Stochastic Approximation Proximal Method of Multipliers for Convex Stochastic Programming

This paper considers the problem of minimizing a convex expectation function over a closed convex set, coupled with a set of inequality convex expectation constraints. We present a new stochastic approximation type algorithm, namely the stochastic approximation proximal method of multipliers (PMMSopt) to solve this convex stochastic optimization problem. We analyze regrets of a stochastic approximation proximal method of multipliers for solving convex stochastic optimization problems. Under mild conditions, we show that this algorithm exhibits rm O(T-1/2) rate of convergence, in terms of both optimality gap and constraint violation if parameters in the algorithm are properly chosen, when the objective and constraint functions are generally convex, where T denotes the number of iterations. Moreover, we show that, with at least 1-e^-T1/4 probability, the algorithm has no more than rm O(T-1/4) objective regret and no more than rm O(T-1/8) constraint violation regret. To the best of our knowledge, this is the first time that such a proximal method for solving expectation constrained stochastic optimization is presented in the literature.