We propose a new distributed algorithm that combines heavy-ball momentum and a consensus-based gradient method to find a Nash equilibrium (NE) in a class of non-cooperative convex games with unconstrained action sets. In this approach, each agent in the game has access to its own smooth local cost function and can exchange information with its neighbors over a communication network. The proposed method is designed to work on a general sequence of time-varying directed graphs and allows for non-identical step-sizes and momentum parameters. Our work is the first to incorporate heavy-ball momentum in the context of non-cooperative games, and we provide a rigorous proof of its geometric convergence to the NE under the common assumptions of strong convexity and Lipschitz continuity of the agents' cost functions. Moreover, we establish explicit bounds for the step-size values and momentum parameters based on the characteristics of the cost functions, mixing matrices, and graph connectivity structures. To showcase the efficacy of our proposed method, we perform numerical simulations on a Nash-Cournot game to demonstrate its accelerated convergence compared to existing methods.

翻译：我们提出了一种新的分布式算法，该算法结合重球动量与基于共识的梯度方法，用于求解一类具有无约束行动集的非合作凸博弈中的纳什均衡（NE）。在该方法中，博弈中的每个智能体可访问自身光滑的局部成本函数，并能通过通信网络与邻居交换信息。所提方法设计适用于一般性的时变有向图序列，且允许使用非一致的步长和动量参数。本研究首次将重球动量引入非合作博弈场景，并在强凸性与利普希茨连续性等常见假设下，严格证明了该算法以几何收敛速率逼近NE。此外，我们基于成本函数特性、混合矩阵和图连通性结构，建立了步长和动量参数的显式边界。为验证所提方法的有效性，我们在纳什-古诺博弈上进行了数值仿真，结果表明该方法相比现有算法具有加速收敛性能。