# Communication-Constrained Distributed Quantile Regression with Optimal Statistical Guarantees

arXiv: https://arxiv.org/abs/2110.13113 · HTML: https://mlsanity.com/paper/2110.13113/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 57 references that are in the map.
- First posted on arXiv 25 Oct 2021.

## Builds on (most cited first, 17 of 17)

- [Federated Learning: Challenges, Methods, and Future Directions](https://mlsanity.com/paper/1908.07873/) (2019)
- [Communication Efficient Distributed Optimization using an Approximate Newton-type Method](https://mlsanity.com/paper/1312.7853/) (2013)
- [Divide and Conquer Kernel Ridge Regression: A Distributed Algorithm with Minimax Optimal Rates](https://mlsanity.com/paper/1305.5029/) (2013)
- [Adaptive Huber Regression](https://mlsanity.com/paper/1706.06991/) (2017)
- [Communication-Efficient Algorithms for Decentralized and Stochastic Optimization](https://mlsanity.com/paper/1701.03961/) (2017)
- [Efficient Distributed Learning with Sparsity](https://mlsanity.com/paper/1605.07991/) (2016)
- [Communication-Efficient Accurate Statistical Estimation](https://mlsanity.com/paper/1906.04870/) (2019)
- [Program Evaluation and Causal Inference with High-Dimensional Data](https://mlsanity.com/paper/1311.2645/) (2013)
- [Smoothed Quantile Regression with Large-Scale Inference](https://mlsanity.com/paper/2012.05187/) (2020)
- [Quantile Regression Under Memory Constraint](https://mlsanity.com/paper/1810.08264/) (2018)
- [First-order Newton-type Estimator for Distributed Estimation and Inference](https://mlsanity.com/paper/1811.11368/) (2018)
- [Smoothing quantile regressions](https://mlsanity.com/paper/1905.08535/) (2019)
- [High-Dimensional Quantile Regression: Convolution Smoothing and Concave Regularization](https://mlsanity.com/paper/2109.05640/) (2021)
- [Semismooth Newton Coordinate Descent Algorithm for Elastic-Net Penalized Huber Loss Regression and Quantile Regression](https://mlsanity.com/paper/1509.02957/) (2015)
- [Uniform Inference for High-dimensional Quantile Regression: Linear Functionals and Regression Rank Scores](https://mlsanity.com/paper/1702.06209/) (2017)
- [Smoothed estimating equations for instrumental variables quantile regression](https://mlsanity.com/paper/1609.09033/) (2016)
- [Simultaneous Inference for Massive Data: Distributed Bootstrap](https://mlsanity.com/paper/2002.08443/) (2020)

## Built on by (most cited first, 8 of 8)

- [Distributed High-Dimensional Quantile Regression: Estimation Efficiency and Support Recovery](https://mlsanity.com/paper/2405.07552/) (2024)
- [SpeedCP: Fast Kernel-based Conditional Conformal Prediction](https://mlsanity.com/paper/2509.24100/) (2025)
- [How Much is Left? LLMs Linearly Encode Their Remaining Output Length](https://mlsanity.com/paper/2607.05316/) (2026)
- [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)
- [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)
- [Renewable Lasso without Batch-Number Constraints: A Gradient-Enhanced Approach](https://mlsanity.com/paper/2606.11738/) (2026)

## Abstract (arXiv)

We address the problem of how to achieve optimal inference in distributed quantile regression without stringent scaling conditions. This is challenging due to the non-smooth nature of the quantile regression (QR) loss function, which invalidates the use of existing methodology. The difficulties are resolved through a double-smoothing approach that is applied to the local (at each data source) and global objective functions. Despite the reliance on a delicate combination of local and global smoothing parameters, the quantile regression model is fully parametric, thereby facilitating interpretation. In the low-dimensional regime, we establish a finite-sample theoretical framework for the sequentially defined distributed QR estimators. This reveals a trade-off between the communication cost and statistical error. We further discuss and compare several alternative confidence set constructions, based on inversion of Wald and score-type tests and resampling techniques, detailing an improvement that is effective for more extreme quantile coefficients. In high dimensions, a sparse framework is adopted, where the proposed doubly-smoothed objective function is complemented with an ℓ1-penalty. We show that the corresponding distributed penalized QR estimator achieves the global convergence rate after a near-constant number of communication rounds. A thorough simulation study further elucidates our findings.