This paper presents a new distributed algorithm that leverages 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 main novelty of our work is the incorporation of heavy-ball momentum in the context of non-cooperative games that operate on fully-decentralized, directed, and time-varying communication graphs, while also accommodating non-identical step-sizes and momentum parameters. Overcoming technical challenges arising from the dynamic and asymmetric nature of mixing matrices and the presence of an additional momentum term, we provide a rigorous proof of the geometric convergence to the NE. 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. We perform numerical simulations on a Nash-Cournot game to demonstrate accelerated convergence of the proposed algorithm compared to that of the existing methods.

翻译：本文提出了一种新的分布式算法，该算法利用重球动量和基于共识的梯度方法，在一类具有无约束行动集的非合作凸博弈中寻找纳什均衡（NE）。在此方法中，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。我们工作的主要创新在于，将重球动量引入到运行于完全去中心化、有向且时变通信图上的非合作博弈中，同时允许非相同的步长和动量参数。克服了混合矩阵动态不对称性及额外动量项带来的技术挑战后，我们严格证明了算法几何收敛到NE。此外，我们基于成本函数、混合矩阵和图连通性结构的特征，建立了步长值和动量参数的显式界限。我们在纳什-古诺博弈上进行了数值模拟，结果表明所提算法相比现有方法具有加速收敛的特性。