Simultaneous Feature- and Sample-Splitting ADMM for Penalized Rank Regression*

This work is concerned with computational issues related to penalized rank regression, which has been proposed for variable selection in rank regression. Although rank regression is an effective analytic tool for dealing with heavy-tailed error and outlier contamination due to its robustness and efficiency properties, rank regression and penalized rank regression are not scalable because rank regression is typically carried out by median regression of the pairwise difference of responses on pairwise difference of covariate vectors. To address computational challenge in penalized rank regression, we proposed simultaneous feature- and sample-splitting ADMM algorithms for high-dimensional penalized rank regression with a large sample size. The proposed algorithm may deal with the memory limitations associated with single-machine algorithms in a parallel computing framework. We systematically study the theoretical property of the proposed algorithms, and show that under mild conditions, the coefficient sequence generated by the proposed algorithm converges to the unique primal minimizer. We conducted Monte Carlo simulation studies to compare the proposed algorithms with existing ones. The numerical comparison implies that the proposed algorithms outperform existing ones in terms of estimation accuracy and capacity to deal with memory limitations.