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Activity Number: 87
Type: Contributed
Date/Time: Sunday, August 9, 2015 : 4:00 PM to 5:50 PM
Sponsor: Section on Statistical Learning and Data Mining
Abstract #317606
Title: A Unified Tightening After Contraction Framework for Sparse Learning Problems: An Algorithmic Approach
Author(s): Qiang Sun* and Jianqing Fan and Han Liu and Tong Zhang
Companies: Princeton University and Princeton University and Princeton University and Rutgers University
Keywords: Algorithmic statistics ; Nonconvex statistical optimization ; Tightenting after contraction ; Optimal rate of convergence ; Geometric convergence
Abstract:

We propose a unified computational framework named tightening after contraction (TAC) for fitting high dimensional statistical models. TAC has two stages: In the contraction stage we approximately solve a convex program to obtain an initial estimator. Then a second tightening stage iteratively refines this initial estimator by solving a sequence of convex programs. Though this framework is completely algorithmic, it provides solutions with optimal statistical and computational guarantees for large families of nonconvex statistical optimization problems. Our unified theoretical treatment relies on a new technique using localized versions of sparse eigenvalue and restricted eigenvalue conditions. Compared to the existing works on nonconvex statistical optimization, our method and theory handle the largest families of loss and penalty functions and provide the strongest guarantees under the weakest assumptions. In addition, our framework provides new theoretical insights that unify the existing multistage convex relaxation method and adaptive Lasso method for solving nonconvex problems. We also provide thorough numerical simulations to back up the theoretical results.


Authors who are presenting talks have a * after their name.

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