Theoretical properties of the overlapping groups lasso

We present two sets of theoretical results on the grouped lasso with overlap due to Jacob, Obozinski and Vert (2009) in the linear regression setting. This method jointly selects 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 demonstrate negative consequences from choosing an increasingly complex set of groups for the procedure, as well for 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 that while the procedure enjoys advantages over the standard lasso, the set of groups must be chosen with caution - an overly complex set of groups will damage the analysis.