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On the finite-sample analysis of Θ-estimators

ELECTRONIC JOURNAL OF STATISTICS 2016年第2期10卷 1874-1895页

作者： She, Yiyuan Florida State Univ Dept Stat Tallahassee FL 32306 USA

In large-scale modern data analysis, first-order optimization methods are usually favored to obtain sparse estimators in high dimensions. This paper performs theoretical analysis of a class of iterative thresholding based estimators defined in this way. Oracle inequalities are built to show the nearly minimax rate optimality of such estimators under a new type of regularity conditions. Moreover, the sequence of iterates is found to be able to approach the statistical truth within the best statistical accuracy geometrically fast. Our results also reveal different benefits brought by convex and nonconvex types of shrinkage.

关键词： Sparsity thresholding nonconvex optimization oracle inequalities statistical algorithmic analysis

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analysis OF GENERALIZED BREGMAN SURROGATE ALGORITHMS FOR NONSMOOTH NONCONVEX statistical LEARNING

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ANNALS OF STATISTICS 2021年第6期49卷 3434-3459页

作者： She, Yiyuan Wang, Zhifeng Jin, Jiuwu Florida State Univ Dept Stat Tallahassee FL 32306 USA

Modern statistical applications often involve minimizing an objective function that may be nonsmooth and/or nonconvex. This paper focuses on a broad Bregman-surrogate algorithm framework including the local linear approximation, mirror descent, iterative thresholding, DC programming and many others as particular instances. The recharacterization via generalized Bregman functions enables us to construct suitable error measures and establish global convergence rates for nonconvex and nonsmooth objectives in possibly high dimensions. For sparse learning problems with a composite objective, under some regularity conditions, the obtained estimators as the surrogate's fixed points, though not necessarily local minimizers, enjoy provable statistical guarantees, and the sequence of iterates can be shown to approach the statistical truth within the desired accuracy geometrically fast. The paper also studies how to design adaptive momentum based accelerations without assuming convexity or smoothness by carefully controlling stepsize and relaxation parameters.

关键词： Nonconvex optimization nonsmooth optimization MM algorithms Bregman divergence statistical algorithmic analysis momentum-based acceleration

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