Model-Consistent Sparse Estimation through the Bootstrap

TL;DR

This paper analyzes the asymptotic model consistency of the Lasso in low-dimensional settings and introduces the Bolasso, a bootstrap-based method for consistent variable selection, extending to high dimensions.

Contribution

It provides a detailed asymptotic analysis of Lasso's model selection probability and proposes the Bolasso, a bootstrap-based variable selection method with proven consistency.

Findings

Lasso selects true variables with probability tending to one exponentially fast under certain conditions.

Intersecting bootstrap Lasso supports yields consistent model selection.

The Bolasso extends to high-dimensional settings with a two-step consistent procedure.

Abstract

We consider the least-square linear regression problem with regularization by the $\ell^1$-norm, a problem usually referred to as the Lasso. In this paper, we first present a detailed asymptotic analysis of model consistency of the Lasso in low-dimensional settings. For various decays of the regularization parameter, we compute asymptotic equivalents of the probability of correct model selection. For a specific rate decay, we show that the Lasso selects all the variables that should enter the model with probability tending to one exponentially fast, while it selects all other variables with strictly positive probability. We show that this property implies that if we run the Lasso for several bootstrapped replications of a given sample, then intersecting the supports of the Lasso bootstrap estimates leads to consistent model selection. This novel variable selection procedure, referred to as the Bolasso, is extended to high-dimensional settings by a provably consistent two-step procedure.