A primal and dual active set algorithm for truncated $L_1$ regularized logistic regression

Truncated $ L_1 $ regularization [2] is one type of approximation to the original $ L_0 $ regularization, and it admits the hard thresholding operator. Thus we consider the truncated $ L_1 $ regularization for variable selection and estimation in the high-dimensional and sparse logistic regression models. Computationally, motivated by the KKT conditions of the truncated $ L_1 $ regularized problem, we propose a primal and dual active set algorithm (PDAS). In PDAS, it first distinguishes the active sets with small size through the primal and dual variables in the previous iteration, then the primal variable is updated by the maximum likelihood estimation limited to the active set and the dual variable is updated explicitly based on the gradient information. Further, we consider a sequential PDAS (SPDAS) with a warm-start and continual strategy. Numerous simulation studies illustrate the effectiveness of the proposed method, and the application is also demonstrate by analysing some binary classification data sets.