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SUMMARY:Strong Oracle Optimality of Folded Concave Penalized Estimation - 
 Jianqing Fan\, Princeton University
DTSTART:20130215T160000Z
DTEND:20130215T170000Z
UID:TALK42445@talks.cam.ac.uk
CONTACT:Richard Samworth
DESCRIPTION:Folded concave penalization methods such as SCAD have been sho
 wn to enjoy the strong oracle property for high-dimensional sparse estimat
 ion.\n\nHowever\, a folded concave penalization problem usually has multip
 le 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 giv
 en optimization algorithm possesses those nice theoretical properties. To 
 close this important theoretical gap in over a decade\, we provide a unifi
 ed theory to show explicitly how to obtain the oracle solution using the l
 ocal linear approximation algorithm.  For a folded concave penalized estim
 ation 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 orac
 le estimator is obtained\, the local linear approximation algorithm conver
 ges\, namely produces the same estimator in the next iteration. We show th
 at the LASSO is a good initial estimator\, which produces the oracle estim
 ator using the one-step LLA algorithm for folded concave penalization meth
 ods.  This is demonstrated by using three classical sparse estimation prob
 lems\, namely\, the sparse linear regression\, the sparse logistic regress
 ion and the sparse precision matrix estimation\, and illustrates the power
  of combining the LASSO and SCAD to solve sparse inartistic estimation pro
 blem.\n\n(Joint work with Lingzhou Xue and Hui Zou.)
LOCATION:MR12\, CMS\, Wilberforce Road\, Cambridge\, CB3 0WB
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