We develop a general framework for margin-based multicategory classification in metric spaces. The basic work-horse is a margin-regularized version of the nearest-neighbor classifier. We prove generalization bounds that match the state of the art in sample size $n$ and significantly improve the dependence on the number of classes $k$. Our point of departure is a nearly Bayes-optimal finite-sample risk bound independent of $k$. Although $k$-free, this bound is unregularized and non-adaptive, which motivates our main result: Rademacher and scale-sensitive margin bounds with a logarithmic dependence on $k$. As the best previous risk estimates in this setting were of order $\sqrt k$, our bound is exponentially sharper. From the algorithmic standpoint, in doubling metric spaces our classifier may be trained on $n$ examples in $O(n^2\log n)$ time and evaluated on new points in $O(\log n)$ time.
University of Genova（热那亚大学）；Istituto Italiano di Tecnologia（意大利理工学院）；MIT（麻省理工学院）
We study the behavior of error bounds for multiclass classification under suitable margin conditions. For a wide variety of methods we prove that the classification error under a hard-margin condition decreases exponentially fast without any bias-variance trade-off. Different convergence rates can be obtained in correspondence of different margin assumptions. With a self-contained and instructive analysis we are able to generalize known results from the binary to the multiclass setting.
Leiden University（莱顿大学）；University of Twente（特文特大学）
For classification problems, trained deep neural networks return probabilities of class memberships. In this work we study convergence of the learned probabilities to the true conditional class probabilities. More specifically we consider sparse deep ReLU network reconstructions minimizing cross-entropy loss in the multiclass classification setup. Interesting phenomena occur when the class membership probabilities are close to zero. Convergence rates are derived that depend on the near-zero behaviour via a margin-type condition.
University of Texas Southwestern Medical Center（德克萨斯大学西南医学中心）
Multiclass problems are often decomposed into multiple binary problems that are solved by individual binary classifiers whose results are integrated into a final answer. Various methods, including all-pairs (APs), one-versus-all (OVA), and error correcting output code (ECOC), have been studied, to decompose multiclass problems into binary problems. However, little study has been made to optimally aggregate binary problems to determine a final answer to the multiclass problem. In this paper we present a convex optimization method for an optimal aggregation of binary classifiers to estimate clas…
Beijing Institute of Technology（北京理工大学）；NICTA（澳大利亚国家信息与通信技术研究院）；Australian National University（澳大利亚国立大学）
In this work, we propose a new optimization framework for multiclass boosting learning. In the literature, AdaBoost.MO and AdaBoost.ECC are the two successful multiclass boosting algorithms, which can use binary weak learners. We explicitly derive these two algorithms' Lagrange dual problems based on their regularized loss functions. We show that the Lagrange dual formulations enable us to design totally-corrective multiclass algorithms by using the primal-dual optimization technique. Experiments on benchmark data sets suggest that our multiclass boosting can achieve a comparable generalizatio…
Ben-Gurion University of the Negev（本-古里安大学）；Weizmann Institute of Science（魏茨曼科学研究所）
We examine the Bayes-consistency of a recently proposed 1-nearest-neighbor-based multiclass learning algorithm. This algorithm is derived from sample compression bounds and enjoys the statistical advantages of tight, fully empirical generalization bounds, as well as the algorithmic advantages of a faster runtime and memory savings. We prove that this algorithm is strongly Bayes-consistent in metric spaces with finite doubling dimension --- the first consistency result for an efficient nearest-neighbor sample compression scheme. Rather surprisingly, we discover that this algorithm continues to …
Los Alamos National Laboratory（洛斯阿拉莫斯国家实验室）；Univ. Grenoble Alpes（格勒诺布尔-阿尔卑斯大学）；CNRS（法国国家科学研究中心）；Grenoble INP（格勒诺布尔理工学院）；University of Washington（华盛顿大学）
We propose Rademacher complexity bounds for multiclass classifiers trained with a two-step semi-supervised model. In the first step, the algorithm partitions the partially labeled data and then identifies dense clusters containing $κ$ predominant classes using the labeled training examples such that the proportion of their non-predominant classes is below a fixed threshold. In the second step, a classifier is trained by minimizing a margin empirical loss over the labeled training set and a penalization term measuring the disability of the learner to predict the $κ$ predominant classes of the i…
