A Primal-Dual Algorithm for Group Sparse Regularization with Overlapping Groups

We deal with the problem of variable selection when variables must be selected group-wise, with possibly overlapping groups defined a priori. In particular we propose a new optimization procedure for solving the regularized algorithm presented in [12], where the group lasso penalty is generalized to overlapping groups of variables. While in [12] the proposed implementation requires explicit replication of the variables belonging to more than one group, our iterative procedure is based on a combination of proximal methods in the primal space and projected Newton method in a reduced dual space, corresponding to the active groups. This procedure provides a scalable alternative with no need for data duplication, and allows to deal with high dimensional problems without pre-processing for dimensionality reduction. The computational advantages of our scheme with respect to state-of-the-art algorithms using data duplication are shown empirically with numerical simulations.