Strong oracle optimality of folded concave penalized estimation

TL;DR

This paper establishes a unified theoretical framework showing how local linear approximation algorithms can reliably find oracle solutions in folded concave penalized estimation, ensuring strong theoretical properties in high-dimensional sparse models.

Contribution

It provides the first comprehensive theory demonstrating that local linear approximation algorithms can achieve oracle solutions in folded concave penalized estimation, addressing a decade-long open problem.

Findings

One-step local linear approximation yields the oracle estimator.

The algorithm converges to the oracle estimator once obtained.

The theory applies to multiple classical sparse estimation problems.

Abstract

Folded concave penalization methods have been shown to enjoy the strong oracle property for high-dimensional sparse estimation. However, a folded concave penalization problem usually has multiple local solutions and the oracle property is established only for one of the unknown local solutions. A challenging fundamental issue still remains that it is not clear whether the local optimum computed by a given optimization algorithm possesses those nice theoretical properties. To close this important theoretical gap in over a decade, we provide a unified theory to show explicitly how to obtain the oracle solution via the local linear approximation algorithm. For a folded concave penalized estimation problem, we show that as long as the problem is localizable and the oracle estimator is well behaved, we can obtain the oracle estimator by using the one-step local linear approximation. In addition, once the oracle estimator is obtained, the local linear approximation algorithm converges, namely it produces the same estimator in the next iteration. The general theory is demonstrated by using four classical sparse estimation problems, that is, sparse linear regression, sparse logistic regression, sparse precision matrix estimation and sparse quantile regression.