Sparse optimization for nonconvex group penalized estimation

Abstract

We consider a linear regression model where there are group structures in covariates. The group LASSO has been proposed for group variable selections. Many nonconvex penalties such as smoothly clipped absolute deviation and minimax concave penalty were extended to group variable selection problems. The group coordinate descent (GCD) algorithm is used popularly for fitting these models. However, the GCD algorithms are hard to be applied to nonconvex group penalties due to computational complexity unless the design matrix is orthogonal. In this paper, we propose an efficient optimization algorithm for nonconvex group penalties by combining the concave convex procedure and the group LASSO algorithm. We also extend the proposed algorithm for generalized linear models. We evaluate numerical efficiency of the proposed algorithm compared to existing GCD algorithms through simulated data and real data sets.

Keywords:

• concave convex procedure
• group LASSO
• nonconvex penalty
• variable selection

Acknowledgments

We would like to thank the editor, the associate editor and the anonymous referees for their insightful comments.

Disclosure statement

No potential conflict of interest was reported by the authors.

Funding statement

This research was supported in part by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education [No. 2013R1A6A3A03026588] and the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) [No. 2014R1A2A2A01004496].