Ranking as Function Approximation

An overview of the problem of learning to rank data is given. Some current machine learning approaches to the problem are described. The cost functions used to assess the quality of a ranking algorithm present particular difficulties: they are non-differentiable (as a function of the scores output by the ranker) and multivariate (in the sense that the cost associated with one ranked object depends on its relations to several other ranked objects). I present some ideas on a general framework for training using such cost functions; the approach has an appealing physical interpretation. The paper is tutorial in the sense that it is not assumed that the reader is familiar with the methods of machine learning; my hope is that the paper will encourage applied mathematicians to explore this topic.