Mind the duality gap: safer rules for the Lasso

TL;DR

This paper introduces improved safe screening rules for Lasso that leverage duality gap considerations, enabling more effective variable elimination and faster convergence in optimization problems.

Contribution

The authors develop new safe rules based on duality gap that ensure convergence of safe test regions and finite-time identification of active sets.

Findings

Faster convergence of Lasso solvers with new safe rules.

More variables screened out across a wider range of parameters.

Significant reductions in computational time compared to previous rules.

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.