Bolasso: model consistent Lasso estimation through the bootstrap

TL;DR

The paper introduces Bolasso, a bootstrap-based method for consistent variable selection in Lasso regression, demonstrating its theoretical properties and superior performance on synthetic and real datasets.

Contribution

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

Findings

Bolasso achieves consistent variable selection through bootstrap support intersection.

Lasso's probability of correct model selection depends on the regularization decay rate.

Bolasso outperforms other linear regression methods on benchmark datasets.

Abstract

We consider the least-square linear regression problem with regularization by the l1-norm, a problem usually referred to as the Lasso. In this paper, we present a detailed asymptotic analysis of model consistency of the Lasso. For various decays of the regularization parameter, we compute asymptotic equivalents of the probability of correct model selection (i.e., variable 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 algorithm, referred to as the Bolasso, is compared favorably to other linear regression methods on synthetic data and datasets from the UCI machine learning repository.