We propose a novel integrated formulation for multiclass and multilabel support vector machines (SVMs). A number of approaches have been proposed to extend the original binary SVM to an all-in-one multiclass SVM. However, its direct extension to a unified multilabel SVM has not been widely investigated. We propose a straightforward extension to the SVM to cope with multiclass and multilabel classification problems within a unified framework. Our framework deviates from the conventional soft margin SVM framework with its direct oppositional structure. In our formulation, class-specific weight v…
Budapest University of Technology and Economics（布达佩斯科技经济大学）；Ariel University（阿里埃勒大学）
We study universal consistency and convergence rates of simple nearest-neighbor prototype rules for the problem of multiclass classification in metric paces. We first show that a novel data-dependent partitioning rule, named Proto-NN, is universally consistent in any metric space that admits a universally consistent rule. Proto-NN is a significant simplification of OptiNet, a recently proposed compression-based algorithm that, to date, was the only algorithm known to be universally consistent in such a general setting. Practically, Proto-NN is simpler to implement and enjoys reduced computatio…
Multiclass classification problems such as image annotation can involve a large number of classes. In this context, confusion between classes can occur, and single label classification may be misleading. We provide in the present paper a general device that, given an unlabeled dataset and a score function defined as the minimizer of some empirical and convex risk, outputs a set of class labels, instead of a single one. Interestingly, this procedure does not require that the unlabeled dataset explores the whole classes. Even more, the method is calibrated to control the expected size of the out…
We study statistical properties of the k-nearest neighbors algorithm for multiclass classification, with a focus on settings where the number of classes may be large and/or classes may be highly imbalanced. In particular, we consider a variant of the k-nearest neighbor classifier with non-uniform class-weightings, for which we derive upper and minimax lower bounds on accuracy, class-weighted risk, and uniform error. Additionally, we show that uniform error bounds lead to bounds on the difference between empirical confusion matrix quantities and their population counterparts across a set of wei…
We consider the problem of cost sensitive multiclass classification, where we would like to increase the sensitivity of an important class at the expense of a less important one. We adopt an {\em apportioned margin} framework to address this problem, which enables an efficient margin shift between classes that share the same boundary. The decision boundary between all pairs of classes divides the margin between them in accordance to a given prioritization vector, which yields a tighter error bound for the important classes while also reducing the overall out-of-sample error. In addition to dem…
Implicit Bias of Spectral Descent and Muon on Multiclass Separable Data
Department of Computer Science University of British Columbia（计算机科学系不列颠哥伦比亚大学）；University of British Columbia Canada（不列颠哥伦比亚大学加拿大）；CIFAR AI Chair (Amii) &（CIFAR人工智能主席（Amii））；Department of Electrical and Computer Engineering University of British Columbia（电气与计算机工程系不列颠哥伦比亚大学）
University of Edinburgh（爱丁堡大学）；University of Cambridge（剑桥大学）
We derive a new asymptotic expansion for the global excess risk of a local-$k$-nearest neighbour classifier, where the choice of $k$ may depend upon the test point. This expansion elucidates conditions under which the dominant contribution to the excess risk comes from the decision boundary of the optimal Bayes classifier, but we also show that if these conditions are not satisfied, then the dominant contribution may arise from the tails of the marginal distribution of the features. Moreover, we prove that, provided the $d$-dimensional marginal distribution of the features has a finite $ρ$th m…
University of California, Berkeley（加州大学伯克利分校）；Queensland University of Technology（昆士兰科技大学）；Cornell University（康奈尔大学）；University of Illinois, Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
This paper presents a margin-based multiclass generalization bound for neural networks that scales with their margin-normalized "spectral complexity": their Lipschitz constant, meaning the product of the spectral norms of the weight matrices, times a certain correction factor. This bound is empirically investigated for a standard AlexNet network trained with SGD on the mnist and cifar10 datasets, with both original and random labels; the bound, the Lipschitz constants, and the excess risks are all in direct correlation, suggesting both that SGD selects predictors whose complexity scales with t…
