Mind the Duality Gap: Safer Rules for the Lasso

Abstract

Screening rules allow to early discard irrelevant variables from the optimization in Lasso problems, or its derivatives, making solvers faster. In this paper, we propose new versions of the so-called \textit{safe} rules for the Lasso. Based on duality gap considerations, our new rules create safe test regions whose diameters converge to zero, provided that one relies on a converging solver. This property helps screening out more variables, for a wider range of regularization parameter values. In addition to faster convergence, we prove that we correctly identify the active sets (supports) of the solutions in finite time. While our proposed strategy can cope with any solver, its performance is demonstrated using a coordinate descent algorithm particularly adapted to machine learning use cases. Significant computing time reductions are obtained with respect to previous safe rules.