Geometric Convergence of Gradient Play Algorithms for Distributed Nash Equilibrium Seeking

We study distributed algorithms for seeking a Nash equilibrium in a class of non-cooperative convex games with strongly monotone mappings. Each player has access to her own smooth local cost function and can communicate to her neighbors in some undirected graph. To deal with fast distributed learning of Nash equilibria under such settings, we introduce a so called augmented game mapping and provide conditions under which this mapping is strongly monotone. We consider a distributed gradient play algorithm for determining a Nash equilibrium (GRANE). The algorithm involves every player performing a gradient step to minimize her own cost function while sharing and retrieving information locally among her neighbors in the network. Using the reformulation of the Nash equilibrium problem based on the strong monotone augmented game mapping, we prove the convergence of this algorithm to a Nash equilibrium with a geometric rate. Further, we introduce the Nesterov type acceleration for the gradient play algorithm. We demonstrate that, similarly to the accelerated algorithms in centralized optimization and variational inequality problems, our accelerated algorithm outperforms GRANE in the convergence rate. Moreover, to relax assumptions required to guarantee the strongly monotone augmented mapping, we analyze the restricted strongly monotone property of this mapping and prove geometric convergence of the distributed gradient play under milder assumptions.