Strong Oracle Optimality of Folded Concave Penalized Estimation

Folded concave penalization methods (Fan and Li, 2001) 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 optimal solution 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 using 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 produces the same estimator in the next iteration. The general theory is demonstrated by using three classical sparse estimation problems, i.e. the sparse linear regression, the sparse logistic regression and the sparse precision matrix estimation, where the LASSO penalized least squares, the LASSO penalized logistic regression and the CLIME are used as the initial estimator, respectively.