Annals of Statistics

On surrogate loss functions and f-divergences

XuanLong Nguyen, Martin J. Wainwright, and Michael I. Jordan

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The goal of binary classification is to estimate a discriminant function γ from observations of covariate vectors and corresponding binary labels. We consider an elaboration of this problem in which the covariates are not available directly but are transformed by a dimensionality-reducing quantizer Q. We present conditions on loss functions such that empirical risk minimization yields Bayes consistency when both the discriminant function and the quantizer are estimated. These conditions are stated in terms of a general correspondence between loss functions and a class of functionals known as Ali-Silvey or f-divergence functionals. Whereas this correspondence was established by Blackwell [Proc. 2nd Berkeley Symp. Probab. Statist. 1 (1951) 93–102. Univ. California Press, Berkeley] for the 0–1 loss, we extend the correspondence to the broader class of surrogate loss functions that play a key role in the general theory of Bayes consistency for binary classification. Our result makes it possible to pick out the (strict) subset of surrogate loss functions that yield Bayes consistency for joint estimation of the discriminant function and the quantizer.

Article information

Ann. Statist., Volume 37, Number 2 (2009), 876-904.

First available in Project Euclid: 10 March 2009

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Zentralblatt MATH identifier

Primary: 62G10: Hypothesis testing 68Q32: Computational learning theory [See also 68T05] 62K05: Optimal designs

Binary classification discriminant analysis surrogate losses f-divergences Ali-Silvey divergences quantizer design nonparametric decentralized detection statistical machine learning Bayes consistency


Nguyen, XuanLong; Wainwright, Martin J.; Jordan, Michael I. On surrogate loss functions and f -divergences. Ann. Statist. 37 (2009), no. 2, 876--904. doi:10.1214/08-AOS595.

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