We present an efficient coreset construction algorithm for large-scale Support Vector Machine (SVM) training in Big Data and streaming applications. A coreset is a small, representative subset of the original data points such that a models trained on the coreset are provably competitive with those trained on the original data set. Since the size of the coreset is generally much smaller than the original set, our preprocess-then-train scheme has potential to lead to significant speedups when training SVM models. We prove lower and upper bounds on the size of the coreset required to obtain small data summaries for the SVM problem. As a corollary, we show that our algorithm can be used to extend the applicability of any off-the-shelf SVM solver to streaming, distributed, and dynamic data settings. We evaluate the performance of our algorithm on real-world and synthetic data sets. Our experimental results reaffirm the favorable theoretical properties of our algorithm and demonstrate its practical effectiveness in accelerating SVM training.
We present a novel coreset construction algorithm for solving classification tasks using Support Vector Machines (SVMs) in a computationally efficient manner. A coreset is a weighted subset of the original data points that provably approximates the original set. We show that coresets of size polylogarithmic in $n$ and polynomial in $d$ exist for a set of $n$ input points with $d$ features and present an $(ε,δ)$-FPRAS for constructing coresets for scalable SVM training. Our method leverages the insight that data points are often redundant and uses an importance sampling scheme based on the sens…
Factorizing tensors has recently become an important optimization module in a number of machine learning pipelines, especially in latent variable models. We show how to do this efficiently in the streaming setting. Given a set of $n$ vectors, each in $\mathbb{R}^d$, we present algorithms to select a sublinear number of these vectors as coreset, while guaranteeing that the CP decomposition of the $p$-moment tensor of the coreset approximates the corresponding decomposition of the $p$-moment tensor computed from the full data. We introduce two novel algorithmic techniques: online filtering and k…
A coreset (or core-set) of an input set is its small summation, such that solving a problem on the coreset as its input, provably yields the same result as solving the same problem on the original (full) set, for a given family of problems (models, classifiers, loss functions). Over the past decade, coreset construction algorithms have been suggested for many fundamental problems in e.g. machine/deep learning, computer vision, graphics, databases, and theoretical computer science. This introductory paper was written following requests from (usually non-expert, but also colleagues) regarding th…
Pennsylvania State University（宾夕法尼亚州立大学）；IBM T. J. Watson Center（IBM托马斯·J·沃森研究中心）；Army Research Laboratory（美国陆军研究实验室）
Coreset, which is a summary of the original dataset in the form of a small weighted set in the same sample space, provides a promising approach to enable machine learning over distributed data. Although viewed as a proxy of the original dataset, each coreset is only designed to approximate the cost function of a specific machine learning problem, and thus different coresets are often required to solve different machine learning problems, increasing the communication overhead. We resolve this dilemma by developing robust coreset construction algorithms that can support a variety of machine lear…
RelationalAI（RelationalAI公司）；UC Merced（加州大学默塞德分校）；Carnegie Mellon University（卡内基梅隆大学）；University of Pittsburgh（匹兹堡大学）
We design and mathematically analyze sampling-based algorithms for regularized loss minimization problems that are implementable in popular computational models for large data, in which the access to the data is restricted in some way. Our main result is that if the regularizer's effect does not become negligible as the norm of the hypothesis scales, and as the data scales, then a uniform sample of modest size is with high probability a coreset. In the case that the loss function is either logistic regression or soft-margin support vector machines, and the regularizer is one of the common reco…
We provide a deterministic data summarization algorithm that approximates the mean $\bar{p}=\frac{1}{n}\sum_{p\in P} p$ of a set $P$ of $n$ vectors in $\REAL^d$, by a weighted mean $\tilde{p}$ of a \emph{subset} of $O(1/\eps)$ vectors, i.e., independent of both $n$ and $d$. We prove that the squared Euclidean distance between $\bar{p}$ and $\tilde{p}$ is at most $\eps$ multiplied by the variance of $P$. We use this algorithm to maintain an approximated sum of vectors from an unbounded stream, using memory that is independent of $d$, and logarithmic in the $n$ vectors seen so far. Our main appl…
