  We formulate a supervised learning problem, referred to as continuous
ranking, where a continuous real-valued label Y is assigned to an observable
r.v. X taking its values in a feature space $\mathcal{X}$ and the goal is to
order all possible observations x in $\mathcal{X}$ by means of a scoring
function $s:\mathcal{X}\rightarrow \mathbb{R}$ so that s(X) and Y tend to
increase or decrease together with highest probability. This problem
generalizes bi/multi-partite ranking to a certain extent and the task of
finding optimal scoring functions s(x) can be naturally cast as optimization of
a dedicated functional criterion, called the IROC curve here, or as
maximization of the Kendall ${\tau}$ related to the pair (s(X), Y ). From the
theoretical side, we describe the optimal elements of this problem and provide
statistical guarantees for empirical Kendall ${\tau}$ maximization under
appropriate conditions for the class of scoring function candidates. We also
propose a recursive statistical learning algorithm tailored to empirical IROC
curve optimization and producing a piecewise constant scoring function that is
fully described by an oriented binary tree. Preliminary numerical experiments
highlight the difference in nature between regression and continuous ranking
and provide strong empirical evidence of the performance of empirical
optimizers of the criteria proposed.
