Strong oracle optimality of folded concave penalized estimation

Jianqing Fan Princeton University Lingzhou Xue Pennsylvania State University Hui Zou University of Minnesota

Statistics Theory and Methods mathscidoc:1909.33002

Best Paper Award in Applied Mathematics in 2019

The Annals of Statistics, 42, (3), 819-849, 2014
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.
Folded concave penalty, local linear approximation, nonconvex optimization, oracle estimator, sparse estimation, strong oracle property
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  title={Strong oracle optimality of folded concave penalized estimation},
  author={Jianqing Fan, Lingzhou Xue, and Hui Zou},
  booktitle={The Annals of Statistics},
Jianqing Fan, Lingzhou Xue, and Hui Zou. Strong oracle optimality of folded concave penalized estimation. 2014. Vol. 42. In The Annals of Statistics. pp.819-849.
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