VC Theory of Large Margin Multi-Category Classifiers
In the context of discriminant analysis, Vapnik's statistical learning theory has mainly been developed in three directions: the computation of dichotomies with binary-valued functions, the computation of dichotomies with real-valued functions, and the computation of polytomies with functions taking their values in finite sets, typically the set of categories itself. The case of classes of vector-valued functions used to compute polytomies has seldom been considered independently, which is unsatisfactory, for three main reasons. First, this case encompasses the other ones. Second, it cannot be treated appropriately through a naïve extension of the results devoted to the computation of dichotomies. Third, most of the classification problems met in practice involve multiple categories. In this paper, a VC theory of large margin multi-category classifiers is introduced. Central in this theory are generalized VC dimensions called the gamma-Psi-dimensions. First, a uniform convergence bound on the risk of the classifiers of interest is derived. The capacity measure involved in this bound is a covering number. This covering number can be upper bounded in terms of the gamma-Psi-dimensions thanks to generalizations of Sauer's lemma, as is illustrated in the specific case of the scale-sensitive Natarajan dimension. A bound on this latter dimension is then computed for the class of functions on which multi-class SVMs are based. This makes it possible to apply the structural risk minimization inductive principle to those machines.