Bayes-Optimal Scorers for Bipartite Ranking

We address the following seemingly simple question: what is the Bayes-optimal scorer for a bipartite ranking risk? The answer to this question helps establish the consistency of the minimisation of surrogate bipartite risks, and elucidates the relationship between bipartite ranking and other established learning problems. We show that the answer is non-trivial in general, but may be easily determined for certain special cases using the theory of proper losses. Our analysis immediately establishes equivalences between several seemingly disparate risks for bipartite ranking, such as minimising a suitable class-probability estimation risk, and minimising the p-norm push risk proposed by Rudin (2009).