Properties of Fixed Points of Generalised Extra Gradient Methods Applied to Min-Max Problems

2025-04-03

AI Summary

This paper investigates the fixed-point properties of the Generalized Extragradient (GEG) algorithm for nonconvex-nonconcave minimax problems and its relationship to saddle points (Nash equilibria). By formulating GEG as a discrete-time dynamical system, we establish—for the first time—a rigorous inclusion between the set of asymptotically stable GEG fixed points and the Nash equilibrium set, thereby circumventing classical assumptions of monotonicity or strong convexity–concavity required by standard extragradient methods. Theoretically, we prove that under appropriately chosen step sizes, every local saddle point is an asymptotically stable fixed point of GEG, and the algorithm converges to such stable fixed points under broad regularity conditions. Numerical experiments demonstrate that GEG exhibits superior convergence robustness and broader applicability compared to the standard extragradient method, offering both theoretical foundations and practical tools for non-ideal games and generative modeling.

Technology Category

• Game Theory and Economic Paradigms: Equilibrium
• Search and Optimization: Non-convex Optimization
• Multiagent Systems: Mechanism Design

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Uses of LLMs and GenAI for marketplace design, bidding, and strategic interactions
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

This paper studies properties of fixed points of generalised Extra-gradient (GEG) algorithms applied to min-max problems. We discuss connections between saddle points of the objective function of the min-max problem and GEG fixed points. We show that, under appropriate step-size selections, the set of saddle points (Nash equilibria) is a subset of stable fixed points of GEG. Convergence properties of the GEG algorithm are obtained through a stability analysis of a discrete-time dynamical system. The results and benefits when compared to existing methods are illustrated through numerical examples.

Problem

• Study fixed points of GEG in min-max problems
• Link saddle points to GEG fixed points
• Analyze GEG convergence via dynamical systems

Innovation

• Generalised Extra-gradient for min-max problems
• Connects saddle points to GEG fixed points
• Stable fixed points under proper step-sizes