Group SLOPE – Adaptive Selection of Groups of Predictors

ABSTRACT

Sorted L-One Penalized Estimation (SLOPE; Bogdan et al. 2013, 2015) is a relatively new convex optimization procedure, which allows for adaptive selection of regressors under sparse high-dimensional designs. Here, we extend the idea of SLOPE to deal with the situation when one aims at selecting whole groups of explanatory variables instead of single regressors. Such groups can be formed by clustering strongly correlated predictors or groups of dummy variables corresponding to different levels of the same qualitative predictor. We formulate the respective convex optimization problem, group SLOPE (gSLOPE), and propose an efficient algorithm for its solution. We also define a notion of the group false discovery rate (gFDR) and provide a choice of the sequence of tuning parameters for gSLOPE so that gFDR is provably controlled at a prespecified level if the groups of variables are orthogonal to each other. Moreover, we prove that the resulting procedure adapts to unknown sparsity and is asymptotically minimax with respect to the estimation of the proportions of variance of the response variable explained by regressors from different groups. We also provide a method for the choice of the regularizing sequence when variables in different groups are not orthogonal but statistically independent and illustrate its good properties with computer simulations. Finally, we illustrate the advantages of gSLOPE in the context of Genome Wide Association Studies. R package grpSLOPE with an implementation of our method is available on The Comprehensive R Archive Network.

KEYWORDS:

• Asymptotic minimax
• False discovery rate
• Group selection
• Model selection
• Multiple regression
• SLOPE

Acknowledgments

The authors thank Ewout van den Berg, Emmanuel J. Candès, Jan Mielniczuk and Chiara Sabatti for helpful remarks and suggestions. D. B. would like to thank Professor Jerzy Ombach for significant help with the process of obtaining access to the data.

Additional information

Funding

D. B. and M. B. are supported by European Union’s 7th Framework Programme for research, technological development and demonstration under Grant Agreement no 602552 and by the Polish Ministry of Science and Higher Education according to agreement 2932/7.PR/2013/2. Additionally, D.B. acknowledges the support from NIMH grant R01MH108467.