Global Convergence to Nash Equilibrium in Nonconvex General-Sum Games under the $n$-Sided PL Condition

2026-02-13 · Source: cs.MA updates on arXiv.org · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Mathematics & Computational Sciences, Robotics & Autonomous Systems · Depth: Expert, extended

Summary

This paper introduces the "n-sided PL condition," an extension of the Polyak-Łojasiewicz (PL) condition and multi-convexity, to analyze the global convergence of gradient-based algorithms in nonconvex general-sum games. The research focuses on finding Nash Equilibria (NE) for games where each player's objective function $f_i(x)$ is nonconvex. The authors demonstrate that this new condition, satisfied by various nonconvex functions like n-player linear quadratic games and linear residual networks, allows for convergence analysis of algorithms such as Block Coordinate Descent (BCD). They propose and evaluate adaptive variants of BCD, namely Ideal Adaptive Random BCD (IA-RBCD) and Adaptive Random BCD (A-RBCD), which provably converge to NEs, even when standard gradient descent methods fail or when objective functions are not lower bounded and have strict saddle points. Experimental results on problems like Cournot competition and infinite horizon n-player Linear-Quadratic (LQ) games show A-RBCD's superior and more stable convergence compared to benchmark BCD methods.

Key takeaway

Research Scientists working on multi-agent optimization in nonconvex settings should consider adopting the Adaptive Random Block Coordinate Descent (A-RBCD) algorithm. This method offers provable global convergence to Nash Equilibria, even in challenging scenarios with strict saddle points or unbounded objective functions, outperforming traditional BCD. Your work on complex game-theoretic models, such as Cournot competition or LQ games, could benefit from A-RBCD's enhanced stability and faster convergence rates, leading to more reliable and efficient equilibrium finding.

Key insights

The n-sided PL condition enables global convergence analysis for gradient-based algorithms in nonconvex general-sum games.

Principles

• n-sided PL condition extends gradient dominance for multi-player games.
• Stationary points are equivalent to Nash Equilibria under n-sided PL.
• Adaptive BCD variants can overcome strict saddle points and unbounded objectives.

Method

The Adaptive Random Block Coordinate Descent (A-RBCD) algorithm refines update directions using a linear combination of partial gradients and a term derived from the sum of objective functions and their best responses, adaptively selecting coefficients.

In practice

• Apply A-RBCD for faster NE convergence in Cournot competition.
• Use A-RBCD for stable convergence in n-player LQ games.
• Consider A-RBCD for nonconvex problems with strict saddle points.

Topics

• Nonconvex Optimization
• General-Sum Games
• Nash Equilibrium
• n-sided PL Condition
• Block Coordinate Descent

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Editorial summary, takeaway, and curation by AIssential. Original article published by cs.MA updates on arXiv.org.