In this paper we propose and explore the k-Nearest Neighbour UCB algorithm for multi-armed bandits with covariates. We focus on a setting where the covariates are supported on a metric space of low intrinsic dimension, such as a manifold embedded within a high dimensional ambient feature space. The algorithm is conceptually simple and straightforward to implement. The k-Nearest Neighbour UCB algorithm does not require prior knowledge of the either the intrinsic dimension of the marginal distribution or the time horizon. We prove a regret bound for the k-Nearest Neighbour UCB algorithm which is…
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）
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…
Institute for Information Transmission Problems RAS（俄罗斯科学院信息传输问题研究所）
We consider a problem of multiclass classification, where the training sample $S_n = \{(X_i, Y_i)\}_{i=1}^n$ is generated from the model $\mathbb P(Y = m | X = x) = η_m(x)$, $1 \leq m \leq M$, and $η_1(x), \dots, η_M(x)$ are unknown $α$-Holder continuous functions.Given a test point $X$, our goal is to predict its label. A widely used $\mathsf k$-nearest-neighbors classifier constructs estimates of $η_1(X), \dots, η_M(X)$ and uses a plug-in rule for the prediction. However, it requires a proper choice of the smoothing parameter $\mathsf k$, which may become tricky in some situations. In our so…
A Robust Twin Parametric Margin Support Vector Machine for Multiclass Classification
School of Science and Technology, University of Camerino（科学与技术学院，坎皮诺大学）；Department of Management, Information and Production Engineering, University of Bergamo（管理、信息与生产工程系，贝拉姆多大学）
As increasing amounts of sensitive personal information is aggregated into data repositories, it has become important to develop mechanisms for processing the data without revealing information about individual data instances. The differential privacy model provides a framework for the development and theoretical analysis of such mechanisms. In this paper, we propose an algorithm for learning a discriminatively trained multi-class Gaussian classifier that satisfies differential privacy using a large margin loss function with a perturbed regularization term. We present a theoretical upper bound…
Tsinghua University（清华大学）；University of California, Berkeley（加州大学伯克利分校）
This paper addresses the problem of unsupervised domain adaption from theoretical and algorithmic perspectives. Existing domain adaptation theories naturally imply minimax optimization algorithms, which connect well with the domain adaptation methods based on adversarial learning. However, several disconnections still exist and form the gap between theory and algorithm. We extend previous theories (Mansour et al., 2009c; Ben-David et al., 2010) to multiclass classification in domain adaptation, where classifiers based on the scoring functions and margin loss are standard choices in algorithm d…
We consider the problem of multi-class classification, where a stream of adversarially chosen queries arrive and must be assigned a label online. Unlike traditional bounds which seek to minimize the misclassification rate, we minimize the total distance from each query to the region corresponding to its correct label. When the true labels are determined via a nearest neighbor partition -- i.e. the label of a point is given by which of $k$ centers it is closest to in Euclidean distance -- we show that one can achieve a loss that is independent of the total number of queries. We complement this …
In this paper, we explore model-based approach to training robust and interpretable binarized regression models for multiclass classification tasks using Mixed-Integer Programming (MIP). Our MIP model balances the optimization of prediction margin and model size by using a weighted objective that: minimizes the total margin of incorrectly classified training instances, maximizes the total margin of correctly classified training instances, and maximizes the overall model regularization. We conduct two sets of experiments to test the classification accuracy of our MIP model over standard and cor…
Université Laval（拉瓦尔大学）；University of Ottawa（渥太华大学）
We significantly improve the generalization bounds for VC classes by using two main ideas. First, we consider the hypergeometric tail inversion to obtain a very tight non-uniform distribution-independent risk upper bound for VC classes. Second, we optimize the ghost sample trick to obtain a further non-negligible gain. These improvements are then used to derive a relative deviation bound, a multiclass margin bound, as well as a lower bound. Numerical comparisons show that the new bound is nearly never vacuous, and is tighter than other VC bounds for all reasonable data set sizes.
