STA / BST 290 Seminar Series
Thursday, January 15, 4:30pm, MSB 1147 (Colloquium Room)
Refreshments at 4:00pm in MSB 4110 (Statistics Lounge)
Speaker: Hui Zou (University of Minnesota)
Title: Computable 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 an estimation problem formulated via a folded concave penalized convex loss, we show that as long as the problem is localizable and the oracle estimator is well behaved, the two step LLA estimator converges to the oracle estimator with overwhelming probability. 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