We introduce a novel framework for classification with a rejection option that consists of simultaneously learning two functions: a classifier along with a rejection function. We present a full theoretical analysis of this framework including new data-dependent learning bounds in terms of the Rademacher complexities of the classifier and rejection families as well as consistency and calibration results. These theoretical guarantees guide us in designing new algorithms that can exploit different kernel-based hypothesis sets for the classifier and rejection functions. We compare and contrast our general framework with the special case of confidence-based rejection for which we devise alternative loss functions and algorithms as well. We report the results of several experiments showing that our kernel-based algorithms can yield a notable improvement over the best existing confidence-based rejection algorithm.
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