Combinatorial and Structural Results for gamma-Psi-dimensions

One of the main open problems of the theory of margin multi-category pattern classification is the characterization of the way the confidence interval of a guaranteed risk should vary as a function of the three basic parameters which are the sample size m, the number C of categories and the scale parameter gamma. This is especially the case when working under minimal learnability hypotheses. In that context, the derivation of a bound is based on the handling of capacity measures belonging to three main families: Rademacher/Gaussian complexities, metric entropies and scale-sensitive combinatorial dimensions. The scale-sensitive combinatorial dimensions dedicated to the classifiers of interest are the gamma-Psi-dimensions. This article introduces the combinatorial and structural results needed to involve them in the derivation of guaranteed risks. Such a bound is then established, under minimal hypotheses regarding the classifier. Its dependence on m, C and gamma is characterized. The special case of multi-class support vector machines is used to illustrate the capacity of the gamma-Psi-dimensions to take into account the specificities of a classifier.