Group selection and shrinkage with application to sparse semiparametric modeling

Sparse regression and classification estimators capable of group selection have application to an assortment of statistical problems, from multitask learning to sparse additive modeling to hierarchical selection. This work introduces a class of group-sparse estimators that combine group subset selection with group lasso or ridge shrinkage. We develop an optimization framework for fitting the nonconvex regularization surface and present finite-sample error bounds for estimation of the regression function. Our methods and analyses accommodate the general setting where groups overlap. As an application of group selection, we study sparse semiparametric modeling, a procedure that allows the effect of each predictor to be zero, linear, or nonlinear. For this task, the new estimators improve across several metrics on synthetic data compared to alternatives. Finally, we demonstrate their efficacy in modeling supermarket foot traffic and economic recessions using many predictors. All of our proposals are made available in the scalable implementation grpsel.