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

TL;DR

The paper addresses finding Nash equilibria in nonconvex general-sum games by introducing the $n$-sided PL condition, which extends gradient-dominance concepts to multi-coordinate settings and ensures that the stationary-point set coincides with the NE set. It analyzes block-coordinate descent (BCD) under this condition and proposes adaptive variants, IA-RBCD and A-RBCD, that leverage the gradient of the aggregate best-response $G_F$ to drive convergence to NE, even when NE are nonunique and the objectives are not globally lower bounded. The authors establish linear convergence under an additional cross-term bound and provide general sublinear rates when a $(\\theta,\\nu)$-PL condition holds for the function $F-G_F$, along with practical schemes to approximate best responses. They validate the approach on potential and general-sum problems, including Cournot competition and infinite-horizon LQ games, demonstrating improved convergence over traditional BCD baselines and robustness to strict saddle points. The results advance theoretical guarantees for gradient-based multi-agent optimization beyond two-block or convex settings and offer algorithmic tools for reliably locating NE in complex nonconvex games with practical impact in economics and control.

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-Łojasiewicz (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.

Figures (7)

• Figure 1: Left is function $f^{(1)}(x_1,x_2)$ and right is $f^{(2)}(x_1,x_2)$.
• Figure 2: The result of R-BCD algorithm applied to potential setting with functions $f^{(3)}(x_1,x_2)$ and $f^{(4)}(x_1,x_2)$. The y-axis is in log scale, thus the R-BCD demonstrates linear convergence for $f^{(4)}$.
• Figure 3: The result of R-BCD algorithm applied to functions $\{f^{(5)}_{1},f^{(5)}_{2}\}$ and $\{f^{(6)}_{1},f^{(6)}_{2}\}$. The y-axis is in log scale, thus the BCD shows linear convergence in the second game.
• Figure 4: Performances of A-RBCD, BM1, and BM2 applied to functions $f_1$ and $f_2$.
• Figure 5: (a) Performances of A-RBCD, BM1, and BM2 applied to the Cournot model with $P(Q)=a-bQ$; (b) with $P(Q)=a-bQ^2$.

Theorems & Definitions (19)

• Definition 2.2: Nash Equilibrium (NE)
• Definition 2.3: $\varepsilon$-Partial Stationary point
• Definition 2.4: $n$-sided PL Condition
• Lemma 2.5
• Lemma 3.1
• Lemma 3.2
• Theorem 3.3
• Definition 3.4: $(\theta,\nu)$-PL condition
• Theorem 3.5
• Theorem 3.6