In this paper, we investigate a distributed learning scheme for a broad class of stochastic optimization problems and games that arise in signal processing and wireless communications. The proposed algorithm relies on the method of matrix exponential learning (MXL) and only requires locally computable gradient observations that are possibly imperfect. To analyze it, we introduce the notion of a stable Nash equilibrium and we show that the algorithm is globally convergent to such equilibria or locally convergent when an equilibrium is only locally stable. To complement our convergence analysis, we also derive explicit bounds for the algorithm’s convergence speed and we test it in realistic multi-carrier/multiple-antenna wireless scenarios where several users seek to maximize their energy efficiency. Our results show that learning allows users to attain a net increase between 100% and 500% in energy efficiency, even under very high uncertainty.