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.
Confident prediction is highly relevant in machine learning; for example, in applications such as medical diagnoses, wrong prediction can be fatal. For classification, there already exist procedures that allow to not classify data when the confidence in their prediction is weak. This approach is known as classification with reject option. In the present paper, we provide new methodology for this approach. Predicting a new instance via a confidence set, we ensure an exact control of the probability of classification. Moreover, we show that this methodology is easily implementable and entails attractive theoretical and numerical properties. A framework for classification is developed with a notion of confidence. In this framework, a classifier consists of two tolerance regions in the predictor space, with a specified coverage level for each class. The classifier also produces an ambiguous region where the classification needs further investigation. Theoretical analysis reveals interesting structures of the confidence-ambiguity trade-off, and the optimal solution is characterized by extending the Neyman–Pearson lemma. We provide general estimating procedures, along with rates of convergence, based on estimates of the conditional probabilities. The method can be easily implemented with good robustness, as illustrated through theory, simulation and a data example.