A Parallel Decomposition Method for Nonconvex Stochastic Multi-Agent Optimization Problems

This paper considers the problem of minimizing the expected value of a (possibly nonconvex) cost function parameterized by a random (vector) variable, when the expectation cannot be computed accurately (e.g., because the statistics of the random variables are unknown and/or the computational complexity is prohibitive). Classical stochastic gradient methods for solving this problem may suffer from slow convergence. In this paper, we propose a stochastic parallel Successive Convex Approximation-based (best-response) algorithm for general nonconvex stochastic sum-utility optimization problems, which arise naturally in the design of multi-agent networks. The proposed novel decomposition approach enables all users to update their optimization variables in parallel by solving a sequence of strongly convex subproblems, one for each user. Almost sure convergence to stationary points is proved. We then customize the algorithmic framework to solve the stochastic sum rate maximization problem over single-input-single-output (SISO) frequency-selective interference channels, multiple-input-multiple-output (MIMO) interference channels, and MIMO multiple-access channels. Numerical results corroborate that the proposed algorithms can converge faster than state-of-the-art stochastic gradient schemes while achieving the same (or better) sum-rates.