In multiclass classification, the goal is to learn how to predict a random label $Y$, valued in $\mathcal{Y}=\{1,\; \ldots,\; K \}$ with $K\geq 3$, based upon observing a r.v. $X$, taking its values in $\mathbb{R}^q$ with $q\geq 1$ say, by means of a classification rule $g:\mathbb{R}^q\to \mathcal{Y}$ with minimum probability of error $\mathbb{P}\{Y\neq g(X) \}$. However, in a wide variety of situations, the task targeted may be more ambitious, consisting in sorting all the possible label values $y$ that may be assigned to $X$ by decreasing order of the posterior probability $η_y(X)=\mathbb{P}\{Y=y \mid X \}$. This article is devoted to the analysis of this statistical learning problem, halfway between multiclass classification and posterior probability estimation (regression) and referred to as label ranking here. We highlight the fact that it can be viewed as a specific variant of ranking median regression (RMR), where, rather than observing a random permutation $Σ$ assigned to the input vector $X$ and drawn from a Bradley-Terry-Luce-Plackett model with conditional preference vector $(η_1(X),\; \ldots,\; η_K(X))$, the sole information available for training a label ranking rule is the label $Y$ ranked on top, namely $Σ^{-1}(1)$. Inspired by recent results in RMR, we prove that under appropriate noise conditions, the One-Versus-One (OVO) approach to multiclassification yields, as a by-product, an optimal ranking of the labels with overwhelming probability. Beyond theoretical guarantees, the relevance of the approach to label ranking promoted in this article is supported by experimental results.
In this paper, we propose online algorithms for multiclass classification using partial labels. We propose two variants of Perceptron called Avg Perceptron and Max Perceptron to deal with the partial labeled data. We also propose Avg Pegasos and Max Pegasos, which are extensions of Pegasos algorithm. We also provide mistake bounds for Avg Perceptron and regret bound for Avg Pegasos. We show the effectiveness of the proposed approaches by experimenting on various datasets and comparing them with the standard Perceptron and Pegasos.
Université Laval（拉瓦尔大学）；Univ. de St-Etienne（圣艾蒂安大学）；Aix-Marseille Université（艾克斯-马赛大学）
The C-bound, introduced in Lacasse et al., gives a tight upper bound on the risk of a binary majority vote classifier. In this work, we present a first step towards extending this work to more complex outputs, by providing generalizations of the C-bound to the multiclass and multi-label settings.
A multiclass classifier is said to be top-label calibrated if the reported probability for the predicted class -- the top-label -- is calibrated, conditioned on the top-label. This conditioning on the top-label is absent in the closely related and popular notion of confidence calibration, which we argue makes confidence calibration difficult to interpret for decision-making. We propose top-label calibration as a rectification of confidence calibration. Further, we outline a multiclass-to-binary (M2B) reduction framework that unifies confidence, top-label, and class-wise calibration, among othe…
Under any Multiclass Classification (MCC) setting defined by a collection of labeled point-cloud specified by a feature-set, we extract only stochastic partial orderings from all possible triplets of point-cloud without explicitly measuring the three cloud-to-cloud distances. We demonstrate that such a collective of partial ordering can efficiently compute a label embedding tree geometry on the Label-space. This tree in turn gives rise to a predictive graph, or a network with precisely weighted linkages. Such two multiscale geometries are taken as the coarse scale information content of MCC. T…
We consider the problem of online multiclass classification with partial feedback, where an algorithm predicts a class for a new instance in each round and only receives its correctness. Although several methods have been developed for this problem, recent challenging real-world applications require further performance improvement. In this paper, we propose a novel online learning algorithm inspired by recent work on learning from complementary labels, where a complementary label indicates a class to which an instance does not belong. This allows us to handle partial feedback deterministically…
Neyman-Pearson multiclass classification under label noise via empirical likelihood
Institute of Statistics and Big Data, Renmin University of China（中国人民大学统计与大数据研究院）；Department of Statistics and Actuarial Science, University of Waterloo（滑铁卢大学统计与精算科学系）
Carnegie Mellon University（卡内基梅隆大学）；Conversant LLC（Conversant有限责任公司）；California Institute of Technology（加州理工学院）；University of Illinois at Chicago（伊利诺伊大学芝加哥分校）；University of Pennsylvania（宾夕法尼亚大学）
We propose a robust adversarial prediction framework for general multiclass classification. Our method seeks predictive distributions that robustly optimize non-convex and non-continuous multiclass loss metrics against the worst-case conditional label distributions (the adversarial distributions) that (approximately) match the statistics of the training data. Although the optimized loss metrics are non-convex and non-continuous, the dual formulation of the framework is a convex optimization problem that can be recast as a risk minimization model with a prescribed convex surrogate loss we call …
