  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.
