Abstract
The authors acknowledge the referees' comments and suggestions that helped to improve the manuscript. This research is based upon work supported in part by the Office of the Director of National Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA), via the Federal Bureau of Investigations, Finance Division. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of ODNI, IARPA, or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright annotation thereon. I.R-L acknowledges partial support by Spain's grants TIN2013-42351-P (MINECO) and S2013/ICE-2845 CASI-CAM-CM (Comunidad de Madrid). The authors gratefully acknowledge the use of the facilities of Centro de Computacion Cientifica (CCC) at Universidad Autonoma de Madrid.
Highlights
Though several multiclass hinge loss functions can be found in the literature [39, 9, 23, 28, 19], only two of them have been shown to be classification calibrated for every multiclass problem: Lee et al.’s loss function [23] and Liu and Yuan’s loss function1 [28]
The reinforced multicategory hinge loss considers both the margin of the positive and negative points of a given class, but experimental results in [28] on two synthetic datasets show that best classification performances are obtained for values of γ that overweight the margin of the positive points, and make the loss function classification uncalibrated
It should be noted that there may exist other points subject to KKT variations inside the same classification calibration region since the solutions for λ ∈ (−1, 0) and λ ∈ ((L − 2)/2, L − 1) depend on the class probability distribution, which in turn depends on λ. This is not the case for λ > (L − 1) since the transition between the two possible solutions is only defined by the class probabilities; that is, the KKT conditions are constant given any λ > (L − 1) and any point. It means that the margin λ = (L − 1) imposed by the reinforced multicategory hinge loss [28], though guaranteeing classification calibration, may slow down the convergence of the the Sequential Minimal Optimization (SMO) algorithm given that the optimizer is searching across different KKT
Summary
Given an L-class classification problem (L ≥ 2), the goal of a multiclass classification algorithm is to find a classifier φ : X → Y such that the class label of every input pattern x ∈ X is correctly estimated. And vectors f ∈ F are known as multicategory margin vectors [44] According to this mathematical framework, the classification function φ is unequivocally defined by the decision function f and, the goal of the classifier is to minimize Eq (1) with respect to f. In practice, the classifier is inferred from the decision function fN that minimizes the empirical Ψ-risk as fN In this framework, Bartlett et al formulate the concept of classification calibration as a necessary and sufficient condition to have Bayes consistency when the empirical risk of a binary loss function Ψy converges to the minimal possible Ψ-risk [3]. They show that multiclass classification calibration is equivalent to Bayes consistency assuming convergence of the empirical Ψ-risk to the minimal possible Ψ-risk, and they characterize classification calibration in terms of geometric properties of the loss function. For a more detailed explanation of the classification calibration framework and the consequent Bayes consistency properties, the reader is referred to [3, 37] and references therein
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