Ranking by pairwise comparison a note on risk minimization

We consider the problem of learning ranking functions in a supervised manner. A ranking function is a mapping from instances to rankings over a finite number of labels and can thus be seen as an extension of a classification function. Our learning method, referred to as ranking by pairwise comparison (RPC), is a two-step procedure. First, a valued preference structure is induced from given preference data, using a natural extension of so-called pairwise classification. A ranking is then derived from that preference structure by means of a simple scoring function. It is shown that, under some idealized assumptions, a prediction thus obtained is a risk minimizer if the distance resp. similarity between rankings is measured by the Spearman rank correlation. We conclude the paper by outlining a potential application of the method in (qualitative) fuzzy classification and identifying some extensions necessary in this context.