Linear convergence for distributed stochastic optimization with coupled inequality constraints

This paper considers the distributed stochastic optimization problem over time-varying networks, in which agents aim to cooperatively minimize the expected value of the sum of their cost functions subject to coupled affine inequalities. Considering various stochastic factors and constraints on decisions inherent in physical environments, this problem has wide applications such as in smart grids, resource allocation and distributed machine learning. To solve this problem, a novel distributed stochastic primal–dual algorithm is devised by applying variance reduction and distributed tracking techniques. A complete and rigorous analysis shows that the developed algorithm linearly converges to the optimal solution in the mean square sense, and an explicit upper bound on the required constant stepsize is presented. Finally, a numerical example is conducted to illustrate the theoretical findings.