Robust Coordinate Descent Algorithm Robust Solution Path for High-dimensional Sparse Regression Modeling

The L-1-type regularization provides a useful tool for variable selection in high-dimensional regression modeling. Various algorithms have been proposed to solve optimization problems for L-1-type regularization. Especially the coordinate descent algorithm has been shown to be effective in sparse regression modeling. Although the algorithm shows a remarkable performance to solve optimization problems for L-1-type regularization, it suffers from outliers, since the procedure is based on the inner product of predictor variables and partial residuals obtained from a non-robust manner. To overcome this drawback, we propose a robust coordinate descent algorithm, especially focusing on the high-dimensional regression modeling based on the principal components space. We show that the proposed robust algorithm converges to the minimum value of its objective function. Monte Carlo experiments and real data analysis are conducted to examine the efficiency of the proposed robust algorithm. We observe that our robust coordinate descent algorithm effectively performs for the high-dimensional regression modeling even in the presence of outliers.