On the $\ell_1-\ell_q$ Regularized Regression

TL;DR

This paper investigates the properties of $\, ext{l}_1- ext{l}_q$ regularized regression for high-dimensional grouped variable selection, demonstrating that many desirable properties of Lasso extend to this broader family even with increasing within-group variables.

Contribution

It provides a unified theoretical analysis of the entire $\, ext{l}_1- ext{l}_q$ regularization family, including new results on estimation and variable selection consistency for high-dimensional data.

Findings

Estimation and selection consistency under fixed design.

Persistency results under random design with weaker conditions.

Unified analysis covering $q=1$ to $q=\infty$ cases.

Abstract

In this paper we consider the problem of grouped variable selection in high-dimensional regression using $\ell_1-\ell_q$ regularization ($1\leq q \leq \infty$), which can be viewed as a natural generalization of the $\ell_1-\ell_2$ regularization (the group Lasso). The key condition is that the dimensionality $p_n$ can increase much faster than the sample size $n$, i.e. $p_n \gg n$ (in our case $p_n$ is the number of groups), but the number of relevant groups is small. The main conclusion is that many good properties from $\ell_1-$regularization (Lasso) naturally carry on to the $\ell_1-\ell_q$ cases ($1 \leq q \leq \infty$), even if the number of variables within each group also increases with the sample size. With fixed design, we show that the whole family of estimators are both estimation consistent and variable selection consistent under different conditions. We also show the persistency result with random design under a much weaker condition. These results provide a unified treatment for the whole family of estimators ranging from $q=1$ (Lasso) to $q=\infty$ (iCAP), with $q=2$ (group Lasso)as a special case. When there is no group structure available, all the analysis reduces to the current results of the Lasso estimator ($q=1$).