A Parallel Algorithm for Large-Scale Nonconvex Penalized Quantile Regression

• Published in: Journal of computational and graphical statistics Vol. 26; no. 4; pp. 935 - 939
• Main Authors: Yu, Liqun, Lin, Nan, Wang, Lan
• Format: Journal Article
• Language: English
• Published: Alexandria Taylor & Francis 02.10.2017; American Statistical Association, Institute of Mathematical Statistics, and Interface Foundation of North America; Taylor & Francis Ltd
• Subjects: ADMM; Algorithms; Computation; Descent; Dimensional analysis; Iterative methods; Nonconvex penalty; Parallelization; Quantile regression and single-loop algorithm; Regression analysis; Short Technical Notes; Statistical analysis; Studies
• ISSN: 1061-8600; 1537-2715; 1537-2715
• DOI: 10.1080/10618600.2017.1328366

Penalized quantile regression (PQR) provides a useful tool for analyzing high-dimensional data with heterogeneity. However, its computation is challenging due to the nonsmoothness and (sometimes) the nonconvexity of the objective function. An iterative coordinate descent algorithm (QICD) was recently proposed to solve PQR with nonconvex penalty. The QICD significantly improves the computational speed but requires a double-loop. In this article, we propose an alternative algorithm based on the alternating direction method of multiplier (ADMM). By writing the PQR into a special ADMM form, we can solve the iterations exactly without using coordinate descent. This results in a new single-loop algorithm, which we refer to as the QPADM algorithm. The QPADM demonstrates favorable performance in both computational speed and statistical accuracy, particularly when the sample size n and/or the number of features p are large. Supplementary material for this article is available online.