Multiclass Linear Perceptrons with Multiplicative Margins
Luleå University of Technology（吕勒奥理工大学）；Institute of Information Technologies and Systems（信息技术与系统研究所）；La Trobe University（拉筹伯大学）；Örebro University（厄勒布鲁大学）；Research Institutes of Sweden（瑞典研究院）
The University of Adelaide（阿德莱德大学）；Australian Centre for Visual Technologies（澳大利亚视觉技术中心）
Random projections have been applied in many machine learning algorithms. However, whether margin is preserved after random projection is non-trivial and not well studied. In this paper we analyse margin distortion after random projection, and give the conditions of margin preservation for binary classification problems. We also extend our analysis to margin for multiclass problems, and provide theoretical bounds on multiclass margin on the projected data.
Multiclass Graph-Based Large Margin Classifiers: Unified Approach for Support Vectors and Neural Networks
Graduate Program in Electrical Engineering, Universidade Federal de Minas Gerais（电气工程联合培养项目，巴西米纳斯吉拉斯联邦大学）；Department of Computer and Systems, Universidade Federal de Ouro Preto（计算机与系统系，巴西欧鲁普穆联邦大学）
The distance metric plays an important role in nearest neighbor (NN) classification. Usually the Euclidean distance metric is assumed or a Mahalanobis distance metric is optimized to improve the NN performance. In this paper, we study the problem of embedding arbitrary metric spaces into a Euclidean space with the goal to improve the accuracy of the NN classifier. We propose a solution by appealing to the framework of regularization in a reproducing kernel Hilbert space and prove a representer-like theorem for NN classification. The embedding function is then determined by solving a semidefini…
Text analytics based on supervised machine learning classifiers has shown great promise in a multitude of domains, but has yet to be applied to Seismology. We test various standard models (Naive Bayes, k-Nearest Neighbors, Support Vector Machines, and Random Forests) on a seismological corpus of 100 articles related to the topic of precursory accelerating seismicity, spanning from 1988 to 2010. This corpus was labelled in Mignan (2011) with the precursor whether explained by critical processes (i.e., cascade triggering) or by other processes (such as signature of main fault loading). We invest…
Beckman Institute, University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校贝克曼研究所）；Adobe Research（奥多比研究院）
Classifying large-scale image data into object categories is an important problem that has received increasing research attention. Given the huge amount of data, non-parametric approaches such as nearest neighbor classifiers have shown promising results, especially when they are underpinned by a learned distance or similarity measurement. Although metric learning has been well studied in the past decades, most existing algorithms are impractical to handle large-scale data sets. In this paper, we present an image similarity learning method that can scale well in both the number of images and th…
Renmin University of China（中国人民大学）；University of Surrey（萨里大学）；The University of Edinburgh（爱丁堡大学）
By transferring knowledge learned from seen/previous tasks, meta learning aims to generalize well to unseen/future tasks. Existing meta-learning approaches have shown promising empirical performance on various multiclass classification problems, but few provide theoretical analysis on the classifiers' generalization ability on future tasks. In this paper, under the assumption that all classification tasks are sampled from the same meta-distribution, we leverage margin theory and statistical learning theory to establish three margin-based transfer bounds for meta-learning based multiclass class…
Ben-Gurion University of the Negev（内盖夫本-古里安大学）；York University（约克大学）
We propose a pool-based non-parametric active learning algorithm for general metric spaces, called MArgin Regularized Metric Active Nearest Neighbor (MARMANN), which outputs a nearest-neighbor classifier. We give prediction error guarantees that depend on the noisy-margin properties of the input sample, and are competitive with those obtained by previously proposed passive learners. We prove that the label complexity of MARMANN is significantly lower than that of any passive learner with similar error guarantees. MARMANN is based on a generalized sample compression scheme, and a new label-effi…
Mississippi State University（密西西比州立大学）；International University of Business Agriculture and Technology（国际商业农业与技术大学）；Rajshahi University of Engineering and Technology（拉杰沙希工程技术大学）
Machine learning qualifies computers to assimilate with data, without being solely programmed [1, 2]. Machine learning can be classified as supervised and unsupervised learning. In supervised learning, computers learn an objective that portrays an input to an output hinged on training input-output pairs [3]. Most efficient and widely used supervised learning algorithms are K-Nearest Neighbors (KNN), Support Vector Machine (SVM), Large Margin Nearest Neighbor (LMNN), and Extended Nearest Neighbor (ENN). The main contribution of this paper is to implement these elegant learning algorithms on ele…
The condensed nearest neighbor (CNN) algorithm is a heuristic for reducing the number of prototypical points stored by a nearest neighbor classifier, while keeping the classification rule given by the reduced prototypical set consistent with the full set. I present an upper bound on the number of prototypical points accumulated by CNN. The bound originates in a bound on the number of times the decision rule is updated during training in the multiclass perceptron algorithm, and thus is independent of training set size.
