Standardization and the Group Lasso Penalty.

We re-examine the original Group Lasso paper of Yuan and Lin (2007). The form of penalty in that paper seems to be designed for problems with uncorrelated features, but the statistical community has adopted it for general problems with correlated features. We show that for this general situation, a Group Lasso with a different choice of penalty matrix is generally more effective. We give insight into this formulation and show that it is intimately related to the uniformly most powerful invariant test for inclusion of a group. We demonstrate the efficacy of this method- the "standardized Group Lasso"- over the usual group lasso on real and simulated data sets. We also extend this to the Ridged Group Lasso to provide within group regularization as needed. We discuss a simple algorithm based on group-wise coordinate descent to fit both this standardized Group Lasso and Ridged Group Lasso.