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.

翻译：本文研究一般和博弈中纳什均衡的求解问题，其中参与者$i$的目标函数为$f_i(x)=f_i(x_1,...,x_n)$，$x_j\in\mathbb{R}^{d_j}$表示参与者$j$的策略变量。我们重点研究基于一阶梯度算法及其变体（如块坐标下降算法）在该问题中的应用。我们提出了一组称为$n$边PL条件的扩展条件，该条件将经典的梯度优势条件（即Polyak-Łojasiewicz条件）与多凸性概念相结合。该条件被多类非凸函数所满足，使我们能够分析各种梯度下降算法的收敛性。此外，本研究深入探讨了标准梯度下降法无法收敛至纳什均衡的场景。针对此类情况，我们提出了能够收敛至纳什均衡的梯度下降改进算法，并分析了其收敛速率。最后，我们通过多组实验评估了所提算法的性能。