On Possibility and Impossibility of Multiclass Classification with Rejection

We investigate the problem of multiclass classification with rejection, where a classifier can choose not to make a prediction to avoid critical misclassification. We consider two approaches for this problem: a traditional one based on confidence scores and a more recent one based on simultaneous training of a classifier and a rejector. An existing method in the former approach focuses on a specific class of losses and its empirical performance is not very convincing. In this paper, we propose confidence-based rejection criteria for multiclass classification, which can handle more general losses and guarantee calibration to the Bayes-optimal solution. The latter approach is relatively new and has been available only for the binary case, to the best of our knowledge. Our second contribution is to prove that calibration to the Bayes-optimal solution is almost impossible by this approach in the multiclass case. Finally, we conduct experiments to validate the relevance of our theoretical findings.