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 proximal method of multipliers to solve this convex stochastic optimization problem. We analyze regrets of the proposed method for solving convex stochastic optimization problems. Under mild conditions, we show that this method exhibits sublinear regret for both objective reduction and constraint violation if parameters in the algorithm are properly chosen. Moreover, we investigate the high probability performance of the proposed method under the standard light-tail assumption.

Funding: This work was supported by Fundamental Research Funds for the Central Universities [Grant DUT20LK33]; Dalian High-Level Talent Innovation Program [Grant 2020RD09]; National Natural Science Foundation of China [Grant 11731013, 11871135, 11971089, and 12071055].