Support vector machine (SVM) has been one of the most popular learning algorithms, with the central idea of maximizing the minimum margin, i.e., the smallest distance from the instances to the classification boundary. Recent theoretical results, however, disclosed that maximizing the minimum margin does not necessarily lead to better generalization performances, and instead, the margin distribution has been proven to be more crucial. Based on this idea, we propose a new method, named Optimal margin Distribution Machine (ODM), which tries to achieve a better generalization performance by optimizing the margin distribution. We characterize the margin distribution by the first- and second-order statistics, i.e., the margin mean and variance. The proposed method is a general learning approach which can be used in any place where SVM can be applied, and their superiority is verified both theoretically and empirically in this paper.
Support vector machine (SVM) has been one of the most popular learning algorithms, with the central idea of maximizing the minimum margin, i.e., the smallest distance from the instances to the classification boundary. Recent theoretical results, however, disclosed that maximizing the minimum margin does not necessarily lead to better generalization performances, and instead, the margin distribution has been proven to be more crucial. In this paper, we propose the Large margin Distribution Machine (LDM), which tries to achieve a better generalization performance by optimizing the margin distrib…
Support Vector Machines (SVMs) are among the most fundamental tools for binary classification. In its simplest formulation, an SVM produces a hyperplane separating two classes of data using the largest possible margin to the data. The focus on maximizing the margin has been well motivated through numerous generalization bounds. In this paper, we revisit and improve the classic generalization bounds in terms of margins. Furthermore, we complement our new generalization bound by a nearly matching lower bound, thus almost settling the generalization performance of SVMs in terms of margins.
Current statistical models for structured prediction make simplifying assumptions about the underlying output graph structure, such as assuming a low-order Markov chain, because exact inference becomes intractable as the tree-width of the underlying graph increases. Approximate inference algorithms, on the other hand, force one to trade off representational power with computational efficiency. In this paper, we propose two new types of probabilistic graphical models, large margin Boltzmann machines (LMBMs) and large margin sigmoid belief networks (LMSBNs), for structured prediction. LMSBNs in …
Toyota Technological Institute at Chicago（芝加哥丰田理工学院）；Ben-Gurion University（本-古里安大学）
We analyze a family of supervised learning algorithms based on sample compression schemes that are stable, in the sense that removing points from the training set which were not selected for the compression set does not alter the resulting classifier. We use this technique to derive a variety of novel or improved data-dependent generalization bounds for several learning algorithms. In particular, we prove a new margin bound for SVM, removing a log factor. The new bound is provably optimal. This resolves a long-standing open question about the PAC margin bounds achievable by SVM.
The support vector machine (SVM) is an important class of learning machines for function approach, pattern recognition, and time-serious prediction, etc. It maps samples into the feature space by so-called support vectors of selected samples, and then feature vectors are separated by maximum margin hyperplane. The present paper presents the general vector machine (GVM) to replace the SVM. The support vectors are replaced by general project vectors selected from the usual vector space, and a Monte Carlo (MC) algorithm is developed to find the general vectors. The general project vectors improve…
This paper studies binary classification problem associated with a family of loss functions called large-margin unified machines (LUM), which offers a natural bridge between distribution-based likelihood approaches and margin-based approaches. It also can overcome the so-called data piling issue of support vector machine in the high-dimension and low-sample size setting. In this paper we establish some new comparison theorems for all LUM loss functions which play a key role in the further error analysis of large-margin learning algorithms.