Booz Allen Hamilton（博思艾伦汉密尔顿公司）；University of Maryland, Baltimore County（马里兰大学巴尔的摩县分校）；Laboratory for Physical Sciences（物理科学实验室）
Subsampling algorithms are a natural approach to reduce data size before fitting models on massive datasets. In recent years, several works have proposed methods for subsampling rows from a data matrix while maintaining relevant information for classification. While these works are supported by theory and limited experiments, to date there has not been a comprehensive evaluation of these methods. In our work, we directly compare multiple methods for logistic regression drawn from the coreset and optimal subsampling literature and discover inconsistencies in their effectiveness. In many cases, …
The automation of posterior inference in Bayesian data analysis has enabled experts and nonexperts alike to use more sophisticated models, engage in faster exploratory modeling and analysis, and ensure experimental reproducibility. However, standard automated posterior inference algorithms are not tractable at the scale of massive modern datasets, and modifications to make them so are typically model-specific, require expert tuning, and can break theoretical guarantees on inferential quality. Building on the Bayesian coresets framework, this work instead takes advantage of data redundancy to s…
Small coresets via negative dependence: DPPs, linear statistics, and concentration
Univ. Lille（里尔大学）；CNRS（法国国家科学研究中心）；Centrale Lille（里尔中央理工学院）；National University of Singapore（新加坡国立大学）；Université Paris-Saclay（巴黎萨克雷大学）；CEA（法国原子能和替代能源委员会）
Adobe Research（奥多比研究院）；University of Massachusetts Amherst（马萨诸塞大学阿默斯特分校）
We give relative error coresets for training linear classifiers with a broad class of loss functions, including the logistic loss and hinge loss. Our construction achieves $(1\pm ε)$ relative error with $\tilde O(d \cdot μ_y(X)^2/ε^2)$ points, where $μ_y(X)$ is a natural complexity measure of the data matrix $X \in \mathbb{R}^{n \times d}$ and label vector $y \in \{-1,1\}^n$, introduced in by Munteanu et al. 2018. Our result is based on subsampling data points with probabilities proportional to their $\ell_1$ $Lewis$ $weights$. It significantly improves on existing theoretical bounds and perfo…
Pruning is one of the predominant approaches for compressing deep neural networks (DNNs). Lately, coresets (provable data summarizations) were leveraged for pruning DNNs, adding the advantage of theoretical guarantees on the trade-off between the compression rate and the approximation error. However, coresets in this domain were either data-dependent or generated under restrictive assumptions on both the model's weights and inputs. In real-world scenarios, such assumptions are rarely satisfied, limiting the applicability of coresets. To this end, we suggest a novel and robust framework for com…
National University of Singapore（新加坡国立大学）；Singapore Management University（新加坡管理大学）
\emph{Coresets} are important tools to generate concise summaries of massive datasets for approximate analysis. A coreset is a small subset of points extracted from the original point set such that certain geometric properties are preserved with provable guarantees. This paper investigates the problem of maintaining a coreset to preserve the minimum enclosing ball (MEB) for a sliding window of points that are continuously updated in a data stream. Although the problem has been extensively studied in batch and append-only streaming settings, no efficient sliding-window solution is available yet…
GraphSculptor: Sculpting Pre-training Coreset for Graph Self-supervised Learning
Sangfor Technologies Inc.（Sangfor 技术公司）；Wuhan University（武汉大学）；The University of Melbourne（墨尔本大学）；Dalian Maritime University（大连海事大学）；Hunan University of Technology and Business（湖南工业大学；湘江实验室）；Xiangjiang Laboratory
Efficient Multi-bit Quantization Network Training via Weight Bias Correction and Bit-wise Coreset Sampling
Department of Electrical and Computer Engineering, Sungkyunkwan University（电气与计算机工程系，全北大学）；Department of Electrical and Computer Engineering, Duke University（电气与计算机工程系，杜克大学）；Kim Jaechul Graduate School of AI, Korea Advanced Institute of Science and Technology (KAIST)（金 Jaechul人工智能研究生院，韩国科学技术院（KAIST））
Support Vector Machine (SVM) is an efficient classification approach, which finds a hyperplane to separate data from different classes. This hyperplane is determined by support vectors. In existing SVM formulations, the objective function uses L2 norm or L1 norm on slack variables. The number of support vectors is a measure of generalization errors. In this work, we propose a Minimal SVM, which uses L0.5 norm on slack variables. The result model further reduces the number of support vectors and increases the classification performance.
