A Parallel Algorithm for Large-Scale Nonconvex Penalized Quantile Regression

Liqun Yu, Nan Lin, Lan Wang

Research output: Contribution to journalComment/debatepeer-review

39 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)935-939
Number of pages5
JournalJournal of Computational and Graphical Statistics
Issue number4
StatePublished - Oct 2 2017

Bibliographical note

Funding Information:
The authors thank the Editor, Dr. Dianne Cook, and an associate editor for their helpful comments and suggestions that greatly improved the article. Lan Wang’s research was partially supported by the National Science Foundation (NSF grant DMS-1512267).


  • ADMM
  • Nonconvex penalty
  • Parallelization
  • Quantile regression and single-loop algorithm


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