Using a support vector machine requires to set two types of hyperparameters: the soft margin parameter C and the parameters of the kernel. To perform this model selection task, the method of choice is cross-validation. Its leave-one-out variant is known to produce an estimator of the generalization error which is almost unbiased. Its major drawback rests in its time requirement. To overcome this difficulty, several upper bounds on the leave-one-out error of the pattern recognition SVM have been derived. Among those bounds, the most popular one is probably the radius-margin bound. It applies to the hard margin pattern recognition SVM, and by extension to the 2-norm SVM. In this report, we introduce a quadratic loss M-SVM, the M-SVM^2, as a direct extension of the 2-norm SVM to the multi-class case. For this machine, a generalized radius-margin bound is then established.
Sparsity-inducing penalties are useful tools to design multiclass support vector machines (SVMs). In this paper, we propose a convex optimization approach for efficiently and exactly solving the multiclass SVM learning problem involving a sparse regularization and the multiclass hinge loss formulated by Crammer and Singer. We provide two algorithms: the first one dealing with the hinge loss as a penalty term, and the other one addressing the case when the hinge loss is enforced through a constraint. The related convex optimization problems can be efficiently solved thanks to the flexibility of…
Support Vector Machines (SVMs) are among the most popular and the best performing classification algorithms. Various approaches have been proposed to reduce the high computation and memory cost when training and predicting based on large-scale datasets with kernel SVMs. A popular one is the linearization framework, which successfully builds a bridge between the $L_1$-loss kernel SVM and the $L_1$-loss linear SVM. For linear SVMs, very recently, a semismooth Newton's method is proposed. It is shown to be very competitive and have low computational cost. Consequently, a natural question is wheth…
University of Michigan（密歇根大学）；Department of Mathematics（数学系）；Department of Statistics（统计系）；Ann Arbor, MI, 48109（密歇根州安娜堡市（48109））
We prove that, under low noise assumptions, the support vector machine with $N\ll m$ random features (RFSVM) can achieve the learning rate faster than $O(1/\sqrt{m})$ on a training set with $m$ samples when an optimized feature map is used. Our work extends the previous fast rate analysis of random features method from least square loss to 0-1 loss. We also show that the reweighted feature selection method, which approximates the optimized feature map, helps improve the performance of RFSVM in experiments on a synthetic data set.
Support vector machine (SVM) is a powerful classification method that has achieved great success in many fields. Since its performance can be seriously impaired by redundant covariates, model selection techniques are widely used for SVM with high dimensional covariates. As an alternative to model selection, significant progress has been made in the area of model averaging in the past decades. Yet no frequentist model averaging method was considered for SVM. This work aims to fill the gap and to propose a frequentist model averaging procedure for SVM which selects the optimal weight by cross va…
Institut für Neuroinformatik, Ruhr-Universität Bochum（波鸿鲁尔大学神经信息学研究所）
Limiting the model size of a kernel support vector machine to a pre-defined budget is a well-established technique that allows to scale SVM learning and prediction to large-scale data. Its core addition to simple stochastic gradient training is budget maintenance through merging of support vectors. This requires solving an inner optimization problem with an iterative method many times per gradient step. In this paper we replace the iterative procedure with a fast lookup. We manage to reduce the merging time by up to 65% and the total training time by 44% without any loss of accuracy.
