The tutorial starts with an overview of the concepts of VC dimension and structural risk minimization. We then describe linear Support Vector Machines (SVMs) for separable and non-separable data, working through a non-trivial example in detail. We describe a mechanical analogy, and discuss when SVM solutions are unique and when they are global. We describe how support vector training can be practically implemented, and discuss in detail the kernel mapping technique which is used to construct SVM solutions which are nonlinear in the data. We show how Support Vector machines can have very large (even infinite) VC dimension by computing the VC dimension for homogeneous polynomial and Gaussian radial basis function kernels. While very high VC dimension would normally bode ill for generalization performance, and while at present there exists no theory which shows that good generalization performance is guaranteed for SVMs, there are several arguments which support the observed high accuracy of SVMs, which we review. Results of some experiments which were inspired by these arguments are also presented. We give numerous examples and proofs of most of the key theorems. There is new material, and I hope that the reader will find that even old material is cast in a fresh light.

In this tutorial we give an overview of the basic ideas underlying Support Vector (SV) machines for function estimation. Furthermore, we include a summary of currently used algorithms for training SV machines, covering both the quadratic (or convex) programming part and advanced methods for dealing with large datasets. Finally, we mention some modifications and extensions that have been applied to the standard SV algorithm, and discuss the aspect of regularization from a SV perspective.

Recently ensemble methods like AdaBoost were successfully applied to character recognition tasks, seemingly defying the problems of overfitting. This paper shows that although AdaBoost rarely overfits in the low noise regime it clearly does so for higher noise levels. Central for understanding this fact is the margin distribution and we find that AdaBoost achieves -- doing gradient descent in an error function with respect to the margin -- asymptotically a hard margin distribution, i.e. the algorithm concentrates its resources on a few hard-to-learn patterns (here an interesting overlap emerge to Support Vectors). This is clearly a sub-optimal strategy in the noisy case, and regularization, i.e. a mistrust in the data, must be introduced in the algorithm to alleviate the distortions that a difficult pattern (e.g. outliers) can cause to the margin distribution. We propose several regularization methods and generalizations of the original AdaBoost algorithm to achieve a soft margin -- a ...

Most real-time computer-controlled systems are built in two separate steps, each in isolation: controller design and its digital implementation. Computational tasks that realize the control algorithms are usually scheduled by treating their execution times and periods as unchangeable parameters. Task scheduling therefore depends only on the limited computing resources available. On the other hand, controller design is primarily based on the continuous-time dynamics of the physical system being controlled. The set of tasks resulting from this controller design may not be schedulable with the limited computing resources available. Even if the given set of tasks is schedulable, the overall control performance may not be optimal in the sense that they do not make a full use of computing resource. In this paper, we propose an integrated approach to controller design and task scheduling. Specifically, task frequencies (or periods) are allowed to vary within a certain range as long as such a ...

Abstract An algorithm is proposed which generates a nonlinear kernel-based separating surface that requires as little as 1 % of a large dataset for its explicit evaluation. To generate this nonlinear surface, the entire dataset is used as a constraint in an optimization problem with very few variables corresponding to the 1%

Wahba's classical representer theorem states that the solutions of certain risk minimization problems involving an empirical risk term and a quadratic regularizer can be written as expansions in terms of the training examples. We generalize the theorem to a larger class of regularizers and empirical risk terms, and give a self-contained proof utilizing the feature space associated with a kernel. The result shows that a wide range of problems have optimal solutions that live in the finite dimensional span of the training examples mapped into feature space, thus enabling us to carry out kernel algorithms independent of the (potentially infinite) dimensionality of the feature space.

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Abstract—A new approach to support vector machine (SVM) classification is proposed wherein each of two data sets are proximal to one of two distinct planes that are not parallel to each other. Each plane is generated such that it is closest to one of the two data sets and as far as possible from the other data set. Each of the two nonparallel proximal planes is obtained by a single MATLAB command as the eigenvector corresponding to a smallest eigenvalue of a generalized eigenvalue problem. Classification by proximity to two distinct nonlinear surfaces generated by a nonlinear kernel also leads to two simple generalized eigenvalue problems. The effectiveness of the proposed method is demonstrated by tests on simple examples as well as on a number of public data sets. These examples show the advantages of the proposed approach in both computation time and test set correctness. Index Terms—Support vector machines, proximal classification, generalized eigenvalues. 1

The origins and some motivational details of a collection of nonlinear mixed complementarity problems are given. This collection serves two purposes. Firstly, it gives a uniform basis for testing currently available and new algorithms for mixed complementarity problems. Function and Jacobian evaluations for the resulting problems are provided via a GAMS interface, making thorough testing of algorithms on practical complementarity problems possible. Secondly, it gives examples of how to formulate many popular problem formats as mixed complementarity problems and how to describe the resulting problems in GAMS format. We demonstrate the ease and power of formulating practical models in the MCP format. Given these examples, it is hoped that this collection will grow to include many problems that test complementarity algorithms more fully. The collection is available by anonymous ftp. Computational results using the PATH solver covering all of these problems are described. 1 Introduction R...

Successive overrelaxation (SOR) for symmetric linear complementarity problems and quadratic programs [11, 12, 9] is used to train a support vector machine (SVM) [20, 3] for discriminating between the elements of two massive datasets, each with millions of points. Because SOR handles one point at a time, similar to Platt's sequential minimal optimization (SMO) algorithm [18] which handles two constraints at a time, it can process very large datasets that need not reside in memory. The algorithm converges linearly to a solution. Encouraging numerical results are presented on datasets with up to 10 million points. Such massive discrimination problems cannot be processed by conventional linear or quadratic programming methods, and to our knowledge have not been solved by other methods. 1 Introduction Successive overrelaxation, originally developed for the solution of large systems of linear equations [16, 15] has been successfully applied to mathematical programming problems [4, 11, 12, 1...