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

TL;DR

This paper introduces the $n$-sided PL condition to analyze the convergence of gradient-based algorithms in nonconvex general-sum games, providing new insights into their effectiveness and proposing variants that ensure convergence to Nash equilibria.

Contribution

The paper extends the PL condition to the $n$-sided setting, enabling convergence analysis of gradient algorithms in complex nonconvex games, and proposes modified algorithms that guarantee convergence where standard methods fail.

Findings

Gradient-based algorithms converge under the $n$-sided PL condition.

Modified gradient descent variants achieve convergence to NE in challenging scenarios.

Experimental results validate the theoretical convergence guarantees.

Abstract

We consider the problem of finding a Nash equilibrium (NE) in a general-sum game, where player $i$'s objective is $f_i(x)=f_i(x_1,...,x_n)$, with $x_j\in\mathbb{R}^{d_j}$ denoting the strategy variables of player $j$. Our focus is on investigating first-order gradient-based algorithms and their variations, such as the block coordinate descent (BCD) algorithm, for tackling this problem. We introduce a set of conditions, called the $n$-sided PL condition, which extends the well-established gradient dominance condition a.k.a Polyak-{\L}ojasiewicz (PL) condition and the concept of multi-convexity. This condition, satisfied by various classes of non-convex functions, allows us to analyze the convergence of various gradient descent (GD) algorithms. Moreover, our study delves into scenarios where the standard gradient descent methods fail to converge to NE. In such cases, we propose adapted variants of GD that converge towards NE and analyze their convergence rates. Finally, we evaluate the performance of the proposed algorithms through several experiments.