Amity University Uttar Pradesh（阿密提 Uttar Pradesh 大学）；Coventry University（考文垂大学）
The support vector clustering algorithm is a well-known clustering algorithm based on support vector machines using Gaussian or polynomial kernels. The classical support vector clustering algorithm works well in general, but its performance degrades when applied on big data. In this paper, we have investigated the performance of support vector clustering algorithm implemented in a quantum paradigm for possible run-time improvements. We have developed and analyzed a quantum version of the support vector clustering algorithm. The proposed approach is based on the quantum support vector machine a…
Rgtsvm provides a fast and flexible support vector machine (SVM) implementation for the R language. The distinguishing feature of Rgtsvm is that support vector classification and support vector regression tasks are implemented on a graphical processing unit (GPU), allowing the libraries to scale to millions of examples with >100-fold improvement in performance over existing implementations. Nevertheless, Rgtsvm retains feature parity and has an interface that is compatible with the popular e1071 SVM package in R. Altogether, Rgtsvm enables large SVM models to be created by both experienced and…
Screening is an effective technique for speeding up the training process of a sparse learning model by removing the features that are guaranteed to be inactive the process. In this paper, we present a efficient screening technique for sparse support vector machine based on variational inequality. The technique is both efficient and safe.
Gdansk University of Technology（格但斯克理工大学）；Polish Academy of Sciences（波兰科学院）
We propose an algorithm for exploring the entire regularization path of asymmetric-cost linear support vector machines. Empirical evidence suggests the predictive power of support vector machines depends on the regularization parameters of the training algorithms. The algorithms exploring the entire regularization paths have been proposed for single-cost support vector machines thereby providing the complete knowledge on the behavior of the trained model over the hyperparameter space. Considering the problem in two-dimensional hyperparameter space though enables our algorithm to maintain great…
It is shown that bootstrap approximations of support vector machines (SVMs) based on a general convex and smooth loss function and on a general kernel are consistent. This result is useful to approximate the unknown finite sample distribution of SVMs by the bootstrap approach.
The main goal of statistical learning theory is to provide a fundamental framework for the problem of decision making and model construction based on sets of data. Here, we present a brief introduction to the fundamentals of statistical learning theory, in particular the difference between empirical and structural risk minimization, including one of its most prominent implementations, i.e. the Support Vector Machine.
EnsembleSVM is a free software package containing efficient routines to perform ensemble learning with support vector machine (SVM) base models. It currently offers ensemble methods based on binary SVM models. Our implementation avoids duplicate storage and evaluation of support vectors which are shared between constituent models. Experimental results show that using ensemble approaches can drastically reduce training complexity while maintaining high predictive accuracy. The EnsembleSVM software package is freely available online at http://esat.kuleuven.be/stadius/ensemblesvm.
Solving different types of optimization models (including parameters fitting) for support vector machines on large-scale training data is often an expensive computational task. This paper proposes a multilevel algorithmic framework that scales efficiently to very large data sets. Instead of solving the whole training set in one optimization process, the support vectors are obtained and gradually refined at multiple levels of coarseness of the data. The proposed framework includes: (a) construction of hierarchy of large-scale data coarse representations, and (b) a local processing of updating t…
The objective of this study is to investigate the efficient determination of $C$ and $γ$ for Support Vector Regression with RBF or mahalanobis kernel based on numerical and statistician considerations, which indicates the connection between $C$ and kernels and demonstrates that the deviation of geometric distance of neighbour observation in mapped space effects the predict accuracy of $ε$-SVR. We determinate the arrange of $γ$ & $C$ and propose our method to choose their best values.
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 …
Humboldt University of Berlin（柏林洪堡大学）；G+J Digital Products Hamburg（汉堡G+J数字产品公司）
We propose a fast inference method for Bayesian nonlinear support vector machines that leverages stochastic variational inference and inducing points. Our experiments show that the proposed method is faster than competing Bayesian approaches and scales easily to millions of data points. It provides additional features over frequentist competitors such as accurate predictive uncertainty estimates and automatic hyperparameter search.