In this paper, we have considered general k-piece-wise linear convex loss functions in SVM model for measuring the empirical risk. The resulting k-Piece-wise Linear loss Support Vector Machine (k-PL-SVM) model is an adaptive SVM model which can learn a suitable piece-wise linear loss function according to nature of the given training set. The k-PL-SVM models are general SVM models and existing popular SVM models, like C-SVM, LS-SVM and Pin-SVM models, are their particular cases. We have performed the extensive numerical experiments with k-PL-SVM models for k = 2 and 3 and shown that they are i…
National Research Institute of Electronics and Cryptology, TUBITAK（土耳其科学技术研究委员会电子学与密码学国家研究所）；Istanbul University（伊斯坦布尔大学）
Although Support Vector Machine (SVM) algorithm has a high generalization property to classify for unseen examples after training phase and it has small loss value, the algorithm is not suitable for real-life classification and regression problems. SVMs cannot solve hundreds of thousands examples in training dataset. In previous studies on distributed machine learning algorithms, SVM is trained over a costly and preconfigured computer environment. In this research, we present a MapReduce based distributed parallel SVM training algorithm for binary classification problems. This work shows how t…
Support vector machines (SVMs) are invaluable tools for many practical applications in artificial intelligence, e.g., classification and event recognition. However, popular SVM solvers are not sufficiently efficient for applications with a great deal of samples as well as a large number of features. In this paper, thus, we present NESVM, a fast gradient SVM solver that can optimize various SVM models, e.g., classical SVM, linear programming SVM and least square SVM. Compared against SVM-Perf \cite{SVM_Perf}\cite{PerfML} (its convergence rate in solving the dual SVM is upper bounded by $\mathca…
Universidad Técnica Federico Santa María（费德里科·圣玛丽亚理工大学）；University of Insubria（因苏布里亚大学）；University of Bologna（博洛尼亚大学）
Recently, there has been a renewed interest in the machine learning community for variants of a sparse greedy approximation procedure for concave optimization known as {the Frank-Wolfe (FW) method}. In particular, this procedure has been successfully applied to train large-scale instances of non-linear Support Vector Machines (SVMs). Specializing FW to SVM training has allowed to obtain efficient algorithms but also important theoretical results, including convergence analysis of training algorithms and new characterizations of model sparsity. In this paper, we present and analyze a novel vari…
Sichuan University（四川大学）；Southwest Jiaotong University（西南交通大学）；Zhejiang University City College（浙江大学城市学院）
Support vector machines (SVMs) are successful modeling and prediction tools with a variety of applications. Previous work has demonstrated the superiority of the SVMs in dealing with the high dimensional, low sample size problems. However, the numerical difficulties of the SVMs will become severe with the increase of the sample size. Although there exist many solvers for the SVMs, only few of them are designed by exploiting the special structures of the SVMs. In this paper, we propose a highly efficient sparse semismooth Newton based augmented Lagrangian method for solving a large-scale convex…
A quadratic approximation of neural network loss landscapes has been extensively used to study the optimization process of these networks. Though, it usually holds in a very small neighborhood of the minimum, it cannot explain many phenomena observed during the optimization process. In this work, we study the structure of neural network loss functions and its implication on optimization in a region beyond the reach of a good quadratic approximation. Numerically, we observe that neural network loss functions possesses a multiscale structure, manifested in two ways: (1) in a neighborhood of mini…
Feature selection with specific multivariate performance measures is the key to the success of many applications, such as image retrieval and text classification. The existing feature selection methods are usually designed for classification error. In this paper, we propose a generalized sparse regularizer. Based on the proposed regularizer, we present a unified feature selection framework for general loss functions. In particular, we study the novel feature selection paradigm by optimizing multivariate performance measures. The resultant formulation is a challenging problem for high-dimension…
The linear Support Vector Machine (SVM) is a classic classification technique in machine learning. Motivated by applications in modern high dimensional statistics, we consider penalized SVM problems involving the minimization of a hinge-loss function with a convex sparsity-inducing regularizer such as: the L1-norm on the coefficients, its grouped generalization and the sorted L1-penalty (aka Slope). Each problem can be expressed as a Linear Program (LP) and is computationally challenging when the number of features and/or samples is large -- the current state of algorithms for these problems i…
University of California San Diego（加州大学圣迭戈分校）；University of Technology Sydney（悉尼科技大学）；IBM Research（IBM研究院）
