Uncertainty Quantification Under Group Sparsity

Quantifying the uncertainty in penalized regression under group sparsity, such as the group Lasso, is an important, yet still 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. Consequently, simulation of bootstrap samples provides a convenient means of statistical inference on potentially large groups of coefficients and on individual coefficients as well. Through extensive numerical comparisons, we demonstrate that our simulation method has substantially better performance than a few popular competitors, highlighting the practical utility of this approach on finite samples. The theoretical result is generalized to other block norm penalization and sub-Gaussian errors, which further broadens the potential applications of this work.