Multiclass Classification, Information, Divergence, and Surrogate Risk

TL;DR

This paper unifies various information measures, divergence concepts, and loss functions in multi-class classification, extending binary results to more general settings and characterizing optimal loss functions for joint classification and data representation.

Contribution

It introduces a comprehensive framework linking divergences, information measures, and losses for multiclass problems, extending existing binary results to multiple classes and joint optimization of classifiers and data representations.

Findings

Established equivalence between divergences, information measures, and losses in multiclass settings.

Characterized convex losses that are consistent for joint classification and data representation.

Extended binary classification calibration results to multiclass scenarios.

Abstract

We provide a unifying view of statistical information measures, multi-way Bayesian hypothesis testing, loss functions for multi-class classification problems, and multi-distribution $f$-divergences, elaborating equivalence results between all of these objects, and extending existing results for binary outcome spaces to more general ones. We consider a generalization of $f$-divergences to multiple distributions, and we provide a constructive equivalence between divergences, statistical information (in the sense of DeGroot), and losses for multiclass classification. A major application of our results is in multi-class classification problems in which we must both infer a discriminant function $\gamma$---for making predictions on a label $Y$ from datum $X$---and a data representation (or, in the setting of a hypothesis testing problem, an experimental design), represented as a quantizer $\mathsf{q}$ from a family of possible quantizers $\mathsf{Q}$. In this setting, we characterize the equivalence between loss functions, meaning that optimizing either of two losses yields an optimal discriminant and quantizer $\mathsf{q}$, complementing and extending earlier results of Nguyen et. al. to the multiclass case. Our results provide a more substantial basis than standard classification calibration results for comparing different losses: we describe the convex losses that are consistent for jointly choosing a data representation and minimizing the (weighted) probability of error in multiclass classification problems.