Optimization with Sparsity-Inducing Penalties

TL;DR

This paper reviews optimization techniques for sparsity-inducing penalties in data and model estimation, covering convex and non-convex methods, with extensive experimental comparisons.

Contribution

It provides a comprehensive overview of optimization tools and algorithms tailored for sparsity-inducing penalties, including recent extensions and practical comparisons.

Findings

Proximal and block-coordinate methods effectively handle non-smooth penalties.

Reweighted and homotopy techniques improve convergence in sparse optimization.

Extensive experiments compare computational efficiency of various algorithms.

Abstract

Sparse estimation methods are aimed at using or obtaining parsimonious representations of data or models. They were first dedicated to linear variable selection but numerous extensions have now emerged such as structured sparsity or kernel selection. It turns out that many of the related estimation problems can be cast as convex optimization problems by regularizing the empirical risk with appropriate non-smooth norms. The goal of this paper is to present from a general perspective optimization tools and techniques dedicated to such sparsity-inducing penalties. We cover proximal methods, block-coordinate descent, reweighted $\ell_2$-penalized techniques, working-set and homotopy methods, as well as non-convex formulations and extensions, and provide an extensive set of experiments to compare various algorithms from a computational point of view.