Geometric Convergence of Distributed Heavy-Ball Nash Equilibrium Algorithm over Time-Varying Digraphs with Unconstrained Actions

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.