Aix Marseille University（艾克斯-马赛大学）；An-Najah National University（纳贾赫国立大学）
In this paper, we consider modelling interaction between a set of variables in the context of time series and high dimension. We suggest two approaches. The first is similar to the neighborhood lasso when the lasso model is replaced by a support vector machine (SVMs). The second is a restricted Bayesian network adapted for time series. We show the efficiency of our approaches by simulations using linear, nonlinear data set and a mixture of both.
Uppsala University（乌普萨拉大学）；Nanjing University of Posts and Telecommunications（南京邮电大学）
We consider the problem of learning a classifier from observed functional data. Here, each data-point takes the form of a single time-series and contains numerous features. Assuming that each such series comes with a binary label, the problem of learning to predict the label of a new coming time-series is considered. Hereto, the notion of {\em margin} underlying the classical support vector machine is extended to the continuous version for such data. The longitudinal support vector machine is also a convex optimization problem and its dual form is derived as well. Empirical results for specifi…
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.
Saint-Petersburg State University（圣彼得堡国立大学）；Research Institute of Influenza（流感研究所）
In this paper there is proposed a generalized version of the SVM for binary classification problems in the case of using an arbitrary transformation x -> y. An approach similar to the classic SVM method is used. The problem is widely explained. Various formulations of primal and dual problems are proposed. For one of the most important cases the formulae are derived in detail. A simple computational example is demonstrated. The algorithm and its implementation is presented in Octave language.
We show that the multi-class support vector machine (MSVM) proposed by Lee et. al. (2004), can be viewed as a MAP estimation procedure under an appropriate probabilistic interpretation of the classifier. We also show that this interpretation can be extended to a hierarchical Bayesian architecture and to a fully-Bayesian inference procedure for multi-class classification based on data augmentation. We present empirical results that show that the advantages of the Bayesian formalism are obtained without a loss in classification accuracy.
Iranian Research Institute for Information Science and Technology (IranDoc)（伊朗信息科学与技术研究院（IranDoc））；Islamic Azad University, North Tehran Branch（伊斯兰阿扎德大学北德黑兰分校）
This paper presents LIBTwinSVM, a free, efficient, and open source library for Twin Support Vector Machines (TSVMs). Our library provides a set of useful functionalities such as fast TSVMs estimators, model selection, visualization, a graphical user interface (GUI) application, and a Python application programming interface (API). The benchmarks results indicate the effectiveness of the LIBTwinSVM library for large-scale classification problems. The source code of LIBTwinSVM library, installation guide, documentation, and usage examples are available at https://github.com/mir-am/LIBTwinSVM.
Multi-label learning has attracted the attention of the machine learning community. The problem conversion method Binary Relevance converts a familiar single label into a multi-label algorithm. The binary relevance method is widely used because of its simple structure and efficient algorithm. But binary relevance does not consider the links between labels, making it cumbersome to handle some tasks. This paper proposes a multi-label learning algorithm that can also be used for single-label classification. It is based on standard support vector machines and changes the original single decision h…
Centro de Investigación en Matemáticas（数学研究中心）；Universidad Militar Nueva Granada（新格拉纳达军事大学）；Universidad de los Andes（安第斯大学）
An importance sampling and bagging approach to solving the support vector machine (SVM) problem in the context of large databases is presented and evaluated. Our algorithm builds on the nearest neighbors ideas presented in Camelo at al. (2015). As in that reference, the goal of the present proposal is to achieve a faster solution of the SVM problem without a significance loss in the prediction error. The performance of the methodology is evaluated in benchmark examples and theoretical aspects of subsample methods are discussed.
Hokkaido University（北海道大学）；Osaka University（大阪大学）；NTT Communication Science Laboratories（NTT通信科学实验室）
In conventional prediction tasks, a machine learning algorithm outputs a single best model that globally optimizes its objective function, which typically is accuracy. Therefore, users cannot access the other models explicitly. In contrast to this, multiple model enumeration attracts increasing interests in non-standard machine learning applications where other criteria, e.g., interpretability or fairness, than accuracy are main concern and a user may want to access more than one non-optimal, but suitable models. In this paper, we propose a K-best model enumeration algorithm for Support Vector…
This study presents a rapid multiple incremental and decremental mechanism based on Weight-Error Curves (WECs) for support-vector analysis. Recursion-free computation is proposed for predicting the Lagrangian multipliers of new samples. This study examines Ridge Support Vector Models, subsequently devising a recursion-free function derived from WECs. With the proposed function, all the new Lagrangian multipliers can be computed at once without using any gradual step sizes. Moreover, such a function relaxes a constraint, where the increment of new multiple Lagrangian multipliers should be the s…
In this paper, we study the support vector machine and introduced the notion of generalized support vector machine for classification of data. We show that the problem of generalized support vector machine is equivalent to the problem of generalized variational inequality and establish various results for the existence of solutions. Moreover, we provide various examples to support our results.
