One of the main open problems of the theory of margin multi-category pattern classification is the characterization of the optimal dependence of the confidence interval of a guaranteed risk on 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. The starting point is a basic supremum inequality whose capacity measure depends on the choice of the margin loss function. Then, transitions are made, from capacity measure to capacity measure. At some level, a structural result performs the transition from the multi-class case to the bi-class one. In this article, we highlight the advantages and drawbacks inherent to the three major options for this decomposition: using Rademacher complexities, covering numbers or scale-sensitive combinatorial dimensions.
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