# Smoothed Quantile Regression with Large-Scale Inference

arXiv: https://arxiv.org/abs/2012.05187 · HTML: https://mlsanity.com/paper/2012.05187/

## Key facts

- Cited by 23 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Dec 2020.

## Built on by (most cited first, 23 of 23)

- [Statistical Properties of the log-cosh Loss Function Used in Machine Learning](https://mlsanity.com/paper/2208.04564/) (2022)
- [Communication-Constrained Distributed Quantile Regression with Optimal Statistical Guarantees](https://mlsanity.com/paper/2110.13113/) (2021)
- [Efficient Sparse Least Absolute Deviation Regression with Differential Privacy](https://mlsanity.com/paper/2401.01294/) (2024)
- [Selective inference using randomized group lasso estimators for general models](https://mlsanity.com/paper/2306.13829/) (2023)
- [Distributed High-Dimensional Quantile Regression: Estimation Efficiency and Support Recovery](https://mlsanity.com/paper/2405.07552/) (2024)
- [Federated Learning of Quantile Inference under Local Differential Privacy](https://mlsanity.com/paper/2509.21800/) (2025)
- [Non-Asymptotic Analysis of Efficiency in Conformalized Regression](https://mlsanity.com/paper/2510.07093/) (2025)
- [Solution to the Non-Monotonicity and Crossing Problems in Quantile Regression](https://mlsanity.com/paper/2111.04805/) (2021)
- [Safe Collaborative Filtering](https://mlsanity.com/paper/2306.05292/) (2023)
- [fastkqr: A Fast Algorithm for Kernel Quantile Regression](https://mlsanity.com/paper/2408.05393/) (2024)
- [High-Dimensional Differentially Private Quantile Regression: Distributed Estimation and Statistical Inference](https://mlsanity.com/paper/2508.05212/) (2025)
- [Distributed Estimation and Inference for Semi-parametric Binary Response Models](https://mlsanity.com/paper/2210.08393/) (2022)
- [High-dimensional Measurement Error Models for Lipschitz Loss](https://mlsanity.com/paper/2210.15008/) (2022)
- [Private Optimal Inventory Policy Learning for Feature-based Newsvendor with Unknown Demand](https://mlsanity.com/paper/2404.15466/) (2024)
- [Fast networked data selection via distributed smoothed quantile estimation](https://mlsanity.com/paper/2406.01929/) (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)
- [Inertial Quadratic Majorization Minimization with Application to Kernel Regularized Learning](https://mlsanity.com/paper/2507.04247/) (2025)
- [Nonparametric efficient inference for network quantile causal effects under partial interference](https://mlsanity.com/paper/2604.13008/) (2026)
- [ConquerNet: Convolution-Smoothed Quantile ReLU Neural Networks with Minimax Guarantees](https://mlsanity.com/paper/2605.06265/) (2026)
- [Byzantine-Robust Distributed Sparse Learning Revisited](https://mlsanity.com/paper/2605.13283/) (2026)
- [Convolution Smoothed Quantile Regression for XGBoost](https://mlsanity.com/paper/2608.15290/) (2026)

## Abstract (arXiv)

Quantile regression is a powerful tool for learning the relationship between a response variable and a multivariate predictor while exploring heterogeneous effects. In this paper, we consider statistical inference for quantile regression with large-scale data in the "increasing dimension" regime. We provide a comprehensive and in-depth analysis of a convolution-type smoothing approach that achieves adequate approximation to computation and inference for quantile regression. This method, which we refer to as itconquer, turns the non-differentiable quantile loss function into a twice-differentiable, convex and locally strongly convex surrogate, which admits a fast and scalable Barzilai-Borwein gradient-based algorithm to perform optimization, and multiplier bootstrap for statistical inference. Theoretically, we establish explicit non-asymptotic bounds on both estimation and Bahadur-Kiefer linearization errors, from which we show that the asymptotic normality of the conquer estimator holds under a weaker requirement on the number of the regressors than needed for conventional quantile regression. Moreover, we prove the validity of multiplier bootstrap confidence constructions. Our numerical studies confirm the conquer estimator as a practical and reliable approach to large-scale inference for quantile regression. Software implementing the methodology is available in the textttR package textttconquer.