On surrogate loss functions and f-divergences
研究了在协变量被降维量化器变换后，替代损失函数需满足什么条件才能保证判别函数和量化器联合估计的贝叶斯一致性，并将0-1损失与f-散度的对应关系推广到更广的替代损失函数类。
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.