Theoretical Properties of the Overlapping Groups Lasso

TL;DR

This paper provides theoretical insights into the overlapping groups lasso, including finite sample bounds and asymptotic behavior, highlighting the importance of careful group set selection to avoid negative effects.

Contribution

It offers the first comprehensive theoretical analysis of the overlapping groups lasso, revealing its properties and limitations in structured sparsity modeling.

Findings

Finite sample bounds on prediction and estimation

Asymptotic distribution and selection results

Complex group sets can impair the method's effectiveness

Abstract

We present two sets of theoretical results on the grouped lasso with overlap of Jacob, Obozinski and Vert (2009) in the linear regression setting. This method allows for joint selection of predictors in sparse regression, allowing for complex structured sparsity over the predictors encoded as a set of groups. This flexible framework suggests that arbitrarily complex structures can be encoded with an intricate set of groups. Our results show that this strategy results in unexpected theoretical consequences for the procedure. In particular, we give two sets of results: (1) finite sample bounds on prediction and estimation, and (2) asymptotic distribution and selection. Both sets of results give insight into the consequences of choosing an increasingly complex set of groups for the procedure, as well as what happens when the set of groups cannot recover the true sparsity pattern. Additionally, these results demonstrate the differences and similarities between the the grouped lasso procedure with and without overlapping groups. Our analysis shows the set of groups must be chosen with caution - an overly complex set of groups will damage the analysis.