Stochastic Approximation for Expectation Objective and Expectation Inequality-Constrained Nonconvex Optimization

TL;DR

This paper introduces a stochastic approximation method for nonconvex optimization problems with noisy objectives and constraints, providing convergence guarantees and demonstrating practical effectiveness.

Contribution

It presents the first stochastic approximation approach for expectation-based constraints and objectives using the Ghost framework with penalty functions.

Findings

Convergence guarantees under stochastic settings.

Effective handling of noisy constraints and objectives.

Successful demonstration on representative examples.

Abstract

Stochastic Approximation has been a prominent set of tools for solving problems with noise and uncertainty. Increasingly, it becomes important to solve optimization problems wherein there is noise in both a set of constraints that a practitioner requires the system to adhere to, as well as the objective, which typically involves some empirical loss. We present the first stochastic approximation approach for solving this class of problems using the Ghost framework of incorporating penalty functions for analysis of a sequential convex programming approach together with a Monte Carlo estimator of nonlinear maps. We provide almost sure convergence guarantees and demonstrate the performance of the procedure on some representative examples.