A General Iterative Shrinkage and Thresholding Algorithm for Non-convex Regularized Optimization Problems

TL;DR

This paper introduces the GIST algorithm, an efficient iterative method for solving large-scale non-convex regularized optimization problems, with proven convergence and practical advantages over traditional convex relaxation methods.

Contribution

The paper proposes the GIST algorithm, a novel iterative shrinkage and thresholding method with a line search for non-convex penalties, improving scalability and convergence analysis.

Findings

GIST algorithm efficiently solves large-scale non-convex optimization problems.

The algorithm has a closed-form solution for many penalties, enhancing computational speed.

Extensive experiments demonstrate superior performance over existing methods.

Abstract

Non-convex sparsity-inducing penalties have recently received considerable attentions in sparse learning. Recent theoretical investigations have demonstrated their superiority over the convex counterparts in several sparse learning settings. However, solving the non-convex optimization problems associated with non-convex penalties remains a big challenge. A commonly used approach is the Multi-Stage (MS) convex relaxation (or DC programming), which relaxes the original non-convex problem to a sequence of convex problems. This approach is usually not very practical for large-scale problems because its computational cost is a multiple of solving a single convex problem. In this paper, we propose a General Iterative Shrinkage and Thresholding (GIST) algorithm to solve the nonconvex optimization problem for a large class of non-convex penalties. The GIST algorithm iteratively solves a proximal operator problem, which in turn has a closed-form solution for many commonly used penalties. At each outer iteration of the algorithm, we use a line search initialized by the Barzilai-Borwein (BB) rule that allows finding an appropriate step size quickly. The paper also presents a detailed convergence analysis of the GIST algorithm. The efficiency of the proposed algorithm is demonstrated by extensive experiments on large-scale data sets.