Recent research shows that the dynamics of an infinitely wide neural network (NN) trained by gradient descent can be characterized by Neural Tangent Kernel (NTK) \citep{jacot2018neural}. Under the squared loss, the infinite-width NN trained by gradient descent with an infinitely small learning rate is equivalent to kernel regression with NTK \citep{arora2019exact}. However, the equivalence is only known for ridge regression currently \citep{arora2019harnessing}, while the equivalence between NN and other kernel machines (KMs), e.g. support vector machine (SVM), remains unknown. Therefore, in t…
Universita’ degli Studi ”Ca’ Foscari” di Venezia（威尼斯“卡·福斯卡里”大学）
We consider some supervised binary classification tasks and a regression task, whereas SVM and Deep Learning, at present, exhibit the best generalization performances. We extend the work [3] on a generalized quadratic loss for learning problems that examines pattern correlations in order to concentrate the learning problem into input space regions where patterns are more densely distributed. From a shallow methods point of view (e.g.: SVM), since the following mathematical derivation of problem (9) in [3] is incorrect, we restart from problem (8) in [3] and we try to solve it with one procedur…
Stochastic Gradient Descent (SGD) has become popular for solving large scale supervised machine learning optimization problems such as SVM, due to their strong theoretical guarantees. While the closely related Dual Coordinate Ascent (DCA) method has been implemented in various software packages, it has so far lacked good convergence analysis. This paper presents a new analysis of Stochastic Dual Coordinate Ascent (SDCA) showing that this class of methods enjoy strong theoretical guarantees that are comparable or better than SGD. This analysis justifies the effectiveness of SDCA for practical a…
Berlin Big Data Center（柏林大数据中心）；Berlin Institute of Technology（柏林理工学院）；Microsoft Research（微软研究院）；Humboldt University of Berlin（柏林洪堡大学）
Training of one-vs.-rest SVMs can be parallelized over the number of classes in a straight forward way. Given enough computational resources, one-vs.-rest SVMs can thus be trained on data involving a large number of classes. The same cannot be stated, however, for the so-called all-in-one SVMs, which require solving a quadratic program of size quadratically in the number of classes. We develop distributed algorithms for two all-in-one SVM formulations (Lee et al. and Weston and Watkins) that parallelize the computation evenly over the number of classes. This allows us to compare these models t…
University of Washington（华盛顿大学）；Rutgers University（罗格斯大学）；University of Wisconsin-Madison（威斯康星大学麦迪逊分校）
Existing results for low-rank matrix recovery largely focus on quadratic loss, which enjoys favorable properties such as restricted strong convexity/smoothness (RSC/RSM) and well conditioning over all low rank matrices. However, many interesting problems involve more general, non-quadratic losses, which do not satisfy such properties. For these problems, standard nonconvex approaches such as rank-constrained projected gradient descent (a.k.a. iterative hard thresholding) and Burer-Monteiro factorization could have poor empirical performance, and there is no satisfactory theory guaranteeing glo…
Xidian University（西安电子科技大学）；Beijing University of Posts and Telecommunications（北京邮电大学）
Over the past two decades, support vector machine (SVM) has become a popular supervised machine learning model, and plenty of distinct algorithms are designed separately based on different KKT conditions of the SVM model for classification/regression with different losses, including the convex loss or nonconvex loss. In this paper, we propose an algorithm that can train different SVM models in a \emph{unified} scheme. First, we introduce a definition of the \emph{LS-DC} (\textbf{l}east \textbf{s}quares type of \textbf{d}ifference of \textbf{c}onvex) loss and show that the most commonly used lo…
In this article, we propose a new Support Vector Machine (SVM) training algorithm based on distributed MapReduce technique. In literature, there are a lots of research that shows us SVM has highest generalization property among classification algorithms used in machine learning area. Also, SVM classifier model is not affected by correlations of the features. But SVM uses quadratic optimization techniques in its training phase. The SVM algorithm is formulated as quadratic optimization problem. Quadratic optimization problem has $O(m^3)$ time and $O(m^2)$ space complexity, where m is the trainin…
Two popular approaches for distributed training of SVMs on big data are parameter averaging and ADMM. Parameter averaging is efficient but suffers from loss of accuracy with increase in number of partitions, while ADMM in the feature space is accurate but suffers from slow convergence. In this paper, we report a hybrid approach called weighted parameter averaging (WPA), which optimizes the regularized hinge loss with respect to weights on parameters. The problem is shown to be same as solving SVM in a projected space. We also demonstrate an $O(\frac{1}{N})$ stability bound on final hypothesis …
Department of Data Science and Artificial Intelligence, The Hong Kong Polytechnic University（数据科学与人工智能系，香港理工大学）；School of Mathematics and Statistics, Beijing Jiaotong University（数学与统计学学院，北京交通大学）；Department of Data Science and Artificial Intelligence, and Department of Applied Mathematics, The Hong Kong Polytechnic University（数据科学与人工智能系和应用数学系，香港理工大学）
