Smoothing Proximal Gradient Methods for Nonsmooth Sparsity Constrained Optimization: Optimality Conditions and Global Convergence

TL;DR

This paper introduces smoothing proximal gradient methods for nonsmooth sparsity constrained optimization, providing stronger optimality conditions, convergence analysis, and demonstrating superior empirical performance over existing methods.

Contribution

It develops two variants of SPGM, offers new convergence theories, and shows that SPGM-BCD finds stronger stationary points than previous approaches.

Findings

SPGM-BCD finds stronger stationary points than prior methods.

Theoretical convergence bounds match the best-known error bounds.

Numerical experiments show SPGM-BCD outperforms existing algorithms.

Abstract

Nonsmooth sparsity constrained optimization encompasses a broad spectrum of applications in machine learning. This problem is generally non-convex and NP-hard. Existing solutions to this problem exhibit several notable limitations, including their inability to address general nonsmooth problems, tendency to yield weaker optimality conditions, and lack of comprehensive convergence analysis. This paper considers Smoothing Proximal Gradient Methods (SPGM) as solutions to nonsmooth sparsity constrained optimization problems. Two specific variants of SPGM are explored: one based on Iterative Hard Thresholding (SPGM-IHT) and the other on Block Coordinate Decomposition (SPGM-BCD). It is shown that the SPGM-BCD algorithm finds stronger stationary points compared to previous methods. Additionally, novel theories for analyzing the convergence rates of both SPGM-IHT and SPGM-BCD algorithms are developed. Our theoretical bounds, capitalizing on the intrinsic sparsity of the optimization problem, are on par with the best-known error bounds available to date. Finally, numerical experiments reveal that SPGM-IHT performs comparably to current IHT-style methods, while SPGM-BCD consistently surpasses them. Keywords: Sparsity Recovery, Nonsmooth Optimization, Nonconvex Optimization, Proximal Gradient Method, Block Coordinate Decomposition, Iterative Hard Thresholding, Convergence Analysis