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Multiclass classification problems such as image annotation can involve alarge number of classes. In this context, confusion between classes can occur,and single label classification may be misleading. We provide in the presentpaper a general device that, given an unlabeled dataset and a score functiondefined as the minimizer of some empirical and convex risk, outputs a set ofclass labels, instead of a single one. Interestingly, this procedure does notrequire that the unlabeled dataset explores the whole classes. Even more, themethod is calibrated to control the expected size of the output set whileminimizing the classification risk. We show the statistical optimality of theprocedure and establish rates of convergence under the Tsybakov margincondition. It turns out that these rates are linear on the number of labels. Weapply our methodology to convex aggregation of confidence sets based on theV-fold cross validation principle also known as the superlearning principle. Weillustrate the numerical performance of the procedure on real data anddemonstrate in particular that with moderate expected size, w.r.t. the numberof labels, the procedure provides significant improvement of the classificationrisk.

Confidence sets with expected sizes for Multiclass Classification