Asymptotic properties for combined $L_1$ and concave regularization

TL;DR

This paper analyzes the asymptotic behavior of a combined $L_1$ and concave regularization method for high-dimensional linear models, demonstrating oracle properties and improved stability over single-penalty approaches.

Contribution

It provides theoretical guarantees for the global optimum of combined regularization, including oracle inequalities and false sign rate bounds, in ultra-high dimensional settings.

Findings

Global optimum enjoys oracle inequalities similar to lasso

False sign rate can asymptotically vanish

Numerical studies show more stable estimates than using concave penalty alone

Abstract

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.