Open Access
October 2018 Variable selection with Hamming loss
Cristina Butucea, Mohamed Ndaoud, Natalia A. Stepanova, Alexandre B. Tsybakov
Ann. Statist. 46(5): 1837-1875 (October 2018). DOI: 10.1214/17-AOS1572


We derive nonasymptotic bounds for the minimax risk of variable selection under expected Hamming loss in the Gaussian mean model in $\mathbb{R}^{d}$ 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.


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Cristina Butucea. Mohamed Ndaoud. Natalia A. Stepanova. Alexandre B. Tsybakov. "Variable selection with Hamming loss." Ann. Statist. 46 (5) 1837 - 1875, October 2018.


Received: 1 December 2015; Revised: 1 March 2017; Published: October 2018
First available in Project Euclid: 17 August 2018

zbMATH: 06964318
MathSciNet: MR3845003
Digital Object Identifier: 10.1214/17-AOS1572

Primary: 62G05 , 62G08 , 62G20

Keywords: Adaptive variable selection , almost full recovery , exact recovery , Hamming loss , minimax selectors , nonasymptotic minimax selection bounds , Phase transitions

Rights: Copyright © 2018 Institute of Mathematical Statistics

Vol.46 • No. 5 • October 2018
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