Debiased distributed quantile regression in high dimensions

This paper concerns the debiased distributed estimation for the linear model in high dimensions with arbitrary noise distribution. Quantile regression (QR) is adopted to safeguard potential heavy-tailed noises. To tackle the computational challenges accompanied by the non-smooth QR loss, we cast the QR loss into a least-squares loss by constructing new pseudo responses. We further equip the new least-squares loss with the $\ell_1$ penalty to accomplish tasks of coefficient estimation and variable selection. To eliminate the bias brought by the $\ell_1$ penalty, we correct the bias of nonzero coefficient estimation for each local machine and aggregate all the local debiased estimators through averaging. Our distributed algorithm is guaranteed to converge in a finite number of iterations. Theoretically, we show that the resulting estimator can consistently recover the sparsity pattern and achieve a near-oracle convergence rate. We conduct extensive numerical studies to demonstrate the competitive finite sample performance of our method.

Keywords

distributed estimation, lasso, high-dimensional quantile regression, bias-correction

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Received 13 July 2022

Accepted 12 September 2022

Published 19 July 2024