Solving Stochastic Optimization with Expectation Constraints Efficiently by a Stochastic Augmented Lagrangian-Type Algorithm

TL;DR

This paper introduces a stochastic approximation algorithm combining proximal methods and Lagrangian techniques to efficiently solve convex stochastic optimization problems with expectation constraints, achieving favorable convergence rates.

Contribution

The paper proposes a novel stochastic linearized proximal method of multipliers for convex expectation-constrained problems, with proven convergence rates and high-probability bounds.

Findings

Expected convergence rate of O(K^{-1/2}) for objective and constraints

High-probability bounds of O(log(K)K^{-1/2}) for constraints

Preliminary numerical results demonstrate effectiveness

Abstract

This paper considers the problem of minimizing a convex expectation function with a set of inequality convex expectation constraints. We present a computable stochastic approximation type algorithm, namely the stochastic linearized proximal method of multipliers, to solve this convex stochastic optimization problem. This algorithm can be roughly viewed as a hybrid of stochastic approximation and the traditional proximal method of multipliers. Under mild conditions, we show that this algorithm exhibits $O(K^{-1/2})$ expected convergence rates for both objective reduction and constraint violation if parameters in the algorithm are properly chosen, where $K$ denotes the number of iterations. Moreover, we show that, with high probability, the algorithm has $O(\log(K)K^{-1/2})$ constraint violation bound and $O(\log^{3/2}(K)K^{-1/2})$ objective bound. Some preliminary numerical results demonstrate the performance of the proposed algorithm.