Feature-splitting algorithms for ultrahigh dimensional quantile regression

This paper is concerned with computational issues related to penalized quantile regression (PQR) with ultrahigh dimensional predictors. Various algorithms have been developed for PQR, but they become ineffective and/or infeasible in the presence of ultrahigh dimensional predictors due to the storage and scalability limitations. The variable updating schema of the feature-splitting algorithm that directly applies the ordinary alternating direction method of multiplier (ADMM) to ultrahigh dimensional PQR may make the algorithm fail to converge. To tackle this hurdle, we propose an efficient and parallelizable algorithm for ultrahigh dimensional PQR based on the three-block ADMM. The compatibility of the proposed algorithm with parallel computing alleviates the storage and scalability limitations of a single machine in the large-scale data processing. We establish the rate of convergence of the newly proposed algorithm. In addition, Monte Carlo simulations are conducted to compare the finite sample performance of the proposed algorithm with that of other existing algorithms. The numerical comparison implies that the proposed algorithm significantly outperforms the existing ones. We further illustrate the proposed algorithm via an empirical analysis of a real-world data set.