Max Planck Institute for Informatics（马克斯·普朗克信息学研究所）；Saarland University（萨尔大学）
Top-k error is currently a popular performance measure on large scale image classification benchmarks such as ImageNet and Places. Despite its wide acceptance, our understanding of this metric is limited as most of the previous research is focused on its special case, the top-1 error. In this work, we explore two directions that shed more light on the top-k error. First, we provide an in-depth analysis of established and recently proposed single-label multiclass methods along with a detailed account of efficient optimization algorithms for them. Our results indicate that the softmax loss and t…
Google Research（谷歌研究院）；CREST ENSAE（法国国家统计与经济管理学院经济与统计研究中心）；MINES ParisTech, PSL University（巴黎文理研究大学巴黎国立高等矿业学院）
We propose a learning algorithm capable of learning from label proportions instead of direct data labels. In this scenario, our data are arranged into various bags of a certain size, and only the proportions of each label within a given bag are known. This is a common situation in cases where per-data labeling is lengthy, but a more general label is easily accessible. Several approaches have been proposed to learn in this setting with linear models in the multiclass setting, or with nonlinear models in the binary classification setting. Here we investigate the more general nonlinear multiclass…
In the domains of dataset construction and crowdsourcing, a notable challenge is to aggregate labels from a heterogeneous set of labelers, each of whom is potentially an expert in some subset of tasks (and less reliable in others). To reduce costs of hiring human labelers or training automated labeling systems, it is of interest to minimize the number of labelers while ensuring the reliability of the resulting dataset. We model this as the problem of performing $K$-class classification using the predictions of smaller classifiers, each trained on a subset of $[K]$, and derive bounds on the num…
Label ranking is a prediction task which deals with learning a mapping between an instance and a ranking (i.e., order) of labels from a finite set, representing their relevance to the instance. Boosting is a well-known and reliable ensemble technique that was shown to often outperform other learning algorithms. While boosting algorithms were developed for a multitude of machine learning tasks, label ranking tasks were overlooked. In this paper, we propose a boosting algorithm which was specifically designed for label ranking tasks. Extensive evaluation of the proposed algorithm on 24 semi-synt…
University of Michigan（密歇根大学）；Department of Statistics（统计系）
We consider the multi-label ranking approach to multi-label learning. Boosting is a natural method for multi-label ranking as it aggregates weak predictions through majority votes, which can be directly used as scores to produce a ranking of the labels. We design online boosting algorithms with provable loss bounds for multi-label ranking. We show that our first algorithm is optimal in terms of the number of learners required to attain a desired accuracy, but it requires knowledge of the edge of the weak learners. We also design an adaptive algorithm that does not require this knowledge and is…
East China University of Science and Technology（华东理工大学）；Shenzhen University（深圳大学）；University of Nottingham（诺丁汉大学）
Label ranking aims to learn a mapping from instances to rankings over a finite number of predefined labels. Random forest is a powerful and one of the most successful general-purpose machine learning algorithms of modern times. In this paper, we present a powerful random forest label ranking method which uses random decision trees to retrieve nearest neighbors. We have developed a novel two-step rank aggregation strategy to effectively aggregate neighboring rankings discovered by the random forest into a final predicted ranking. Compared with existing methods, the new random forest method has …
We propose to solve a label ranking problem as a structured output regression task. We adopt a least square surrogate loss approach that solves a supervised learning problem in two steps: the regression step in a well-chosen feature space and the pre-image step. We use specific feature maps/embeddings for ranking data, which convert any ranking/permutation into a vector representation. These embeddings are all well-tailored for our approach, either by resulting in consistent estimators, or by solving trivially the pre-image problem which is often the bottleneck in structured prediction. We als…
INESCTEC；LIACS；HasLab, INESC TEC, Departamento de Informática, Universidade do Minho（Minho大学信息学院HasLab, INESC TEC）；DCC - Faculdade de Ciencias, Universidade do Porto（波尔图大学科学学院DCC）；LIACS, Leiden, Netherlands（莱顿大学LIACS）
