Variable selection with Hamming loss

Cristina Butucea (), Natalia Stepanova () and Alexandre Tsybakov ()
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Cristina Butucea: Université Paris-Est Marne-la-Vallée, LAMA (UMR8050), UPEM, UPEC, CNRS, ENSAE
Natalia Stepanova: School of Mathematics and Statistics (Carleton University)
Alexandre Tsybakov: CREST, ENSAE, CNRS

No 2016-10, Working Papers from Center for Research in Economics and Statistics

Abstract: We derive non-asymptotic bounds for the minimax risk of variable selection under expected Hamming loss in the Gaussian mean model in Rd for classes of 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. Analogous results 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 adaptive to the parameters of the classes.

Keywords: adaptive variable selection; almost full recovery; exact recovery; Hamming loss; minimax selectors; nonasymptotic minimax selection bounds; phase transitions (search for similar items in EconPapers)
Pages: 23
Date: 2016-01
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