Communication-efficient and distributed-oracle estimation for high-dimensional quantile regression

Abstract

In this article, we present a novel communication-efficient estimator for distributed high-dimensional quantile regression with folded-concave penalties. An Iterative Multi-Step Algorithm (IMSA) is employed to tackle the nonconvex challenge of the objective function, taking into account both the statistical accuracy and the communication constraints. We demonstrate that the proposed IMSA estimators share similar properties with the global folded-concave penalized estimator. To establish the theoretical results, we introduce a new concept called the distributed-oracle estimator. We prove that the proposed estimator converges to the distributed-oracle estimator with high probability. Compared to the ℓ1-penalized method, the IMSA estimator possesses a faster rate of convergence and requires milder conditions to achieve support recovery. Furthermore, we extend our framework to facilitate distributed inference for the preconceived low-dimensional components within the high-dimensional model. We derive the limiting distribution of the corresponding test statistic under the null hypothesis and the local alternatives. In addition, a new feature-splitting algorithm is devised to accommodate the high-dimensional data within the distributed system. Extensive numerical studies demonstrate the effectiveness and validity of our proposed estimation and inference methods. A real example is also presented for illustration.