Gamma-Phi losses constitute a family of multiclass classification loss functions that generalize the logistic and other common losses, and have found application in the boosting literature. We establish the first general sufficient condition for the classification-calibration (CC) of such losses. To our knowledge, this sufficient condition gives the first family of nonconvex multiclass surrogate losses for which CC has been fully justified. In addition, we show that a previously proposed sufficient condition is in fact not sufficient. This contribution highlights a technical issue that is important in the study of multiclass CC but has been neglected in prior work.
Google Research（谷歌研究院）；Rutgers University（罗格斯大学）；Courant Institute of Mathematical Sciences（库朗数学科学研究所）
Adversarial robustness is an increasingly critical property of classifiers in applications. The design of robust algorithms relies on surrogate losses since the optimization of the adversarial loss with most hypothesis sets is NP-hard. But which surrogate losses should be used and when do they benefit from theoretical guarantees? We present an extensive study of this question, including a detailed analysis of the H-calibration and H-consistency of adversarial surrogate losses. We show that, under some general assumptions, convex loss functions, or the supremum-based convex losses often used in…
In this paper we refine the process of computing calibration functions for a number of multiclass classification surrogate losses. Calibration functions are a powerful tool for easily converting bounds for the surrogate risk (which can be computed through well-known methods) into bounds for the true risk, the probability of making a mistake. They are particularly suitable in non-parametric settings, where the approximation error can be controlled, and provide tighter bounds than the common technique of upper-bounding the 0-1 loss by the surrogate loss. The abstract nature of the more sophistic…
University of Amsterdam（阿姆斯特丹大学）；Universidad Carlos III de Madrid（马德里卡洛斯三世大学）
We study the statistical properties of learning to defer (L2D) to multiple experts. In particular, we address the open problems of deriving a consistent surrogate loss, confidence calibration, and principled ensembling of experts. Firstly, we derive two consistent surrogates -- one based on a softmax parameterization, the other on a one-vs-all (OvA) parameterization -- that are analogous to the single expert losses proposed by Mozannar and Sontag (2020) and Verma and Nalisnick (2022), respectively. We then study the frameworks' ability to estimate P( m_j = y | x ), the probability that the jth…
We study consistency properties of surrogate loss functions for general multiclass learning problems, defined by a general multiclass loss matrix. We extend the notion of classification calibration, which has been studied for binary and multiclass 0-1 classification problems (and for certain other specific learning problems), to the general multiclass setting, and derive necessary and sufficient conditions for a surrogate loss to be calibrated with respect to a loss matrix in this setting. We then introduce the notion of convex calibration dimension of a multiclass loss matrix, which measures …
The University of Tokyo（东京大学）；RIKEN Center for Advanced Intelligence Project（理化学研究所高级智能项目中心）
We investigate the problem of multiclass classification with rejection, where a classifier can choose not to make a prediction to avoid critical misclassification. First, we consider an approach based on simultaneous training of a classifier and a rejector, which achieves the state-of-the-art performance in the binary case. We analyze this approach for the multiclass case and derive a general condition for calibration to the Bayes-optimal solution, which suggests that calibration is hard to achieve by general loss functions unlike the binary case. Next, we consider another traditional approach…
We investigate the calibration of estimations to increase performance with an optimal monotone transform on the estimator outputs. We start by studying the traditional square error setting with its weighted variant and show that the optimal monotone transform is in the form of a unique staircase function. We further show that this staircase behavior is preserved for general strictly convex loss functions. Their optimal monotone transforms are also unique, i.e., there exist a single staircase transform that achieves the minimum loss. We propose a linear time and space algorithm that can find su…
We present surrogate regret bounds for arbitrary surrogate losses in the context of binary classification with label-dependent costs. Such bounds relate a classifier's risk, assessed with respect to a surrogate loss, to its cost-sensitive classification risk. Two approaches to surrogate regret bounds are developed. The first is a direct generalization of Bartlett et al. [2006], who focus on margin-based losses and cost-insensitive classification, while the second adopts the framework of Steinwart [2007] based on calibration functions. Nontrivial surrogate regret bounds are shown to exist preci…
University of California, Berkeley（加州大学伯克利分校）；Google Research Accra（谷歌阿克拉研究院）；University of Illinois at Urbana Champaign（伊利诺伊大学厄巴纳-香槟分校）
The top-$k$ error is often employed to evaluate performance for challenging classification tasks in computer vision as it is designed to compensate for ambiguity in ground truth labels. This practical success motivates our theoretical analysis of consistent top-$k$ classification. Surprisingly, it is not rigorously understood when taking the $k$-argmax of a vector is guaranteed to return the $k$-argmax of another vector, though doing so is crucial to describe Bayes optimality; we do both tasks. Then, we define top-$k$ calibration and show it is necessary and sufficient for consistency. Based o…
The University of Tokyo（东京大学）；RIKEN Center of Artificial Intelligence Project（理化学研究所人工智能项目中心）
This paper aims to provide a better understanding of a symmetric loss. First, we emphasize that using a symmetric loss is advantageous in the balanced error rate (BER) minimization and area under the receiver operating characteristic curve (AUC) maximization from corrupted labels. Second, we prove general theoretical properties of symmetric losses, including a classification-calibration condition, excess risk bound, conditional risk minimizer, and AUC-consistency condition. Third, since all nonnegative symmetric losses are non-convex, we propose a convex barrier hinge loss that benefits signif…
The University of Tokyo（东京大学）；RIKEN（理化学研究所）；AIP（应用物理研究所）；University of Michigan（密歇根大学）
Adversarially robust classification seeks a classifier that is insensitive to adversarial perturbations of test patterns. This problem is often formulated via a minimax objective, where the target loss is the worst-case value of the 0-1 loss subject to a bound on the size of perturbation. Recent work has proposed convex surrogates for the adversarial 0-1 loss, in an effort to make optimization more tractable. A primary question is that of consistency, that is, whether minimization of the surrogate risk implies minimization of the adversarial 0-1 risk. In this work, we analyze this question thr…
Calibeating for general proper losses: A Bregman divergence approach
Independent researcher（独立研究者）；Institute for Mathematical and Computational Engineering, Faculty of Mathematics and School of Engineering, Pontificia Universidad Católica de Chile（数学与计算工程学院，数学系和工程学院，天主教智利大学）；Department of Computer Science, University of Victoria（计算机科学系，维多利亚大学）
本文提出了一种基于懊悔最小化的通用校准框架，考虑了包括α-Tsallis损失（α∈[1,2]）和Lipschitz损失在内的广泛恰当损失家族，同时展示了新的关于Be The Regularized Leader的懊悔等式。
Unified Binary and Multiclass Margin-Based Classification
University of Michigan（密歇根大学）；Department of Electrical Engineering and Computer Science（电气工程与计算机科学系）；Michigan Institute of Data Science（密歇根数据科学研究所）；Department of Statistics（统计学系）
University of California, San Diego（加州大学圣地亚哥分校）；Halıcıoğlu Data Science Institute（哈利哲奥卢数据科学研究所）；Wisconsin Institute for Discovery（威斯康星发现研究所）
Nearly all practical neural models for classification are trained using cross-entropy loss. Yet this ubiquitous choice is supported by little theoretical or empirical evidence. Recent work (Hui & Belkin, 2020) suggests that training using the (rescaled) square loss is often superior in terms of the classification accuracy. In this paper we propose the "squentropy" loss, which is the sum of two terms: the cross-entropy loss and the average square loss over the incorrect classes. We provide an extensive set of experiments on multi-class classification problems showing that the squentropy loss ou…
