Smola-tutorial on support vector regression

# Smola-tutorial on support vector regression - Statistics...

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Statistics and Computing 14 : 199–222, 2004 C ° 2004 Kluwer Academic Publishers. Manufactured in The Netherlands . A tutorial on support vector regression ALEX J. SMOLA and BERNHARD SCH ¨ OLKOPF RSISE, Australian National University, Canberra 0200, Australia [email protected] Max-Planck-Institut f¨ur biologische Kybernetik, 72076 T¨ubingen, Germany [email protected] Received July 2002 and accepted November 2003 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 modi±cations and extensions that have been applied to the standard SV algorithm, and discuss the aspect of regularization from a SV perspective. Keywords: machine learning, support vector machines, regression estimation 1. Introduction The purpose of this paper is twofold. It should serve as a self- contained introduction to Support Vector regression for readers new to this rapidly developing ±eld of research. 1 On the other hand, it attempts to give an overview of recent developments in the ±eld. To this end, we decided to organize the essay as follows. We start by giving a brief overview of the basic techniques in Sections 1, 2 and 3, plus a short summary with a number of ±gures and diagrams in Section 4. Section 5 reviews current algorithmic techniques used for actually implementing SV machines. This may be of most interest for practitioners. The following section covers more advanced topics such as extensions of the basic SV algorithm, connections between SV machines and regularization and brie²y mentions methods for carrying out model selection. We conclude with a discussion of open questions and problems and current directions of SV research. Most of the results presented in this review paper already have been published elsewhere, but the comprehensive presentations and some details are new. 1.1. Historic background The SV algorithm is a nonlinear generalization of the Gener- alized Portrait algorithm developed in Russia in the sixties 2 An extended version of this paper is available as NeuroCOLT Technical Report TR-98-030. (Vapnik and Lerner 1963, Vapnik and Chervonenkis 1964). As such, it is ±rmly grounded in the framework of statistical learn- ing theory, or VC theory ,which has been developed over the last three decades by Vapnik and Chervonenkis (1974) and Vapnik (1982, 1995). In a nutshell, VC theory characterizes properties of learning machines which enable them to generalize well to unseen data. In its present form, the SV machine was largely developed at AT&T Bell Laboratories by Vapnik and co-workers (Boser, Guyon and Vapnik 1992, Guyon, Boser and Vapnik 1993, Cortes and Vapnik, 1995, Sch¨olkopf, Burges and Vapnik 1995, 1996, Vapnik, Golowich and Smola 1997). Due to this industrial con- text, SV research has up to date had a sound orientation towards real-world applications. Initial work focused on OCR (optical

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Smola-tutorial on support vector regression - Statistics...

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