[Paper Review] Communication-Constrained Distributed Quantile Regression with Optimal Statistical Guarantees

Kean Ming Tan, Heather Battey+1 more|arXiv (Cornell University)|Oct 25, 2021

Statistical Methods and InferenceMathematics18 citations

TL;DR

This paper proposes a communication-efficient distributed quantile regression method using a double-smoothing approach to handle the non-smooth loss function, achieving optimal statistical guarantees with minimal communication. It establishes finite-sample theory for low-dimensional settings and shows that a sparse, penalized version achieves global convergence rates in near-constant communication rounds in high dimensions.

ABSTRACT

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 $\ell_1$-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.

Motivation & Objective

• Address the challenge of achieving optimal statistical inference in distributed quantile regression under communication constraints.
• Overcome the non-smooth nature of the quantile regression loss function, which invalidates standard distributed methods.
• Develop a theoretically grounded framework for distributed inference that balances communication cost and statistical accuracy.
• Extend the methodology to high-dimensional settings using an ℓ1-penalized, doubly-smoothed objective function.
• Provide practical confidence set constructions with improved coverage, especially for extreme quantile coefficients.

Proposed method

• Introduce a double-smoothing technique: smoothing both the local (per-data-source) and global objective functions to handle non-smoothness.
• Use a sequentially defined distributed estimator that iteratively refines estimates across communication rounds.
• Apply multiplier bootstrap and score-type test inversion for constructing confidence sets, with improvements for extreme quantiles.
• In high dimensions, combine the doubly-smoothed objective with an ℓ1-penalty to induce sparsity and achieve optimal convergence rates.
• Control the trade-off between communication cost and statistical error through adaptive smoothing parameters.
• Establish theoretical guarantees via finite-sample analysis in low dimensions and asymptotic convergence rates in high dimensions.

Experimental results

Research questions

• RQ1 Can optimal statistical inference be achieved in distributed quantile regression without stringent scaling conditions on the number of data sources?
• RQ2 How can the non-smooth quantile regression loss be effectively handled in a distributed setting to ensure statistical optimality?
• RQ3 What is the trade-off between communication cost and statistical error in distributed quantile regression?
• RQ4 How do different confidence set construction methods—Wald, score, bootstrap—perform in terms of coverage and width under distributed computation?
• RQ5 Can a sparse, penalized distributed quantile regression estimator achieve the global convergence rate with near-constant communication rounds?

Key findings

• The double-smoothing approach successfully mitigates the non-smoothness of the quantile regression loss, enabling optimal inference in distributed settings.
• In low dimensions, the method achieves optimal convergence rates with a finite-sample theoretical framework, revealing a clear trade-off between communication cost and statistical error.
• For confidence intervals, the CE-Boot (b) and CE-Score methods show superior coverage, especially for extreme quantiles, with coverage probabilities near 0.95 in simulations.
• In high dimensions, the ℓ1-penalized, doubly-smoothed estimator achieves the global convergence rate after only a near-constant number of communication rounds.
• Simulation results show that the DC-Normal method has poor coverage (e.g., ~0.25 for n=200, m=200), while CE-Boot and CE-Score achieve coverage near 0.95–0.98.
• The mean width of confidence intervals is minimized under the CE-Boot (b) and CE-Score methods, indicating efficient precision trade-offs.