A Unified Algorithm for Penalized Convolution Smoothed Quantile Regression

Abstract

Penalized quantile regression (QR) is widely used for studying the relationship between a response variable and a set of predictors under data heterogeneity in high-dimensional settings. Compared to penalized least squares, scalable algorithms for fitting penalized QR are lacking due to the non-differentiable piecewise linear loss function. To overcome the lack of smoothness, a recently proposed convolution-type smoothed method brings an interesting tradeoff between statistical accuracy and computational efficiency for both standard and penalized quantile regressions. In this article, we propose a unified algorithm for fitting penalized convolution smoothed quantile regression with various commonly used convex penalties, accompanied by an R-language package conquer available from the Comprehensive R Archive Network. We perform extensive numerical studies to demonstrate the superior performance of the proposed algorithm over existing methods in both statistical and computational aspects. We further exemplify the proposed algorithm by fitting a fused lasso additive QR model on the world happiness data.

Keywords:

• Convolution smoothing
• Lasso
• Majorize-minimization algorithm
• Penalized optimization
• Quantile estimation regression

Acknowledgments

We are grateful to the Editor, the Associate Editor, and anonymous reviewers for their constructive comments and suggestions that have significantly improved the article.

Disclosure Statement

No potential conflict of interest was reported by the author(s).

Notes

1 The rqPen package does not have the elastic net penalty option.

Additional information

Funding

K. M. Tan was supported by NSF Grants DMS-2113356 and NSF DMS-2238428. W.-X. Zhou acknowledges the support of the NSF Grant DMS-2113409.