Statistics and Its Interface

Volume 2 (2009)

Number 3

On consistency and robustness properties of support vector machines for heavy-tailed distributions

Pages: 311 – 327



Andreas Christmann (Department of Mathematics, University of Bayreuth, Germany)

Ingo Steinwart (Information Sciences Group (CCS-3), Los Alamos National Laboratory, Los Alamos, New Mexico, U.S.A.)

Arnout van Messem (Department of Mathematics, Vrije Universiteit Brussel, Brussels, Belgium)


Support Vector Machines (SVMs) are known to be consistent and robust for classification and regression if they are based on a Lipschitz continuous loss function and on a bounded kernel with a dense and separable reproducing kernel Hilbert space. These facts are even true in the regression context for unbounded output spaces, if the target function $f$ is integrable with respect to the marginal distribution of the input variable $X$ and if the output variable $Y$ has a finite first absolute moment. The latter assumption clearly excludes distributions with heavy tails, e.g., several stable distributions or some extreme value distributions which occur in financial or insurance projects. The main point of this paper is that we can enlarge the applicability of SVMs even to heavy-tailed distributions, which violate this moment condition. Results on existence, uniqueness, representation, consistency, and statistical robustness are given.

2010 Mathematics Subject Classification

Primary 62G05. Secondary 62G08, 62G20, 62G35, 62J02, 68Q32, 68T10.

Published 1 January 2009