The binomial deviance and the SVM hinge loss functions are two of the most widely used loss functions in machine learning. While there are many similarities between them, they also have their own strengths when dealing with different types of data. In this work, we introduce a new exponential family based on a convex relaxation of the hinge loss function using softness and class-separation parameters. This new family, denoted Soft-SVM, allows us to prescribe a generalized linear model that effectively bridges between logistic regression and SVM classification. This new model is interpretable a…
Dhirubhai Ambani Institute of Information Technology（迪鲁巴伊·安巴尼信息技术学院）；South Asian University（南亚大学）
Recently, there have been several papers that discuss the extension of the Pinball loss Support Vector Machine (Pin-SVM) model, originally proposed by Huang et al.,[1][2]. Pin-SVM classifier deals with the pinball loss function, which has been defined in terms of the parameter $τ$. The parameter $τ$ can take values in $[ -1,1]$. The existing Pin-SVM model requires to solve the same optimization problem for all values of $τ$ in $[ -1,1]$. In this paper, we improve the existing Pin-SVM model for the binary classification task. At first, we note that there is major difficulty in Pin-SVM model (Hu…
The support vector machines (SVM) is one of the most widely used and practical optimization based classification models in machine learning because of its interpretability and flexibility to produce high quality results. However, the big data imposes a certain difficulty to the most sophisticated but relatively slow versions of SVM, namely, the nonlinear SVM. The complexity of nonlinear SVM solvers and the number of elements in the kernel matrix quadratically increases with the number of samples in training data. Therefore, both runtime and memory requirements are negatively affected. Moreover…
INSERM U556（法国国家健康与医学研究院 U556）；CREATIS, UMR CNRS 5220（CREATIS，法国国家科学研究中心混合研究单位 5220）；INSERM U1044（法国国家健康与医学研究院 U1044）；INSA-Lyon（里昂国立应用科学学院）；UCBL（里昂第一大学）；LITIS EA 4108, INSA-Universite de Rouen（鲁昂国立应用科学学院 LITIS EA 4108）
This paper addresses the pattern classification problem arising when available target data include some uncertainty information. Target data considered here is either qualitative (a class label) or quantitative (an estimation of the posterior probability). Our main contribution is a SVM inspired formulation of this problem allowing to take into account class label through a hinge loss as well as probability estimates using epsilon-insensitive cost function together with a minimum norm (maximum margin) objective. This formulation shows a dual form leading to a quadratic problem and allows the u…
Support Vector Machine (SVM) is an effective model for many classification problems. However, SVM needs the solution of a quadratic program which require specialized code. In addition, SVM has many parameters, which affects the performance of SVM classifier. Recently, the Generalized Eigenvalue Proximal SVM (GEPSVM) has been presented to solve the SVM complexity. In real world applications data may affected by error or noise, working with this data is a challenging problem. In this paper, an approach has been proposed to overcome this problem. This method is called DSA-GEPSVM. The main improve…
Shiraz University（设拉子大学）；University of California, San Diego（加州大学圣地亚哥分校）
A new procedure for learning cost-sensitive SVM(CS-SVM) classifiers is proposed. The SVM hinge loss is extended to the cost sensitive setting, and the CS-SVM is derived as the minimizer of the associated risk. The extension of the hinge loss draws on recent connections between risk minimization and probability elicitation. These connections are generalized to cost-sensitive classification, in a manner that guarantees consistency with the cost-sensitive Bayes risk, and associated Bayes decision rule. This ensures that optimal decision rules, under the new hinge loss, implement the Bayes-optimal…
Nagoya University（名古屋大学）；The University of Tokyo（东京大学）
The support vector machine (SVM) is one of the most successful learning methods for solving classification problems. Despite its popularity, SVM has a serious drawback, that is sensitivity to outliers in training samples. The penalty on misclassification is defined by a convex loss called the hinge loss, and the unboundedness of the convex loss causes the sensitivity to outliers. To deal with outliers, robust variants of SVM have been proposed, such as the robust outlier detection algorithm and an SVM with a bounded loss called the ramp loss. In this paper, we propose a robust variant of SVM a…
Dropout and other feature noising schemes have shown promising results in controlling over-fitting by artificially corrupting the training data. Though extensive theoretical and empirical studies have been performed for generalized linear models, little work has been done for support vector machines (SVMs), one of the most successful approaches for supervised learning. This paper presents dropout training for linear SVMs. To deal with the intractable expectation of the non-smooth hinge loss under corrupting distributions, we develop an iteratively re-weighted least square (IRLS) algorithm by e…
Wenzhou University（温州大学）；Jiangxi Normal University（江西师范大学）
