Generalizing Consistent Multi-Class Classification with Rejection to be Compatible with Arbitrary Losses

\emph{Classification with rejection} (CwR) refrains from making a prediction to avoid critical misclassification when encountering test samples that are difficult to classify. Though previous methods for CwR have been provided with theoretical guarantees, they are only compatible with certain loss functions, making them not flexible enough when the loss needs to be changed with the dataset in practice. In this paper, we derive a novel formulation for CwR that can be equipped with arbitrary loss functions while maintaining the theoretical guarantees. First, we show that $K$-class CwR is equivalent to a $(K\!+\!1)$-class classification problem on the original data distribution with an augmented class, and propose an empirical risk minimization formulation to solve this problem with an estimation error bound. Then, we find necessary and sufficient conditions for the learning \emph{consistency} of the surrogates constructed on our proposed formulation equipped with any classification-calibrated multi-class losses, where consistency means the surrogate risk minimization implies the target risk minimization for CwR. Finally, experiments on benchmark datasets validate the effectiveness of our proposed method.