  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
$1-e^{-T^{1/4}}$ 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.