The insensitive parameter in support vector regression determines the set of support vectors that greatly impacts the prediction. A data-driven approach is proposed to determine an approximate value for this insensitive parameter by minimizing a generalized loss function originating from the likelihood principle. This data-driven support vector regression also statistically standardizes samples using the scale of noises. Nonlinear and linear numerical simulations with three types of noises ($ε$-Laplacian distribution, normal distribution, and uniform distribution), and in addition, five real b…
The least-squares support vector machine is a frequently used kernel method for non-linear regression and classification tasks. Here we discuss several approximation algorithms for the least-squares support vector machine classifier. The proposed methods are based on randomized block kernel matrices, and we show that they provide good accuracy and reliable scaling for multi-class classification problems with relatively large data sets. Also, we present several numerical experiments that illustrate the practical applicability of the proposed methods.
School of Computational Science & Engineering, Georgia Tech（佐治亚理工学院计算科学与工程学院）
We provide a formulation for Local Support Vector Machines (LSVMs) that generalizes previous formulations, and brings out the explicit connections to local polynomial learning used in nonparametric estimation literature. We investigate the simplest type of LSVMs called Local Linear Support Vector Machines (LLSVMs). For the first time we establish conditions under which LLSVMs make Bayes consistent predictions at each test point $x_0$. We also establish rates at which the local risk of LLSVMs converges to the minimum value of expected local risk at each point $x_0$. Using stability arguments we…
Columbia University（哥伦比亚大学）；Georgia Institute of Technology（佐治亚理工学院）
The support vector machine (SVM) is a well-established classification method whose name refers to the particular training examples, called support vectors, that determine the maximum margin separating hyperplane. The SVM classifier is known to enjoy good generalization properties when the number of support vectors is small compared to the number of training examples. However, recent research has shown that in sufficiently high-dimensional linear classification problems, the SVM can generalize well despite a proliferation of support vectors where all training examples are support vectors. In th…
The support vector machines (SVM) algorithm is a popular classification technique in data mining and machine learning. In this paper, we propose a distributed SVM algorithm and demonstrate its use in a number of applications. The algorithm is named high-performance support vector machines (HPSVM). The major contribution of HPSVM is two-fold. First, HPSVM provides a new way to distribute computations to the machines in the cloud without shuffling the data. Second, HPSVM minimizes the inter-machine communications in order to maximize the performance. We apply HPSVM to some real-world classificat…
Universidad Autónoma de Madrid（马德里自治大学）；KU Leuven（鲁汶大学）
This work proposes a new algorithm for training a re-weighted L2 Support Vector Machine (SVM), inspired on the re-weighted Lasso algorithm of Candès et al. and on the equivalence between Lasso and SVM shown recently by Jaggi. In particular, the margin required for each training vector is set independently, defining a new weighted SVM model. These weights are selected to be binary, and they are automatically adapted during the training of the model, resulting in a variation of the Frank-Wolfe optimization algorithm with essentially the same computational complexity as the original algorithm. As…
La Trobe University（拉筹伯大学）；University of Queensland（昆士兰大学）
Support vector machines (SVMs) are an important tool in modern data analysis. Traditionally, support vector machines have been fitted via quadratic programming, either using purpose-built or off-the-shelf algorithms. We present an alternative approach to SVM fitting via the majorization--minimization (MM) paradigm. Algorithms that are derived via MM algorithm constructions can be shown to monotonically decrease their objectives at each iteration, as well as be globally convergent to stationary points. We demonstrate the construction of iteratively-reweighted least-squares (IRLS) algorithms, vi…
Support Vector Data Description (SVDD) is a popular outlier detection technique which constructs a flexible description of the input data. SVDD computation time is high for large training datasets which limits its use in big-data process-monitoring applications. We propose a new iterative sampling-based method for SVDD training. The method incrementally learns the training data description at each iteration by computing SVDD on an independent random sample selected with replacement from the training data set. The experimental results indicate that the proposed method is extremely fast and prov…