This paper aims at refined error analysis for binary classification using support vector machine (SVM) with Gaussian kernel and convex loss. Our first result shows that for some loss functions such as the truncated quadratic loss and quadratic loss, SVM with Gaussian kernel can reach the almost optimal learning rate, provided the regression function is smooth. Our second result shows that, for a large number of loss functions, under some Tsybakov noise assumption, if the regression function is infinitely smooth, then SVM with Gaussian kernel can achieve the learning rate of order $m^{-1}$, whe…
Multiclass extensions of the support vector machine (SVM) have been formulated in a variety of ways. A recent empirical comparison of nine such formulations [Doǧan et al. 2016] recommends the variant proposed by Weston and Watkins (WW), despite the fact that the WW-hinge loss is not calibrated with respect to the 0-1 loss. In this work we introduce a novel discrete loss function for multiclass classification, the ordered partition loss, and prove that the WW-hinge loss is calibrated with respect to this loss. We also argue that the ordered partition loss is maximally informative among discrete…
The lack of interpretability often makes black-box models difficult to be applied to many practical domains. For this reason, the current work, from the black-box model input port, proposes to incorporate data-based prior information into the black-box soft-margin SVM model to enhance its interpretability. The concept and incorporation mechanism of data-based prior information are successively developed, based on which the interpretable or partly interpretable SVM optimization model is designed and then solved through handily rewriting the optimization problem as a nonlinear quadratic programm…
Faster Algorithms for Structured Linear and Kernel Support Vector Machines
New York University（纽约大学）；Simons Institute for the Theory of Computing（西蒙斯理论计算研究所）；UC Berkeley（加州大学伯克利分校）；Massachusetts Institute of Technology（麻省理工学院）
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…
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…
University of Chicago（芝加哥大学）；Australian National University（澳大利亚国立大学）；NICTA（国家信息与通信技术研究院）；Purdue University（普渡大学）
Regularized risk minimization with the binary hinge loss and its variants lies at the heart of many machine learning problems. Bundle methods for regularized risk minimization (BMRM) and the closely related SVMStruct are considered the best general purpose solvers to tackle this problem. It was recently shown that BMRM requires $O(1/ε)$ iterations to converge to an $ε$ accurate solution. In the first part of the paper we use the Hadamard matrix to construct a regularized risk minimization problem and show that these rates cannot be improved. We then show how one can exploit the structure of th…
Graz University of Technology（格拉茨工业大学）；University of Bath（巴斯大学）；University of Trieste（的里雅斯特大学）；Inria（法国国家信息与自动化研究所）；CentraleSupélec（巴黎中央理工-高等电力学院）；University Paris Saclay（巴黎萨克雷大学）
Lehigh University（理海大学）；University of Edinburgh（爱丁堡大学）；Toyota Technological Institute at Chicago（芝加哥丰田技术研究院）
We present an improved analysis of mini-batched stochastic dual coordinate ascent for regularized empirical loss minimization (i.e. SVM and SVM-type objectives). Our analysis allows for flexible sampling schemes, including where data is distribute across machines, and combines a dependence on the smoothness of the loss and/or the data spread (measured through the spectral norm).
School of Liberal Arts, Korea University of Technology and Education（韩国技术教育大学文理学院）
Classification is the most important process in data analysis. However, due to the inherent non-convex and non-smooth structure of the zero-one loss function of the classification model, various convex surrogate loss functions such as hinge loss, squared hinge loss, logistic loss, and exponential loss are introduced. These loss functions have been used for decades in diverse classification models, such as SVM (support vector machine) with hinge loss, logistic regression with logistic loss, and Adaboost with exponential loss and so on. In this work, we present a Perceptron-augmented convex clas…
Recently, fully-connected and convolutional neural networks have been trained to achieve state-of-the-art performance on a wide variety of tasks such as speech recognition, image classification, natural language processing, and bioinformatics. For classification tasks, most of these "deep learning" models employ the softmax activation function for prediction and minimize cross-entropy loss. In this paper, we demonstrate a small but consistent advantage of replacing the softmax layer with a linear support vector machine. Learning minimizes a margin-based loss instead of the cross-entropy loss. …
Sun Yat-sen University（中山大学）；School of Intelligent Systems Engineering（智能系统工程学院）
This paper presents a Multiple Kernel Learning (abbreviated as MKL) framework for the Support Vector Machine (SVM) with the $(0, 1)$ loss function. Some KKT-like first-order optimality conditions are provided and then exploited to develop a fast ADMM algorithm to solve the nonsmooth nonconvex optimization problem. Numerical experiments on real data sets show that the performance of our MKL-$L_{0/1}$-SVM is comparable with the one of the leading approaches called SimpleMKL developed by Rakotomamonjy, Bach, Canu, and Grandvalet [Journal of Machine Learning Research, vol. 9, pp. 2491-2521, 2008].