Boosting algorithms produce a classifier by iteratively combining base hypotheses. It has been observed experimentally that the generalization error keeps improving even after achieving zero training error. One popular explanation attributes this to improvements in margins. A common goal in a long line of research, is to maximize the smallest margin using as few base hypotheses as possible, culminating with the AdaBoostV algorithm by (R{ä}tsch and Warmuth [JMLR'04]). The AdaBoostV algorithm was later conjectured to yield an optimal trade-off between number of hypotheses trained and the minimal…
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）
Pontifical Catholic University of Rio de Janeiro（里约热内卢天主教大学）；Technical University of Munich（慕尼黑工业大学）；Polytechnique Montréal（蒙特利尔理工学院）
The classical hinge-loss support vector machines (SVMs) model is sensitive to outlier observations due to the unboundedness of its loss function. To circumvent this issue, recent studies have focused on non-convex loss functions, such as the hard-margin loss, which associates a constant penalty to any misclassified or within-margin sample. Applying this loss function yields much-needed robustness for critical applications but it also leads to an NP-hard model that makes training difficult, since current exact optimization algorithms show limited scalability, whereas heuristics are not able to …
Hainan University（海南大学）；Zhijiang College, Zhejiang University of Technology（浙江工业大学之江学院）
Recent advance on linear support vector machine with the 0-1 soft margin loss ($L_{0/1}$-SVM) shows that the 0-1 loss problem can be solved directly. However, its theoretical and algorithmic requirements restrict us extending the linear solving framework to its nonlinear kernel form directly, the absence of explicit expression of Lagrangian dual function of $L_{0/1}$-SVM is one big deficiency among of them. In this paper, by applying the nonparametric representation theorem, we propose a nonlinear model for support vector machine with 0-1 soft margin loss, called $L_{0/1}$-KSVM, which cunningl…
The foundational concept of Max-Margin in machine learning is ill-posed for output spaces with more than two labels such as in structured prediction. In this paper, we show that the Max-Margin loss can only be consistent to the classification task under highly restrictive assumptions on the discrete loss measuring the error between outputs. These conditions are satisfied by distances defined in tree graphs, for which we prove consistency, thus being the first losses shown to be consistent for Max-Margin beyond the binary setting. We finally address these limitations by correcting the concept o…
We study the task of agnostically learning halfspaces under the Gaussian distribution. Specifically, given labeled examples $(\mathbf{x},y)$ from an unknown distribution on $\mathbb{R}^n \times \{ \pm 1\}$, whose marginal distribution on $\mathbf{x}$ is the standard Gaussian and the labels $y$ can be arbitrary, the goal is to output a hypothesis with 0-1 loss $\mathrm{OPT}+ε$, where $\mathrm{OPT}$ is the 0-1 loss of the best-fitting halfspace. We prove a near-optimal computational hardness result for this task, under the widely believed sub-exponential time hardness of the Learning with Errors…
Beijing Jiaotong University（北京交通大学）；Hainan University（海南大学）；University of Southampton（南安普顿大学）
Support vector machine (SVM) has attracted great attentions for the last two decades due to its extensive applications, and thus numerous optimization models have been proposed. To distinguish all of them, in this paper, we introduce a new model equipped with an $L_{0/1}$ soft-margin loss (dubbed as $L_{0/1}$-SVM) which well captures the nature of the binary classification. Many of the existing convex/non-convex soft-margin losses can be viewed as a surrogate of the $L_{0/1}$ soft-margin loss. Despite the discrete nature of $L_{0/1}$, we manage to establish the existence of global minimizer of…
We present a flow-based approach to the optimal transport (OT) problem between two continuous distributions $π_0,π_1$ on $\mathbb{R}^d$, of minimizing a transport cost $\mathbb{E}[c(X_1-X_0)]$ in the set of couplings $(X_0,X_1)$ whose marginal distributions on $X_0,X_1$ equals $π_0,π_1$, respectively, where $c$ is a cost function. Our method iteratively constructs a sequence of neural ordinary differentiable equations (ODE), each learned by solving a simple unconstrained regression problem, which monotonically reduce the transport cost while automatically preserving the marginal constraints. T…
In this paper, we explore and detail our experiments in a high-dimensionality, multi-class image classification problem often found in the automatic recognition of Sign Languages. Here, our efforts are directed towards comparing the characteristics, advantages and drawbacks of creating and training Support Vector Machines disposed in a Directed Acyclic Graph and Artificial Neural Networks to classify signs from the Brazilian Sign Language (LIBRAS). We explore how the different heuristics, hyperparameters and multi-class decision schemes affect the performance, efficiency and ease of use for ea…
