A PAC-Bayesian Margin Bound for Linear Classifiers: Why SVMs work

Herbrich, Ralf, Graepel, Thore

Neural Information Processing Systems 

We present a bound on the generalisation error of linear classifiers in terms of a refined margin quantity on the training set. The result is obtained in a PAC-Bayesian framework and is based on geometrical arguments in the space of linear classifiers. The new bound constitutes an exponential improvement of the so far tightest margin bound by Shawe-Taylor et al. [8] and scales logarithmically in the inverse margin. Even in the case of less training examples than input dimensions sufficiently large margins lead to nontrivial bound values and - for maximum margins - to a vanishing complexity term.Furthermore, the classical margin is too coarse a measure for the essential quantity that controls the generalisation error: the volume ratio between the whole hypothesis space and the subset of consistent hypotheses. The practical relevance of the result lies in the fact that the well-known support vector machine is optimal w.r.t. the new bound only if the feature vectors are all of the same length. As a consequence we recommend to use SVMs on normalised feature vectors only - a recommendation that is well supported by our numerical experiments on two benchmark data sets. 1 Introduction Linear classifiers are exceedingly popular in the machine learning community due to their straightforward applicability and high flexibility which has recently been boosted by the so-called kernel methods [13]. A natural and popular framework for the theoretical analysis of classifiers is the PAC (probably approximately correct) framework[11] which is closely related to Vapnik's work on the generalisation error [12]. For binary classifiers it turned out that the growth function is an appropriate measureof "complexity" and can tightly be upper bounded by the VC (Vapnik-Chervonenkis) dimension [14].