A PROXIMAL DUAL SEMISMOOTH NEWTON METHOD FOR ZERO-NORM PENALIZED QUANTILE REGRESSION ESTIMATOR

作者：Zhang, Dongdong; Pan, Shaohua; Bi, Shujun^*

来源：Statistica Sinica, 2022, 32(2): 1121-1141.

DOI：10.5705/ss.202019.0415

• Key veords and phrases
• High -dimension
• proximal dual semismooth Newton method
• variable selection
• zero -norm penalized quantile regression
• nonGaussian error distributions

This study examines the computation of the high-dimensional zero-norm penalized quantile regression estimator, defined as the global minimizer of the zeronorm penalized check loss function. To seek a desirable approximation to the estimator, we reformulate this NP-hard problem as an equivalent augmented Lipschitz optimization problem. Then, we exploit its coupled structure to propose a multistage convex relaxation approach (MSCRA PPA), each step of which solves inexactly a weighted `1-regularized check loss minimization problem using a proximal dual semismooth Newton method. Under a restricted strong convexity condition, we provide a theoretical guarantee for the MSCRA PPA by establishing the error bound of each iterate to the true estimator and the rate of linear convergence in a statistical sense. Numerical comparisons using synthetic and real data show that the MSCRA PPA exhibits comparable or better estimation performance and requires much less CPU time.