Combinatorial and Structural Results for gamma-Psi-dimensions

This article deals with the generalization performance of margin multi-category classifiers, when minimal learnability hypotheses are made. In that context, the derivation of a guaranteed risk 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 this kind are the gamma-Psi-dimensions. We introduce the combinatorial and structural results needed to involve them in the derivation of upper bounds on the metric entropies and the Rademacher complexity. Their incidence on the guaranteed risks is characterized, which establishes that in the theoretical framework of interest, performing the transition from the multi-class case to the binary one with combinatorial dimensions is a promising alternative to proceeding with covering numbers.