Uncertainty Quantification Under Group Sparsity

TL;DR

This paper develops and validates a bootstrap method for quantifying uncertainty in high-dimensional group lasso regression, providing reliable simultaneous inference for large groups of coefficients.

Contribution

It introduces a modified parametric bootstrap approach with proven asymptotic validity under Gaussian errors and extends it to other block norm penalizations and sub-Gaussian errors.

Findings

Bootstrap method outperforms competitors in simulations

Provides valid simultaneous inference for large groups

Applicable to various block norm penalizations

Abstract

Quantifying the uncertainty in penalized regression under group sparsity is an important open question. We establish, under a high-dimensional scaling, the asymptotic validity of a modified parametric bootstrap method for the group lasso, assuming a Gaussian error model and mild conditions on the design matrix and the true coefficients. Simulation of bootstrap samples provides simultaneous inferences on large groups of coefficients. Through extensive numerical comparisons, we demonstrate that our bootstrap method performs much better than popular competitors, highlighting its practical utility. The theoretical result is generalized to other block norm penalization and sub-Gaussian errors, which further broadens the potential applications.