  One of the main open problems in the theory of multi-category margin
classification is the form of the optimal dependency of a guaranteed risk on
the number C of categories, the sample size m and the margin parameter gamma.
From a practical point of view, the theoretical analysis of generalization
performance contributes to the development of new learning algorithms. In this
paper, we focus only on the theoretical aspect of the question posed. More
precisely, under minimal learnability assumptions, we derive a new risk bound
for multi-category margin classifiers. We improve the dependency on C over the
state of the art when the margin loss function considered satisfies the
Lipschitz condition. We start with the basic supremum inequality that involves
a Rademacher complexity as a capacity measure. This capacity measure is then
linked to the metric entropy through the chaining method. In this context, our
improvement is based on the introduction of a new combinatorial metric entropy
bound.
