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In many real applications of statistical learning, a decision made frommisclassification can be too costly to afford; in this case, a reject option,which defers the decision until further investigation is conducted, is oftenpreferred. In recent years, there has been much development for binaryclassification with a reject option. Yet, little progress has been made for themulticategory case. In this article, we propose margin-based multicategoryclassification methods with a reject option. In addition, and more importantly,we introduce a new and unique refine option for the multicategory problem,where the class of an observation is predicted to be from a set of classlabels, whose cardinality is not necessarily one. The main advantage of bothoptions lies in their capacity of identifying error-prone observations.Moreover, the refine option can provide more constructive information forclassification by effectively ruling out implausible classes. Efficientimplementations have been developed for the proposed methods. On thetheoretical side, we offer a novel statistical learning theory and show a fastconvergence rate of the excess $\ell$-risk of our methods with emphasis ondiverging dimensionality and number of classes. The results can be furtherimproved under a low noise assumption. A set of comprehensive simulation andreal data studies has shown the usefulness of the new learning tools comparedto regular multicategory classifiers. Detailed proofs of theorems and extendednumerical results are included in the supplemental materials available online.

On Reject and Refine Options in Multicategory Classification