Distributed high-dimensional quantile regression with the elastic net

This paper investigates the problem of high-dimensional quantile regression in a distributed framework. Using the Newton–Raphson method, we transform the quantile regression problem to a ridge regression problem by creating a new response variable, and then finally transform it to an Elastic Net regression problem by adding an ℓ 1 -norm penalty. This not only solves the problem of the non-smooth quantile loss function, but also selects the Elastic Net regression as an effective tool for robust estimation and variable selection in high-dimensional data analysis. Then, we propose two distributed estimators that are computationally and communicationally efficient, the naive Elastic Net distributed estimator and the Elastic Net distributed estimator, respectively. Theoretically, after a certain number of iterations, both of our proposed estimators can accomplish a near-oracle convergence rates. Simulation analysis shows that our methods have good convergence rates and model selection capabilities. We further apply our methods to a real data to show their effectiveness.