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June 2014 Strong oracle optimality of folded concave penalized estimation
Jianqing Fan, Lingzhou Xue, Hui Zou
Ann. Statist. 42(3): 819-849 (June 2014). DOI: 10.1214/13-AOS1198

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.

Citation

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Jianqing Fan. Lingzhou Xue. Hui Zou. "Strong oracle optimality of folded concave penalized estimation." Ann. Statist. 42 (3) 819 - 849, June 2014. https://doi.org/10.1214/13-AOS1198

Information

Published: June 2014
First available in Project Euclid: 20 May 2014

zbMATH: 1305.62252
MathSciNet: MR3210988
Digital Object Identifier: 10.1214/13-AOS1198

Subjects:
Primary: 62J07

Keywords: Folded concave penalty , local linear approximation , nonconvex optimization , oracle estimator , Sparse estimation , strong oracle property

Rights: Copyright © 2014 Institute of Mathematical Statistics

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Vol.42 • No. 3 • June 2014
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