Efficient Smoothed Concomitant Lasso Estimation for High Dimensional Regression

TL;DR

This paper introduces a numerically stable and computationally efficient variant of the Smoothed Concomitant Lasso for high-dimensional regression, enabling joint estimation of regression parameters and noise level.

Contribution

It proposes a modified Smoothed Concomitant Lasso with improved numerical stability and an efficient solver using coordinate descent and screening rules.

Findings

The new method is as fast as standard Lasso solvers.

It demonstrates increased numerical stability over previous formulations.

The approach effectively estimates noise level alongside regression coefficients.

Abstract

In high dimensional settings, sparse structures are crucial for efficiency, both in term of memory, computation and performance. It is customary to consider $\ell_1$ penalty to enforce sparsity in such scenarios. Sparsity enforcing methods, the Lasso being a canonical example, are popular candidates to address high dimension. For efficiency, they rely on tuning a parameter trading data fitting versus sparsity. For the Lasso theory to hold this tuning parameter should be proportional to the noise level, yet the latter is often unknown in practice. A possible remedy is to jointly optimize over the regression parameter as well as over the noise level. This has been considered under several names in the literature: Scaled-Lasso, Square-root Lasso, Concomitant Lasso estimation for instance, and could be of interest for confidence sets or uncertainty quantification. In this work, after illustrating numerical difficulties for the Smoothed Concomitant Lasso formulation, we propose a modification we coined Smoothed Concomitant Lasso, aimed at increasing numerical stability. We propose an efficient and accurate solver leading to a computational cost no more expansive than the one for the Lasso. We leverage on standard ingredients behind the success of fast Lasso solvers: a coordinate descent algorithm, combined with safe screening rules to achieve speed efficiency, by eliminating early irrelevant features.