Asymptotic properties for combined $L_1$ and concave regularization
This work addresses variable selection and prediction accuracy for statisticians and data scientists in high-dimensional settings, representing an incremental improvement over existing regularization methods.
The paper tackles the problem of prediction and variable selection in high-dimensional linear models by combining L1 and concave penalties, proving that the global optimum achieves oracle prediction risk and asymptotically vanishing false sign rates, with numerical studies showing more stable estimates than using concave penalty alone.
Two important goals of high-dimensional modeling are prediction and variable selection. In this article, we consider regularization with combined $L_1$ and concave penalties, and study the sampling properties of the global optimum of the suggested method in ultra-high dimensional settings. The $L_1$-penalty provides the minimum regularization needed for removing noise variables in order to achieve oracle prediction risk, while concave penalty imposes additional regularization to control model sparsity. In the linear model setting, we prove that the global optimum of our method enjoys the same oracle inequalities as the lasso estimator and admits an explicit bound on the false sign rate, which can be asymptotically vanishing. Moreover, we establish oracle risk inequalities for the method and the sampling properties of computable solutions. Numerical studies suggest that our method yields more stable estimates than using a concave penalty alone.