High-Dimensional Quantile Regression: Convolution Smoothing and Concave Regularization

2021/09/12 by Kean Ming Tan, Tan, Kean Ming, Lan Wang +4 · 8 citations
Engineering · Mathematics · Medicine · #Applied mathematics #Artificial intelligence #Computer science #Convexity #Estimator #FOS: Computer and information sciences #FOS: Mathematics #Liver Disease Diagnosis and Treatment #Mathematical optimization #Mathematics #Methodology (stat.ME) #Quantile #Quantile regression #Rate of convergence #Regularization (linguistics) #Smoothing #Sparse and Compressive Sensing Techniques #Statistical Methods and Inference #Statistics #Statistics Theory (math.ST) #math.ST #stat.ME #stat.TH

paper · pdf · doi:10.48550/arxiv.2109.05640

published in arXiv (Cornell University) (Cornell University) · Main text is 27 pages, online supplementary materials are attached after the main text

arxiv created 2021/09/12 · openalex publication_date 2021/09/12 · arxiv updated 2021/09/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/08

Abstract

ℓ₁-penalized quantile regression is widely used for analyzing high-dimensional data with heterogeneity. It is now recognized that the ℓ₁-penalty introduces non-negligible estimation bias, while a proper use of concave regularization may lead to estimators with refined convergence rates and oracle properties as the signal strengthens. Although folded concave penalized M-estimation with strongly convex loss functions have been well studied, the extant literature on quantile regression is relatively silent. The main difficulty is that the quantile loss is piecewise linear: it is non-smooth and has curvature concentrated at a single point. To overcome the lack of smoothness and strong convexity, we propose and study a convolution-type smoothed quantile regression with iteratively reweighted ℓ₁-regularization. The resulting smoothed empirical loss is twice continuously differentiable and (provably) locally strongly convex with high probability. We show that the iteratively reweighted ℓ₁-penalized smoothed quantile regression estimator, after a few iterations, achieves the optimal rate of convergence, and moreover, the oracle rate and the strong oracle property under an almost necessary and sufficient minimum signal strength condition. Extensive numerical studies corroborate our theoretical results.

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