Single timescale stochastic approximation for stochastic Nash games in cognitive radio systems

Abstract

In the face of increasing demand for wireless services, the design of spectrum assignment policies has gained enormous relevance. We consider one such instance in cognitive radio systems where recent efforts have focused on the application of game-theoretic approaches. Much of this work has been restricted to deterministic regimes and this paper considers a stochastic generalizations. The corresponding problems are seen to be stochastic Nash games over continuous strategy sets. Notably, the gradient map of player utilities is seen to be a monotone mapping over the cartesian product of strategy sets, leading to a monotone stochastic variational inequality. We consider the application of projection-based stochastic approximation schemes. However, such techniques are characterized by a key shortcoming in that the mappings were required to be be strongly monotone. Standard extensions of stochastic approximation schemes for merely monotone mappings rely on obtaining a sequence of increasingly exact solutions, a natively two-timescale scheme. However, obtaining solutions with increasing accuracy remains a challenging task in a simulation-based setting. Accordingly, we consider the development of single timescale techniques for computing equilibria when the associated gradient map does not admit strong monotonicity. We develop convergence theory for distributed single-timescale stochastic approximation schemes, namely stochastic iterative proximal point method which requires exactly one projection step at every step. Finally we apply this framework to the design of cognitive radio systems in uncertain regimes under temperature-interference constraints.