In this paper we investigate two variants of association rules for preference data, Label Ranking Association Rules and Pairwise Association Rules. Label Ranking Association Rules (LRAR) are the equivalent of Class Association Rules (CAR) for the Label Ranking task. In CAR, the consequent is a single class, to which the example is expected to belong to. In LRAR, the consequent is a ranking of the labels. The generation of LRAR requires special support and confidence measures to assess the similarity of rankings. In this work, we carry out a sensitivity analysis of these similarity-based measur…
A comparative analysis of rank aggregation methods for the partial label ranking problem
Laboratorio de Sistemas Inteligentes y Minería de Datos, Universidad de Castilla-La Mancha, Albacete, Spain（智能系统与数据挖掘实验室，卡斯蒂利亚-拉曼查大学，阿尔巴塞特，西班牙）；German Research Center for Artificial Intelligence, Kaiserslautern, Germany（德国人工智能研究中心，凯泽斯劳滕，德国）
Claremont Graduate University（克莱蒙特研究生大学）；Institute of Mathematical Sciences（数学科学研究所）；Naval Air Warfare Center（海军空战中心）
We present a graph-based variational algorithm for multiclass classification of high-dimensional data, motivated by total variation techniques. The energy functional is based on a diffuse interface model with a periodic potential. We augment the model by introducing an alternative measure of smoothness that preserves symmetry among the class labels. Through this modification of the standard Laplacian, we construct an efficient multiclass method that allows for sharp transitions between classes. The experimental results demonstrate that our approach is competitive with the state of the art amon…
Institute of Computer Science, Polish Academy of Sciences（波兰科学院计算机科学研究所）
We propose a simple and efficient method for ranking features in multi-label classification. The method produces a ranking of features showing their relevance in predicting labels, which in turn allows to choose a final subset of features. The procedure is based on Markov Networks and allows to model the dependencies between labels and features in a direct way. In the first step we build a simple network using only labels and then we test how much adding a single feature affects the initial network. More specifically, in the first step we use the Ising model whereas the second step is based on…
Northeastern University（东北大学）；JD.com Inc.（京东公司）；North Carolina State University（北卡罗来纳州立大学）
Extreme Multi-label classification (XML) is an important yet challenging machine learning task, that assigns to each instance its most relevant candidate labels from an extremely large label collection, where the numbers of labels, features and instances could be thousands or millions. XML is more and more on demand in the Internet industries, accompanied with the increasing business scale / scope and data accumulation. The extremely large label collections yield challenges such as computational complexity, inter-label dependency and noisy labeling. Many methods have been proposed to tackle th…
In this paper, we consider the problem of making skeptical inferences for the multi-label ranking problem. We assume that our uncertainty is described by a convex set of probabilities (i.e. a credal set), defined over the set of labels. Instead of learning a singleton prediction (or, a completed ranking over the labels), we thus seek for skeptical inferences in terms of set-valued predictions consisting of completed rankings.
University of Chinese Academy of Sciences（中国科学院大学）；Chinese Academy of Sciences（中国科学院）；Beijing Union University（北京联合大学）
Multi-label classification studies the task where each example belongs to multiple labels simultaneously. As a representative method, Ranking Support Vector Machine (Rank-SVM) aims to minimize the Ranking Loss and can also mitigate the negative influence of the class-imbalance issue. However, due to its stacking-style way for thresholding, it may suffer error accumulation and thus reduces the final classification performance. Binary Relevance (BR) is another typical method, which aims to minimize the Hamming Loss and only needs one-step learning. Nevertheless, it might have the class-imbalance…
The rising interest in pattern recognition and data analytics has spurred the development of innovative machine learning algorithms and tools. However, as each algorithm has its strengths and limitations, one is motivated to judiciously fuse multiple algorithms in order to find the "best" performing one, for a given dataset. Ensemble learning aims at such high-performance meta-algorithm, by combining the outputs from multiple algorithms. The present work introduces a blind scheme for learning from ensembles of classifiers, using a moment matching method that leverages joint tensor and matrix f…
RTL Nederland B.V.（RTL荷兰公司）；University of Amsterdam（阿姆斯特丹大学）
Multiclass multilabel classification is the task of attributing multiple labels to examples via predictions. Current models formulate a reduction of the multilabel setting into either multiple binary classifications or multiclass classification, allowing for the use of existing loss functions (sigmoid, cross-entropy, logistic, etc.). Multilabel classification reductions do not accommodate for the prediction of varying numbers of labels per example and the underlying losses are distant estimates of the performance metrics. We propose a loss function, sigmoidF1, which is an approximation of the …
