  We formulate the local ranking problem in the framework of bipartite ranking
where the goal is to focus on the best instances. We propose a methodology
based on the construction of real-valued scoring functions. We study empirical
risk minimization of dedicated statistics which involve empirical quantiles of
the scores. We first state the problem of finding the best instances which can
be cast as a classification problem with mass constraint. Next, we develop
special performance measures for the local ranking problem which extend the
Area Under an ROC Curve (AUC/AROC) criterion and describe the optimal elements
of these new criteria. We also highlight the fact that the goal of ranking the
best instances cannot be achieved in a stage-wise manner where first, the best
instances would be tentatively identified and then a standard AUC criterion
could be applied. Eventually, we state preliminary statistical results for the
local ranking problem.
