Ranking and empirical minimization of U-statistics
Stephan Clemencon, Gábor Lugosi and Nicolas Vayatis
Annals of Statistics -159, 2008.

Abstract

The problem of ranking/ordering instances, instead of simply classifying them, has recently gained much attention in machine learning. In this paper we formulate the \textit{ranking problem} in a rigorous statistical framework. The goal is to learn a ranking rule for deciding, among two instances, which one is ``better,'' with minimum ranking risk. Since the natural estimates of the risk are of the form of a $U$-statistic, results of the theory of $U$-processes are required for investigating the consistency of empirical risk minimizers. We establish, in particular, a tail inequality for degenerate U-processes, and apply it for showing that fast rates of convergence may be achieved under specific noise assumptions, just like in classification. Convex risk minimization methods are also studied.

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