Harvard-MIT Health Sciences and Technology（哈佛-麻省理工学院健康科学与技术）；Massachusetts Institute of Technology（麻省理工学院）
The support vector machine (SVM) is a widely used machine learning tool for classification based on statistical learning theory. Given a set of training data, the SVM finds a hyperplane that separates two different classes of data points by the largest distance. While the standard form of SVM uses L2-norm regularization, other regularization approaches are particularly attractive for biomedical datasets where, for example, sparsity and interpretability of the classifier's coefficient values are highly desired features. Therefore, in this paper we consider different types of regularization appr…
Tampere University（坦佩雷大学）；Finnish Environment Institute（芬兰环境研究所）；Aarhus University（奥胡斯大学）
In this paper, we propose a novel method for transforming data into a low-dimensional space optimized for one-class classification. The proposed method iteratively transforms data into a new subspace optimized for ellipsoidal encapsulation of target class data. We provide both linear and non-linear formulations for the proposed method. The method takes into account the covariance of the data in the subspace; hence, it yields a more generalized solution as compared to Subspace Support Vector Data Description for a hypersphere. We propose different regularization terms expressing the class varia…
Institute for Mathematics and its Applications, University of Minnesota（明尼苏达大学数学及其应用研究所）；College of Information Science and Engineering, Northeastern University（东北大学信息科学与工程学院）；Precima, a NielsenIQ Company（普雷西玛（尼耳森艾克集团旗下公司））
We propose $\ell_1$ norm regularized quadratic surface support vector machine models for binary classification in supervised learning. We establish their desired theoretical properties, including the existence and uniqueness of the optimal solution, reduction to the standard SVMs over (almost) linearly separable data sets, and detection of true sparsity pattern over (almost) quadratically separable data sets if the penalty parameter of $\ell_1$ norm is large enough. We also demonstrate their promising practical efficiency by conducting various numerical experiments on both synthetic and public…
In this paper, we study the hard and soft support vector regression techniques applied to a set of $n$ linear measurements of the form $y_i=\boldsymbolβ_\star^{T}{\bf x}_i +n_i$ where $\boldsymbolβ_\star$ is an unknown vector, $\left\{{\bf x}_i\right\}_{i=1}^n$ are the feature vectors and $\left\{{n}_i\right\}_{i=1}^n$ model the noise. Particularly, under some plausible assumptions on the statistical distribution of the data, we characterize the feasibility condition for the hard support vector regression in the regime of high dimensions and, when feasible, derive an asymptotic approximation f…
Jilin University（吉林大学）；University of Aberdeen（阿伯丁大学）
We propose a novel support vector regression approach called e-Distance Weighted Support Vector Regression (e-DWSVR).e-DWSVR specifically addresses two challenging issues in support vector regression: first, the process of noisy data; second, how to deal with the situation when the distribution of boundary data is different from that of the overall data. The proposed e-DWSVR optimizes the minimum margin and the mean of functional margin simultaneously to tackle these two issues. In addition, we use both dual coordinate descent (CD) and averaged stochastic gradient descent (ASGD) strategies to …
The support vector machine (SVM) and minimum Euclidean norm least squares regression are two fundamentally different approaches to fitting linear models, but they have recently been connected in models for very high-dimensional data through a phenomenon of support vector proliferation, where every training example used to fit an SVM becomes a support vector. In this paper, we explore the generality of this phenomenon and make the following contributions. First, we prove a super-linear lower bound on the dimension (in terms of sample size) required for support vector proliferation in independen…
This paper adapts a recently developed regularized stochastic version of the Broyden, Fletcher, Goldfarb, and Shanno (BFGS) quasi-Newton method for the solution of support vector machine classification problems. The proposed method is shown to converge almost surely to the optimal classifier at a rate that is linear in expectation. Numerical results show that the proposed method exhibits a convergence rate that degrades smoothly with the dimensionality of the feature vectors.