(Partial) ranking loss is a commonly used evaluation measure for multi-label classification, which is usually optimized with convex surrogates for computational efficiency. Prior theoretical work on multi-label ranking mainly focuses on (Fisher) consistency analyses. However, there is a gap between existing theory and practice -- some pairwise losses can lead to promising performance but lack consistency, while some univariate losses are consistent but usually have no clear superiority in practice. In this paper, we attempt to fill this gap through a systematic study from two complementary per…
Claremont Graduate University（克莱蒙特研究生大学）；Naval Air Warfare Center（海军空战中心）
We present a graph-based variational algorithm for classification of high-dimensional data, generalizing the binary diffuse interface model to the case of multiple classes. Motivated by total variation techniques, the method involves minimizing an energy functional made up of three terms. The first two terms promote a stepwise continuous classification function with sharp transitions between classes, while preserving symmetry among the class labels. The third term is a data fidelity term, allowing us to incorporate prior information into the model in a semi-supervised framework. The performanc…
Department of Computer Science, University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校计算机科学系）；Google Inc.（谷歌公司）；Independent Researcher（独立研究者）
We propose a framework for constructing and analyzing multiclass and multioutput classification metrics, i.e., involving multiple, possibly correlated multiclass labels. Our analysis reveals novel insights on the geometry of feasible confusion tensors -- including necessary and sufficient conditions for the equivalence between optimizing an arbitrary non-decomposable metric and learning a weighted classifier. Further, we analyze averaging methodologies commonly used to compute multioutput metrics and characterize the corresponding Bayes optimal classifiers. We show that the plug-in estimator b…
We consider the problem of personalization of online services from the viewpoint of ad targeting, where we seek to find the best ad categories to be shown to each user, resulting in improved user experience and increased advertisers' revenue. We propose to address this problem as a task of ranking the ad categories depending on a user's preference, and introduce a novel label ranking approach capable of efficiently learning non-linear, highly accurate models in large-scale settings. Experiments on a real-world advertising data set with more than 3.2 million users show that the proposed algorit…
College of Science and Engineering, James Cook University（詹姆斯·库克大学理工学院）
Robotic weed control has seen increased research of late with its potential for boosting productivity in agriculture. Majority of works focus on developing robotics for croplands, ignoring the weed management problems facing rangeland stock farmers. Perhaps the greatest obstacle to widespread uptake of robotic weed control is the robust classification of weed species in their natural environment. The unparalleled successes of deep learning make it an ideal candidate for recognising various weed species in the complex rangeland environment. This work contributes the first large, public, multicl…
As larger and more comprehensive datasets become standard in contemporary machine learning, it becomes increasingly more difficult to obtain reliable, trustworthy label information with which to train sophisticated models. To address this problem, crowdsourcing has emerged as a popular, inexpensive, and efficient data mining solution for performing distributed label collection. However, crowdsourced annotations are inherently untrustworthy, as the labels are provided by anonymous volunteers who may have varying, unreliable expertise. Worse yet, some participants on commonly used platforms such…
Department of Computer Science and Engineering, Texas A&M University（德克萨斯农工大学计算机科学与工程系）
Adversarial examples are delicately perturbed inputs, which aim to mislead machine learning models towards incorrect outputs. While most of the existing work focuses on generating adversarial perturbations in multi-class classification problems, many real-world applications fall into the multi-label setting in which one instance could be associated with more than one label. For example, a spammer may generate adversarial spams with malicious advertising while maintaining the other labels such as topic labels unchanged. To analyze the vulnerability and robustness of multi-label learning models,…
We formulate a supervised learning problem, referred to as continuous ranking, where a continuous real-valued label Y is assigned to an observable r.v. X taking its values in a feature space $\mathcal{X}$ and the goal is to order all possible observations x in $\mathcal{X}$ by means of a scoring function $s:\mathcal{X}\rightarrow \mathbb{R}$ so that s(X) and Y tend to increase or decrease together with highest probability. This problem generalizes bi/multi-partite ranking to a certain extent and the task of finding optimal scoring functions s(x) can be naturally cast as optimization of a dedic…