Multi-marginal optimal transport (MOT) is a generalization of optimal transport to multiple marginals. Optimal transport has evolved into an important tool in many machine learning applications, and its multi-marginal extension opens up for addressing new challenges in the field of machine learning. However, the usage of MOT has been largely impeded by its computational complexity which scales exponentially in the number of marginals. Fortunately, in many applications, such as barycenter or interpolation problems, the cost function adheres to structures, which has recently been exploited for d…
University of Georgia（佐治亚大学）；Zhongnan University of Economics and Law（中南财经政法大学）
Margin-based classifiers have been popular in both machine learning and statistics for classification problems. Since a large number of classifiers are available, one natural question is which type of classifiers should be used given a particular classification task. We answer this question by investigating the asymptotic performance of a family of large-margin classifiers under the two component mixture models in situations where the data dimension $p$ and the sample $n$ are both large. This family covers a broad range of classifiers including support vector machine, distance weighted discrim…
Rensselaer Polytechnic Institute（伦斯勒理工学院）；IBM T.J. Watson Research Center（IBM托马斯·J·沃森研究中心）
Let X be a data matrix of rank ρ, whose rows represent n points in d-dimensional space. The linear support vector machine constructs a hyperplane separator that maximizes the 1-norm soft margin. We develop a new oblivious dimension reduction technique which is precomputed and can be applied to any input matrix X. We prove that, with high probability, the margin and minimum enclosing ball in the feature space are preserved to within ε-relative error, ensuring comparable generalization as in the original space in the case of classification. For regression, we show that the margin is preserved to…
Computationally solving multi-marginal optimal transport (MOT) with squared Euclidean costs for $N$ discrete probability measures has recently attracted considerable attention, in part because of the correspondence of its solutions with Wasserstein-$2$ barycenters, which have many applications in data science. In general, this problem is NP-hard, calling for practical approximative algorithms. While entropic regularization has been successfully applied to approximate Wasserstein barycenters, this loses the sparsity of the optimal solution, making it difficult to solve the MOT problem directly …
Shanghai Tenth People's Hospital（上海市第十人民医院）；Tongji University（同济大学）；University of Texas at Arlington（德克萨斯大学阿灵顿分校）；Jilin University（吉林大学）；The Ohio State University（俄亥俄州立大学）
Support vector regression (SVR) is one of the most popular machine learning algorithms aiming to generate the optimal regression curve through maximizing the minimal margin of selected training samples, i.e., support vectors. Recent researchers reveal that maximizing the margin distribution of whole training dataset rather than the minimal margin of a few support vectors, is prone to achieve better generalization performance. However, the margin distribution support vector regression machines suffer difficulties resulted from solving a non-convex quadratic optimization, compared to the margin …
Indian Institute of Science（印度科学学院）；Washington University in St. Louis（圣路易斯华盛顿大学）
The new generation of machine learning processors have evolved from multi-core and parallel architectures that were designed to efficiently implement matrix-vector-multiplications (MVMs). This is because at the fundamental level, neural network and machine learning operations extensively use MVM operations and hardware compilers exploit the inherent parallelism in MVM operations to achieve hardware acceleration on GPUs and FPGAs. However, many IoT and edge computing platforms require embedded ML devices close to the network in order to compensate for communication cost and latency. Hence a nat…
University of Wisconsin-Madison（威斯康星大学麦迪逊分校）；University of California, San Diego（加利福尼亚大学圣迭戈分校）
We study the fundamental problems of agnostically learning halfspaces and ReLUs under Gaussian marginals. In the former problem, given labeled examples $(\mathbf{x}, y)$ from an unknown distribution on $\mathbb{R}^d \times \{ \pm 1\}$, whose marginal distribution on $\mathbf{x}$ is the standard Gaussian and the labels $y$ can be arbitrary, the goal is to output a hypothesis with 0-1 loss $\mathrm{OPT}+ε$, where $\mathrm{OPT}$ is the 0-1 loss of the best-fitting halfspace. In the latter problem, given labeled examples $(\mathbf{x}, y)$ from an unknown distribution on $\mathbb{R}^d \times \mathb…
EPFL（洛桑联邦理工学院）；Singapore University of Technology and Design（新加坡科技设计大学）
We study the computational complexity of the optimal transport problem that evaluates the Wasserstein distance between the distributions of two K-dimensional discrete random vectors. The best known algorithms for this problem run in polynomial time in the maximum of the number of atoms of the two distributions. However, if the components of either random vector are independent, then this number can be exponential in K even though the size of the problem description scales linearly with K. We prove that the described optimal transport problem is #P-hard even if all components of the first rando…
