Most existing classification methods aim to minimize the overall misclassification error rate. However, in applications such as loan default prediction, different types of errors can have varying consequences. To address this asymmetry issue, two popular paradigms have been developed: the Neyman-Pearson (NP) paradigm and the cost-sensitive (CS) paradigm. Previous studies on the NP paradigm have primarily focused on the binary case, while the multi-class NP problem poses a greater challenge due to its unknown feasibility. In this work, we tackle the multi-class NP problem by establishing a connection with the CS problem via strong duality and propose two algorithms. We extend the concept of NP oracle inequalities, crucial in binary classifications, to NP oracle properties in the multi-class context. Our algorithms satisfy these NP oracle properties under certain conditions. Furthermore, we develop practical algorithms to assess the feasibility and strong duality in multi-class NP problems, which can offer practitioners the landscape of a multi-class NP problem with various target error levels. Simulations and real data studies validate the effectiveness of our algorithms. To our knowledge, this is the first study to address the multi-class NP problem with theoretical guarantees. The proposed algorithms have been implemented in the R package \texttt{npcs}, which is available on CRAN.

翻译：现有的大多数分类方法旨在最小化总体误分类错误率。然而，在贷款违约预测等应用中，不同类型的错误可能产生截然不同的后果。为解决这种不对称性问题，学界发展出两种主流范式：Neyman-Pearson（NP）范式和代价敏感（CS）范式。以往关于NP范式的研究主要聚焦于二分类情况，而多类NP问题因其可行性未知而更具挑战性。本研究通过强对偶性建立多类NP问题与CS问题的联系，并提出两种算法。我们将二分类中至关重要的NP预言不等式概念扩展至多类情境下的NP预言性质，所提算法在特定条件下满足这些NP预言性质。此外，我们开发了实用算法来评估多类NP问题的可行性与强对偶性，从而为实践者呈现不同目标错误水平下多类NP问题的全景图。仿真实验与真实数据研究验证了算法的有效性。据我们所知，这是首项为多类NP问题提供理论保障的研究。所提算法已部署于CRAN平台的R包\texttt{npcs}中。