We consider the online multiclass linear classification under the bandit feedback setting. Beygelzimer, Pál, Szörényi, Thiruvenkatachari, Wei, and Zhang [ICML'19] considered two notions of linear separability, weak and strong linear separability. When examples are strongly linearly separable with margin $γ$, they presented an algorithm based on Multiclass Perceptron with mistake bound $O(K/γ^2)$, where $K$ is the number of classes. They employed rational kernel to deal with examples under the weakly linearly separable condition, and obtained the mistake bound of $\min(K\cdot 2^{\tilde{O}(K\log…
Yahoo Research（雅虎研究院）；New York University（纽约大学）；University of Southern California（南加州大学）；Microsoft Research（微软研究院）
We study the problem of efficient online multiclass linear classification with bandit feedback, where all examples belong to one of $K$ classes and lie in the $d$-dimensional Euclidean space. Previous works have left open the challenge of designing efficient algorithms with finite mistake bounds when the data is linearly separable by a margin $γ$. In this work, we take a first step towards this problem. We consider two notions of linear separability: strong and weak. 1. Under the strong linear separability condition, we design an efficient algorithm that achieves a near-optimal mistake bound o…
Beijing University of Posts and Telecommunications（北京邮电大学）；China Telecom Corporation Limited Research Institute（中国电信股份有限公司研究院）
As deep learning becomes the mainstream in the field of natural language processing, the need for suitable active learning method are becoming unprecedented urgent. Active Learning (AL) methods based on nearest neighbor classifier are proposed and demonstrated superior results. However, existing nearest neighbor classifier are not suitable for classifying mutual exclusive classes because inter-class discrepancy cannot be assured by nearest neighbor classifiers. As a result, informative samples in the margin area can not be discovered and AL performance are damaged. To this end, we propose a no…
Ariel University（阿里尔大学）；Ben-Gurion University of the Negev（内盖夫本-古里安大学）；Weizmann Institute of Science（魏茨曼科学研究所）
Recent advances in large-margin classification of data residing in general metric spaces (rather than Hilbert spaces) enable classification under various natural metrics, such as string edit and earthmover distance. A general framework developed for this purpose by von Luxburg and Bousquet [JMLR, 2004] left open the questions of computational efficiency and of providing direct bounds on generalization error. We design a new algorithm for classification in general metric spaces, whose runtime and accuracy depend on the doubling dimension of the data points, and can thus achieve superior classif…
École Polytechnique（巴黎综合理工学院）；Sony Computer Science Laboratories（索尼计算机科学实验室）；Data61；The Australian National University (ANU)（澳大利亚国立大学）；The University of Sydney（悉尼大学）
We consider the supervised classification problem of machine learning in Cayley-Klein projective geometries: We show how to learn a curved Mahalanobis metric distance corresponding to either the hyperbolic geometry or the elliptic geometry using the Large Margin Nearest Neighbor (LMNN) framework. We report on our experimental results, and further consider the case of learning a mixed curved Mahalanobis distance. Besides, we show that the Cayley-Klein Voronoi diagrams are affine, and can be built from an equivalent (clipped) power diagrams, and that Cayley-Klein balls have Mahalanobis shapes wi…
Large margin nearest neighbor (LMNN) is a metric learner which optimizes the performance of the popular $k$NN classifier. However, its resulting metric relies on pre-selected target neighbors. In this paper, we address the feasibility of LMNN's optimization constraints regarding these target points, and introduce a mathematical measure to evaluate the size of the feasible region of the optimization problem. We enhance the optimization framework of LMNN by a weighting scheme which prefers data triplets which yield a larger feasible region. This increases the chances to obtain a good metric as t…
