VARIABLE SELECTION WITH HAMMING LOSS

• 成果类型: Article
• 署名作者: Butucea, Cristina; Ndaoud, Mohamed; Stepanova, Natalia A.; Tsybakov, Alexandre B.
• 署名单位: Universite Paris-Est-Creteil-Val-de-Marne (UPEC); Universite Gustave-Eiffel; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Institut Polytechnique de Paris; Ecole Polytechnique; Universite Paris Saclay; ENSAE Paris; Carleton University
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/17-AOS1572
• 发表日期: 2018
• 页码: 1837-1875
• 关键词: false discovery rate; Nonparametric Regression; Lasso; SPARSE; Consistency; models

摘要

We derive nonasymptotic bounds for the minimax risk of variable selection under expected Hamming loss in the Gaussian mean model in R-d for classes of at most s-sparse vectors separated from 0 by a constant a > 0. In some cases, we get exact expressions for the nonasymptotic minimax risk as a function of d, s, a and find explicitly the minimax selectors. These results are extended to dependent or non-Gaussian observations and to the problem of crowdsourcing. Analogous conclusions are obtained for the probability of wrong recovery of the sparsity pattern. As corollaries, we derive necessary and sufficient conditions for such asymptotic properties as almost full recovery and exact recovery. Moreover, we propose data-driven selectors that provide almost full and exact recovery adaptively to the parameters of the classes.