Convergence to Nash equilibrium in continuous games with noisy first-order feedback

This paper examines the convergence of a broad class of distributed learning dynamics for games with continuous action sets. The dynamics under study comprise a multi-agent generalization of Nesterov's dual averaging (DA) method, a primal-dual mirror descent method that has recently seen a major resurgence in the field of large-scale optimization and machine learning. To account for settings with high temporal variability and uncertainty, we adopt a continuous-time formulation of dual averaging and we investigate the dynamics' long-run behavior when players have either noiseless or noisy information on their payoff gradients. In both the deterministic and stochastic regimes, we establish sublinear rates of convergence of actual and averaged trajectories to Nash equilibrium under a variational stability condition.