Optimal Convergence for Stochastic Optimization with Multiple Expectation Constraints

TL;DR

This paper develops an optimized stochastic approximation algorithm for convex expectation-constrained problems, achieving optimal convergence rates and demonstrating improved empirical performance over existing methods.

Contribution

It extends the cooperative stochastic approximation algorithm to handle multiple expectation constraints and provides a novel proof for optimal convergence rates.

Findings

Achieves optimal convergence rates for both optimality gap and constraint violation.

Demonstrates improved empirical convergence compared to state-of-the-art methods.

Provides a new proof technique for analyzing stochastic optimization with constraints.

Abstract

In this paper, we focus on the problem of stochastic optimization where the objective function can be written as an expectation function over a closed convex set. We also consider multiple expectation constraints which restrict the domain of the problem. We extend the cooperative stochastic approximation algorithm from Lan and Zhou [2016] to solve the particular problem. We close the gaps in the previous analysis and provide a novel proof technique to show that our algorithm attains the optimal rate of convergence for both optimality gap and constraint violation when the functions are generally convex. We also compare our algorithm empirically to the state-of-the-art and show improved convergence in many situations.