Optimization with Sparsity-Inducing Penalties

Francis Bach (1, 2) , Rodolphe Jenatton (1, 2) , Julien Mairal (3) , Guillaume Obozinski (2, 1)

1 LIENS - Laboratoire d'informatique de l'école normale supérieure
2 SIERRA - Statistical Machine Learning and Parsimony
3 Department of Statistics [Berkeley]

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.

Mots clés: Convex optimization; sparsity

Domaines: Apprentissage [cs.LG]; Optimisation et contrôle [math.OC]; Autres [stat.ML]

Francis Bach, Rodolphe Jenatton, Julien Mairal, Guillaume Obozinski. Optimization with Sparsity-Inducing Penalties. Foundations and Trends in Machine Learning, 2011, ⟨10.1561/2200000015⟩. ⟨hal-00613125v2⟩