Ranking Multi-Class Data : Optimality and Pairwise Aggregation

It is the primary purpose of this paper to set the goals of ranking in a multiple-class context rigorously, following in the footsteps of recent results in the bipartite framework. Under specific likelihood ratio monotonicity conditions, optimal solutions for this global learning problem are described in the ordinal situation, i.e. when there exists a natural order on the set of labels. Criteria reflecting ranking performance under these conditions such as the ROC surface and its natural summary, the volume under the ROC surface (VUS), are next considered as targets for empirical optimization. Whereas plug-in techniques or the Empirical Risk Maximization principle can be then easily extended to the ordinal multi-class setting, reducing the K-partite ranking task to the solving of a collection of bipartite ranking problems, following in the footsteps of the pairwise comparison approach in classification, is in contrast more challenging. Here we consider a concept of ranking rule consensus based on the Kendall τ distance and show that, when it exists and is based on consistent ranking rules for the bipartite ranking subproblems defined by all consecutive pairs of labels, the latter forms a consistent ranking rule in the VUS sense under adequate conditions. This result paves the way for extending the use of recently developed learning algorithms, tailored for bipartite ranking, to multi-class data in a valid theoretical framework. Preliminary experimental results are presented for illustration purpose.