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In this work we study the semi-supervised framework of confidence setclassification with controlled expected size in minimax settings. We obtainsemi-supervised minimax rates of convergence under the margin assumption and aH{\"o}lder condition on the regression function. Besides, we show that if nofurther assumptions are made, there is no supervised method that outperformsthe semi-supervised estimator proposed in this work. We establish that the bestachievable rate for any supervised method is n^{--1/2} , even if the marginassumption is extremely favorable. On the contrary, semi-supervised estimatorscan achieve faster rates of convergence provided that sufficiently manyunlabeled samples are available. We additionally perform numerical evaluationof the proposed algorithms empirically confirming our theoretical findings.

Minimax semi-supervised confidence sets for multi-class classification