Metric learning is one of the techniques in manifold learning with the goal of finding a projection subspace for increasing and decreasing the inter- and intra-class variances, respectively. Some of the metric learning methods are based on triplet learning with anchor-positive-negative triplets. Large margin metric learning for nearest neighbor classification is one of the fundamental methods to do this. Recently, Siamese networks have been introduced with the triplet loss. Many triplet mining methods have been developed for Siamese networks; however, these techniques have not been applied on …
Indian Institute of Information Technology Design and Manufacturing Jabalpur（印度信息技术设计与制造学院）；Birla Institute of Technology and Science – Pilani（比尔拉理工学院）；National Institute of Technology Karnataka（卡纳塔克国立理工学院）；VIT University（韦洛尔理工大学）
Gear drives are one of the most widely used transmission system in many machinery. Sound signals of a rotating machine contain the dynamic information about its health conditions. Not much information available in the literature reporting suitability of sound signals for fault diagnosis applications. Maximum numbers of literature are based on FFT (Fast Fourier Transform) analysis and have its own limitations with non-stationary signals like the ones from gears. In this paper, attempt has been made in using sound signals acquired from gears in good and simulated faulty conditions for the purpos…
University of Arkansas（阿肯色大学）；University of Washington（华盛顿大学）；Washington University（华盛顿大学）；Klinikum rechts der Isar（伊萨尔河右岸医院）；Technical University of Munich（慕尼黑工业大学）
Deep representation learning using triplet network for classification suffers from a lack of theoretical foundation and difficulty in tuning both the network and classifiers for performance. To address the problem, local-margin triplet loss along with local positive and negative mining strategy is proposed with theory on how the strategy integrate nearest-neighbor hyper-parameter with triplet learning to increase subsequent classification performance. Results in experiments with 2 public datasets, MNIST and Cifar-10, and 2 small medical image datasets demonstrate that proposed strategy outperf…
Ariel University（阿里埃勒大学）；Ben-Gurion University of the Negev（内盖夫本-古里安大学）；Tel-Aviv University（特拉维夫大学）
We present the first sample compression algorithm for nearest neighbors with non-trivial performance guarantees. We complement these guarantees by demonstrating almost matching hardness lower bounds, which show that our bound is nearly optimal. Our result yields new insight into margin-based nearest neighbor classification in metric spaces and allows us to significantly sharpen and simplify existing bounds. Some encouraging empirical results are also presented.
The nearest neighbor rule is a classic yet essential classification model, particularly in problems where the supervising information is given by pairwise dissimilarities and the embedding function are not easily obtained. Prototype selection provides means of generalization and improving efficiency of the nearest neighbor model, but many existing methods assume and rely on the analyses of the input vector space. In this paper, we explore a dissimilarity-based, parametrized model of the nearest neighbor rule. In the proposed model, the selection of the nearest prototypes is influenced by the p…
Univ. of Milan（米兰大学）；Google（谷歌）；Istituto Italiano di Tecnologia（意大利技术研究院）；TU Wien（维也纳工业大学）
We study exact active learning of binary and multiclass classifiers with margin. Given an $n$-point set $X \subset \mathbb{R}^m$, we want to learn any unknown classifier on $X$ whose classes have finite strong convex hull margin, a new notion extending the SVM margin. In the standard active learning setting, where only label queries are allowed, learning a classifier with strong convex hull margin $γ$ requires in the worst case $Ω\big(1+\frac{1}γ\big)^{(m-1)/2}$ queries. On the other hand, using the more powerful seed queries (a variant of equivalence queries), the target classifier could be l…
Robustness is an increasingly important property of machine learning models as they become more and more prevalent. We propose a defense against adversarial examples based on a k-nearest neighbor (kNN) on the intermediate activation of neural networks. Our scheme surpasses state-of-the-art defenses on MNIST and CIFAR-10 against l2-perturbation by a significant margin. With our models, the mean perturbation norm required to fool our MNIST model is 3.07 and 2.30 on CIFAR-10. Additionally, we propose a simple certifiable lower bound on the l2-norm of the adversarial perturbation using a more spec…
