BANDWIDTH SELECTION IN KERNEL EMPIRICAL RISK MINIMIZATION VIA THE GRADIENT

• 成果类型: Article
• 署名作者: Chichignoud, Michael; Loustau, Sebastien
• 署名单位: Swiss Federal Institutes of Technology Domain; ETH Zurich; Universite d'Angers
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/15-AOS1318
• 发表日期: 2015
• 页码: 1617-1646
• 关键词: oracle inequalities; Adaptive estimation; DENSITY-ESTIMATION; fast rates; adaptation; performance; robust; bounds

摘要

In this paper, we deal with the data-driven selection of multidimensional and possibly anisotropic bandwidths in the general framework of kernel empirical risk minimization. We propose a universal selection rule, which leads to optimal adaptive results in a large variety of statistical models such as nonparametric robust regression and statistical learning with errors in variables. These results are stated in the context of smooth loss functions, where the gradient of the risk appears as a good criterion to measure the performance of our estimators. The selection rule consists of a comparison of gradient empirical risks. It can be viewed as a nontrivial improvement of the so-called Goldenshluger-Lepski method to nonlinear estimators. Furthermore, one main advantage of our selection rule is the nondependency on the Hessian matrix of the risk, usually involved in standard adaptive procedures.