# High-Dimensional Quantile Regression: Convolution Smoothing and Concave Regularization

arXiv: https://arxiv.org/abs/2109.05640 · HTML: https://mlsanity.com/paper/2109.05640/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 Sep 2021.

## Built on by (most cited first, 15 of 15)

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- [Communication-Constrained Distributed Quantile Regression with Optimal Statistical Guarantees](https://mlsanity.com/paper/2110.13113/) (2021)
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- [fastkqr: A Fast Algorithm for Kernel Quantile Regression](https://mlsanity.com/paper/2408.05393/) (2024)
- [Distributed Estimation and Inference for Semi-parametric Binary Response Models](https://mlsanity.com/paper/2210.08393/) (2022)
- [Private Optimal Inventory Policy Learning for Feature-based Newsvendor with Unknown Demand](https://mlsanity.com/paper/2404.15466/) (2024)
- [Smoothed Robust Phase Retrieval](https://mlsanity.com/paper/2409.01570/) (2024)
- [A Pathwise Coordinate Descent Algorithm for LASSO Penalized Quantile Regression](https://mlsanity.com/paper/2502.12363/) (2025)
- [Efficient Distributed Learning over Decentralized Networks with Convoluted Support Vector Machine](https://mlsanity.com/paper/2503.07563/) (2025)
- [High-dimensional online learning via asynchronous decomposition: Non-divergent results, dynamic regularization, and beyond](https://mlsanity.com/paper/2603.20696/) (2026)
- [ConquerNet: Convolution-Smoothed Quantile ReLU Neural Networks with Minimax Guarantees](https://mlsanity.com/paper/2605.06265/) (2026)
- [Convolution Smoothed Quantile Regression for XGBoost](https://mlsanity.com/paper/2608.15290/) (2026)
- [Revisiting Tabular Deep Regression: Objectives and the Limits of Global Early Stopping](https://mlsanity.com/paper/2610.13909/) (2026)

## Abstract (arXiv)

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.