The Annals of Statistics

Variable selection with Hamming loss

Cristina Butucea, Mohamed Ndaoud, Natalia A. Stepanova, and Alexandre B. Tsybakov

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Abstract

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.

Article information

Source
Ann. Statist., Volume 46, Number 5 (2018), 1837-1875.

Dates
Received: December 2015
Revised: March 2017
First available in Project Euclid: 17 August 2018

Permanent link to this document
https://projecteuclid.org/euclid.aos/1534492821

Digital Object Identifier
doi:10.1214/17-AOS1572

Mathematical Reviews number (MathSciNet)
MR3845003

Zentralblatt MATH identifier
06964318

Subjects
Primary: 62G05: Estimation 62G08: Nonparametric regression 62G20: Asymptotic properties

Keywords
Adaptive variable selection almost full recovery exact recovery Hamming loss minimax selectors nonasymptotic minimax selection bounds phase transitions

Citation

Butucea, Cristina; Ndaoud, Mohamed; Stepanova, Natalia A.; Tsybakov, Alexandre B. Variable selection with Hamming loss. Ann. Statist. 46 (2018), no. 5, 1837--1875. doi:10.1214/17-AOS1572. https://projecteuclid.org/euclid.aos/1534492821


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Supplemental materials

  • Supplement to “Variable selection with Hamming loss”. We derive a general lower bound for the minimax risk over all selectors on the class of at most $s$-sparse vectors. The main term of this bound is a Bayes risk with arbitrary prior and the non-asymptotic remainder term is given explicitly. Using this, we prove the lower bounds of Theorems 2.2, 3.2 and 3.3.