## The Annals of Statistics

### Strong oracle optimality of folded concave penalized estimation

#### 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.

#### Article information

Source
Ann. Statist., Volume 42, Number 3 (2014), 819-849.

Dates
First available in Project Euclid: 20 May 2014

Permanent link to this document
https://projecteuclid.org/euclid.aos/1400592644

Digital Object Identifier
doi:10.1214/13-AOS1198

Mathematical Reviews number (MathSciNet)
MR3210988

Zentralblatt MATH identifier
1305.62252

Subjects
Primary: 62J07: Ridge regression; shrinkage estimators

#### Citation

Fan, Jianqing; Xue, Lingzhou; Zou, Hui. Strong oracle optimality of folded concave penalized estimation. Ann. Statist. 42 (2014), no. 3, 819--849. doi:10.1214/13-AOS1198. https://projecteuclid.org/euclid.aos/1400592644

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#### Supplemental materials

• Supplementary material: Supplement to “Strong oracle optimality of folded concave penalized estimation”. In this supplementary note, we give the complete proof of Theorem 5 and some comments on the simulation studies.