Distributed High-dimensional Regression Under a Quantile Loss Function

TL;DR

This paper introduces a distributed high-dimensional regression method using quantile loss to handle heavy-tailed noise, achieving efficient computation and support recovery with theoretical guarantees.

Contribution

It develops a novel distributed estimator based on quantile regression, connecting it to linear regression, and provides theoretical analysis for convergence and support recovery.

Findings

Estimator achieves near-oracle convergence rate after a few iterations

Supports recovery guarantees are established

Simulation results demonstrate effectiveness

Abstract

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.