Minimax learning rates for bipartite ranking and plug-in rules

While it is now well-known in the standard binary classification setup, that, under suitable margin assumptions and complexity conditions on the regression function, fast or even super-fast rates (i.e. rates faster than n^-1/2 or even faster than n^-1) can be achieved by plug-in classifiers, no result of this nature has been proved yet in the context of bipartite ranking, though akin to that of classification. It is the main purpose of the present paper to investigate this issue, by considering bipartite ranking as a nested continuous collection of cost-sensitive classification problems. A global low noise condition is exhibited under which certain (plug-in) ranking rules are proved to achieve fast (but not super-fast) rates over a wide non-parametric class of models. A lower bound result is also stated in a specific situation, establishing that such rates are optimal from a minimax perspective.