Semismooth Newton Coordinate Descent Algorithm for Elastic-Net Penalized Huber Loss Regression and Quantile Regression
This work addresses computational bottlenecks for statisticians and data scientists working with high-dimensional robust regression models, representing an incremental improvement over existing coordinate descent methods.
The authors tackled the computational challenges of high-dimensional elastic-net penalized Huber loss and quantile regression by proposing the semismooth Newton coordinate descent (SNCD) algorithm, which updates coefficients and subgradients simultaneously, and demonstrated its efficiency and scalability to ultra-high dimensions in numerical experiments.
We propose an algorithm, semismooth Newton coordinate descent (SNCD), for the elastic-net penalized Huber loss regression and quantile regression in high dimensional settings. Unlike existing coordinate descent type algorithms, the SNCD updates each regression coefficient and its corresponding subgradient simultaneously in each iteration. It combines the strengths of the coordinate descent and the semismooth Newton algorithm, and effectively solves the computational challenges posed by dimensionality and nonsmoothness. We establish the convergence properties of the algorithm. In addition, we present an adaptive version of the "strong rule" for screening predictors to gain extra efficiency. Through numerical experiments, we demonstrate that the proposed algorithm is very efficient and scalable to ultra-high dimensions. We illustrate the application via a real data example.