Inexact best-response schemes for stochastic Nash games: Linear convergence and Iteration complexity analysis

We consider a subclass of N-player stochastic Nash games in which each player solves a parametrized stochastic optimization problem. In deterministic regimes, best response schemes have been shown to be convergent under a suitable spectral property associated with the proximal-response map. However, a direct application of this scheme to stochastic settings requires obtaining exact solutions to stochastic optimization problems at every step. Instead, we propose an inexact generalization of this scheme in which an inexact solution to the best response problem is computed where the player-specific inexactness sequence is assumed to be separable. Notably, this scheme is an implementable single-loop scheme that requires a fixed (but increasing) number of stochastic gradient steps to compute an inexact solution to the best response problem. On the basis of this framework, we make several contributions: (i) The presented inexact best-response scheme produces iterates that converge to the unique equilibrium in mean; (ii) Surprisingly, we show that the iterates converge at a prescribed linear rate with a prescribed constant rather than a sub-linear rate; and (iii) Finally, by assuming that an inexact solution is computed by a stochastic approximation scheme, the overall iteration complexity for computing an -Nash equilibrium less that O(√N/)^2+δ where δ is a positive scalar. Additionally, we show that the upper bound of this effort is shown to be N=²).