Exact Coordinate Descent for High-Dimensional Regularized Huber Regression

Abstract

In this study, an exact coordinate descent algorithm is developed for high-dimensional Huber regression regularized with an elastic net penalty. Unlike existing gradient descent or coordinate descent-type methods, this algorithm remains effective even when the Hessian becomes ill-conditioned due to high correlations between covariates drawn from heavy-tailed distributions. For each coordinate, marginal increments arise solely from inlier observations, while the derivatives remain monotonically increasing over a grid constructed from the partial residuals. Building on conventional coordinate descent frameworks, adaptive variable screening rules are proposed to selectively determine which variables to update at each iteration, thereby accelerating convergence. The convergence of the proposed algorithm is formally analyzed, and practical computational strategies are presented to speed up its execution. These enhancements ensure that the algorithm operates rapidly and stably even under challenging scenarios. Extensive simulation studies involving heavy-tailed noise and highly correlated predictors, along with a real-world data application, demonstrate both the practical efficiency of this method and the benefits of the computational enhancements.