Exterior-point Optimization for Sparse and Low-rank Optimization

TL;DR

This paper introduces NExOS, a first-order exterior-point algorithm for sparse and low-rank optimization that effectively finds local minima of nonconvex problems by solving a sequence of penalized problems, outperforming existing methods.

Contribution

The paper presents NExOS, a novel nonconvex exterior-point optimization algorithm tailored for sparse and low-rank problems, with proven convergence properties and empirical superiority.

Findings

NExOS converges linearly to local minima under regularity conditions.

The algorithm outperforms specialized methods on diverse sparse and low-rank problems.

Local minima of penalized problems approach those of the original problem as penalties decrease.

Abstract

Many problems of substantial current interest in machine learning, statistics, and data science can be formulated as sparse and low-rank optimization problems. In this paper, we present the nonconvex exterior-point optimization solver NExOS -- a first-order algorithm tailored to sparse and low-rank optimization problems. We consider the problem of minimizing a convex function over a nonconvex constraint set, where the set can be decomposed as the intersection of a compact convex set and a nonconvex set involving sparse or low-rank constraints. Unlike the convex relaxation approaches, NExOS finds a locally optimal point of the original problem by solving a sequence of penalized problems with strictly decreasing penalty parameters by exploiting the nonconvex geometry. NExOS solves each penalized problem by applying a first-order algorithm, which converges linearly to a local minimum of the corresponding penalized formulation under regularity conditions. Furthermore, the local minima of the penalized problems converge to a local minimum of the original problem as the penalty parameter goes to zero. We then implement and test NExOS on many instances from a wide variety of sparse and low-rank optimization problems, empirically demonstrating that our algorithm outperforms specialized methods.