An ADMM approach for multi-response regression with overlapping groups and interaction effects

In this paper, we consider the regularized multi-response regression problem where there exists some structural relation within the responses and also between the covariates and a set of modifying variables. To handle this problem, we propose MADMMplasso, a novel regularized regression method. This method is able to find covariates and their corresponding interactions, with some joint association with multiple related responses. We allow the interaction term between covariate and modifying variable to be included in a (weak) asymmetrical hierarchical manner by first considering whether the corresponding covariate main term is in the model. For parameter estimation, we develop an ADMM algorithm that allows us to implement the overlapping groups in a simple way. The results from the simulations and analysis of a pharmacogenomic screen data set show that the proposed method has an advantage in handling correlated responses and interaction effects, both with respect to prediction and variable selection performance.