Whereas a variety of e fficient learning algorithms have been recently proposed to perform bipartite ranking tasks, cast as M-estimation problems, when K>2, no method for optimizing the ROC manifold, or criteria summarizing the latter such as its volume, the gold standard for assessing performance in K-partite ranking, have been introduced in the statistical learning literature yet. It is the main purpose of this paper to describe at length an e fficient approach to recursive maximization of the ROC surface, extending the TreeRank methodology originally tailored for the bipartite situation (i.e. when K = 2). The main barrier arises from the fact that, in contrast to the bipartite case, the VUS criterion of any scoring rule taking K 3 values cannot be interpreted as a cost-sensitive misclassi cation error and no method is readily available to perform the recursive optimization stage. The learning algorithm we propose, called TreeRank Tournament, breaks it and builds recursively an ordered partition of the feature space, de ning a piecewise scoring function whose ROC manifold can be remarkably interpreted as a statistical version of an adaptive piecewise linear approximant of the optimal ROC manifold. Rate bounds in sup norm desccribing the generalization ability of the scoring rule thus built are established and numerical results illustrating the performance of the TreeRank Tournament approach, compared to that of natural competitors such as aggregation methods, are also displayed.
The TreeRank Tournament} Algorithm for Multipartite Ranking