In this work we study the semi-supervised framework of confidence set classification with controlled expected size in minimax settings. We obtain semi-supervised minimax rates of convergence under the margin assumption and a H o lder condition on the regression function. Besides, we show that if no further assumptions are made, there is no supervised method that outperforms the semi-supervised estimator proposed in this work. We establish that the best achievable rate for any supervised method is n^ --1 2 , even if the margin assumption is extremely favorable. On the contrary, semi-supervised estimators can achieve faster rates of convergence provided that sufficiently many unlabeled samples are available. We additionally perform numerical evaluation of the proposed algorithms empirically confirming our theoretical findings.
On the other part, the confidence set estimation problem is directly related to the standard classification settings. This problem has been widely studied from a theoretical point of view in the binary classification framework. @cite_1 study the statistical performance of plug-in classification rules under assumptions which involve the smoothness of the regression function and the margin condition. In particular, they derive fast rates of convergence for plug-in classifiers based on local polynomial estimators and show their optimality in the minimax sense. One of the aim of present work is to extend these results to the confidence set classification framework.
@cite_1
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It has been recently shown that, under the margin (or low noise) assumption, there exist classifiers attaining fast rates of convergence of the excess Bayes risk, that is, rates faster than n -1 2 . The work on this subject has suggested the following two conjectures: (i) the best achievable fast rate is of the order n -1 , and (ii) the plug-in classifiers generally converge more slowly than the classifiers based on empirical risk minimization. We show that both conjectures are not correct. In particular, we construct plug-in classifiers that can achieve not only fast, but also super-fast rates, that is, rates faster than n -1 . We establish minimax lower bounds showing that the obtained rates cannot be improved.