.

Problem

• co-coercive games
• last-iterate convergence
• stochastic gradient descent
• noisy feedback
• Nash equilibria

Innovation

• last-iterate convergence
• co-coercive games
• stochastic gradient descent
• non-vanishing noise
• Nash equilibrium