Nonconvex Dantzig selector and its parallel computing algorithm

The Dantzig selector is a popular \(\ell _1\)-type variable selection method widely used across various research fields. However, \(\ell _1\)-type methods may not perform well for variable selection without complex irrepresentable conditions. In this article, we introduce a nonconvex Dantzig selector for ultrahigh-dimensional linear models. We begin by demonstrating that the oracle estimator serves as a local optimum for the nonconvex Dantzig selector. In addition, we propose a one-step local linear approximation estimator, called the Dantzig-LLA estimator, for the nonconvex Dantzig selector, and establish its strong oracle property. The proposed regularization method avoids the restrictive conditions imposed by \(\ell _1\) regularization methods to guarantee the model selection consistency. Furthermore, we propose an efficient and parallelizable computing algorithm based on feature-splitting to address the computational challenges associated with the nonconvex Dantzig selector in high-dimensional settings. A comprehensive numerical study is conducted to evaluate the performance of the nonconvex Dantzig selector and the computing efficiency of the feature-splitting algorithm. The results demonstrate that the Dantzig selector with nonconvex penalty outperforms the \(\ell _1\) penalty-based selector, and the feature-splitting algorithm performs well in high-dimensional settings where linear programming solver may fail. Finally, we generalize the concept of nonconvex Dantzig selector to deal with more general loss functions.