Random Multi-Constraint Projection: Stochastic Gradient Methods for Convex Optimization with Many Constraints

TL;DR

This paper introduces stochastic gradient methods with random multi-constraint projections for convex optimization problems involving many constraints, providing convergence guarantees and efficiency analysis.

Contribution

It proposes new algorithms combining stochastic gradient descent with random feasibility updates, including a novel polyhedral-set projection scheme, and establishes their convergence and rate benchmarks.

Findings

Polyhedral-set projection scheme is most efficient among tested methods.

Algorithms converge almost surely under the proposed schemes.

New convergence rate benchmarks for stochastic convex optimization with many constraints.

Abstract

Consider convex optimization problems subject to a large number of constraints. We focus on stochastic problems in which the objective takes the form of expected values and the feasible set is the intersection of a large number of convex sets. We propose a class of algorithms that perform both stochastic gradient descent and random feasibility updates simultaneously. At every iteration, the algorithms sample a number of projection points onto a randomly selected small subsets of all constraints. Three feasibility update schemes are considered: averaging over random projected points, projecting onto the most distant sample, projecting onto a special polyhedral set constructed based on sample points. We prove the almost sure convergence of these algorithms, and analyze the iterates' feasibility error and optimality error, respectively. We provide new convergence rate benchmarks for stochastic first-order optimization with many constraints. The rate analysis and numerical experiments reveal that the algorithm using the polyhedral-set projection scheme is the most efficient one within known algorithms.