# Distributed High-dimensional Regression Under a Quantile Loss Function

arXiv: https://arxiv.org/abs/1906.05741 · HTML: https://mlsanity.com/paper/1906.05741/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 13 Jun 2019.

## Built on by (most cited first, 12 of 12)

- [Fast Composite Optimization and Statistical Recovery in Federated Learning](https://mlsanity.com/paper/2207.08204/) (2022)
- [Variance Reduced Median-of-Means Estimator for Byzantine-Robust Distributed Inference](https://mlsanity.com/paper/2103.02860/) (2021)
- [Fast and Robust Sparsity Learning over Networks: A Decentralized Surrogate Median Regression Approach](https://mlsanity.com/paper/2202.05498/) (2022)
- [Distributed Sparse Feature Selection in Communication-Restricted Networks](https://mlsanity.com/paper/2111.02802/) (2021)
- [Privacy-Preserving Community Detection for Locally Distributed Multiple Networks](https://mlsanity.com/paper/2306.15709/) (2023)
- [Distributed High-Dimensional Quantile Regression: Estimation Efficiency and Support Recovery](https://mlsanity.com/paper/2405.07552/) (2024)
- [Distributed Semi-Supervised Sparse Statistical Inference](https://mlsanity.com/paper/2306.10395/) (2023)
- [High-Dimensional Differentially Private Quantile Regression: Distributed Estimation and Statistical Inference](https://mlsanity.com/paper/2508.05212/) (2025)
- [Distributed Sparse Linear Regression under Communication Constraints](https://mlsanity.com/paper/2301.04022/) (2023)
- [Communication-Efficient Distributed Estimation and Inference for Cox's Model](https://mlsanity.com/paper/2302.12111/) (2023)
- [Distributed Linear Regression with Compositional Covariates](https://mlsanity.com/paper/2310.13969/) (2023)
- [A Selective Review on Statistical Methods for Massive Data Computation: Distributed Computing, Subsampling, and Minibatch Techniques](https://mlsanity.com/paper/2403.11163/) (2024)

## Abstract (arXiv)

This paper studies distributed estimation and support recovery for high-dimensional linear regression model with heavy-tailed noise. To deal with heavy-tailed noise whose variance can be infinite, we adopt the quantile regression loss function instead of the commonly used squared loss. However, the non-smooth quantile loss poses new challenges to high-dimensional distributed estimation in both computation and theoretical development. To address the challenge, we transform the response variable and establish a new connection between quantile regression and ordinary linear regression. Then, we provide a distributed estimator that is both computationally and communicationally efficient, where only the gradient information is communicated at each iteration. Theoretically, we show that, after a constant number of iterations, the proposed estimator achieves a near-oracle convergence rate without any restriction on the number of machines. Moreover, we establish the theoretical guarantee for the support recovery. The simulation analysis is provided to demonstrate the effectiveness of our method.