# High-Dimensional Differentially Private Quantile Regression: Distributed Estimation and Statistical Inference

arXiv: https://arxiv.org/abs/2508.05212 · HTML: https://mlsanity.com/paper/2508.05212/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 37 of its 68 references that are in the map.
- First posted on arXiv 7 Aug 2025.

## Builds on (most cited first, 30 of 37)

- [Coordinate Descent Algorithms](https://mlsanity.com/paper/1502.04759/) (2015)
- [cpSGD: Communication-efficient and differentially-private distributed SGD](https://mlsanity.com/paper/1805.10559/) (2018)
- [Minimax Optimal Procedures for Locally Private Estimation](https://mlsanity.com/paper/1604.02390/) (2016)
- [Robust and Differentially Private Mean Estimation](https://mlsanity.com/paper/2102.09159/) (2021)
- [DP-ADMM: ADMM-based Distributed Learning with Differential Privacy](https://mlsanity.com/paper/1808.10101/) (2018)
- [Encode, Shuffle, Analyze Privacy Revisited: Formalizations and Empirical Evaluation](https://mlsanity.com/paper/2001.03618/) (2020)
- [Differentially Private Ordinary Least Squares](https://mlsanity.com/paper/1507.02482/) (2015)
- [Optimal Algorithms for Mean Estimation under Local Differential Privacy](https://mlsanity.com/paper/2205.02466/) (2022)
- [Tight Lower Bounds for Differentially Private Selection](https://mlsanity.com/paper/1704.03024/) (2017)
- [Differentially private inference via noisy optimization](https://mlsanity.com/paper/2103.11003/) (2021)
- [Smoothed Quantile Regression with Large-Scale Inference](https://mlsanity.com/paper/2012.05187/) (2020)
- [Score Attack: A Lower Bound Technique for Optimal Differentially Private Learning](https://mlsanity.com/paper/2303.07152/) (2023)
- [Quantile Regression Under Memory Constraint](https://mlsanity.com/paper/1810.08264/) (2018)
- [Differentially Private False Discovery Rate Control](https://mlsanity.com/paper/1807.04209/) (2018)
- [On robustness and local differential privacy](https://mlsanity.com/paper/2201.00751/) (2022)
- [Differentially Private Covariance Revisited](https://mlsanity.com/paper/2205.14324/) (2022)
- [Distributed High-dimensional Regression Under a Quantile Loss Function](https://mlsanity.com/paper/1906.05741/) (2019)
- [A Bias-Accuracy-Privacy Trilemma for Statistical Estimation](https://mlsanity.com/paper/2301.13334/) (2023)
- [Distributed inference for quantile regression processes](https://mlsanity.com/paper/1701.06088/) (2017)
- [Optimal Federated Learning for Nonparametric Regression with Heterogeneous Distributed Differential Privacy Constraints](https://mlsanity.com/paper/2406.06755/) (2024)
- [Federated Transfer Learning with Differential Privacy](https://mlsanity.com/paper/2403.11343/) (2024)
- [Communication-Constrained Distributed Quantile Regression with Optimal Statistical Guarantees](https://mlsanity.com/paper/2110.13113/) (2021)
- [Private Heterogeneous Federated Learning Without a Trusted Server Revisited: Error-Optimal and Communication-Efficient Algorithms for Convex Losses](https://mlsanity.com/paper/2407.09690/) (2024)
- [Differentially Private High Dimensional Sparse Covariance Matrix Estimation](https://mlsanity.com/paper/1901.06413/) (2019)
- [Private Mean Estimation with Person-Level Differential Privacy](https://mlsanity.com/paper/2405.20405/) (2024)
- [Federated Nonparametric Hypothesis Testing with Differential Privacy Constraints: Optimal Rates and Adaptive Tests](https://mlsanity.com/paper/2406.06749/) (2024)
- [Uniform Inference for High-dimensional Quantile Regression: Linear Functionals and Regression Rank Scores](https://mlsanity.com/paper/1702.06209/) (2017)
- [Minimax And Adaptive Transfer Learning for Nonparametric Classification under Distributed Differential Privacy Constraints](https://mlsanity.com/paper/2406.20088/) (2024)
- [A General Framework for Robust Testing and Confidence Regions in High-Dimensional Quantile Regression](https://mlsanity.com/paper/1412.8724/) (2014)
- [Distributed Bootstrap for Simultaneous Inference Under High Dimensionality](https://mlsanity.com/paper/2102.10080/) (2021)

## Built on by (most cited first, 1 of 1)

- [How Much is Left? LLMs Linearly Encode Their Remaining Output Length](https://mlsanity.com/paper/2607.05316/) (2026)

## Abstract (arXiv)

With the development of big data and machine learning, privacy concerns have become increasingly critical, especially when handling heterogeneous datasets containing sensitive personal information. Differential privacy provides a rigorous framework for safeguarding individual privacy while enabling meaningful statistical analysis. In this paper, we propose a differentially private quantile regression method for high-dimensional data in a distributed setting. Quantile regression is a powerful and robust tool for modeling the relationships between the covariates and responses in the presence of outliers or heavy-tailed distributions. To address the computational challenges due to the non-smoothness of the quantile loss function, we introduce a Newton-type transformation that reformulates the quantile regression task into an ordinary least squares problem. Building on this, we develop a differentially private estimation algorithm with iterative updates, ensuring both near-optimal statistical accuracy and formal privacy guarantees. For inference, we further propose a differentially private debiased estimator, which enables valid confidence interval construction and hypothesis testing. Additionally, we propose a communication-efficient and differentially private bootstrap for simultaneous hypothesis testing in high-dimensional quantile regression, suitable for distributed settings with both small and abundant local data. Extensive simulations demonstrate the robustness and effectiveness of our methods in practical scenarios.