 
Summary: Proceedings of the 18th Annual Conference on Learning Theory, 2005
Learnability of Bipartite Ranking Functions
Shivani Agarwal and Dan Roth
Department of Computer Science
University of Illinois at UrbanaChampaign
201 N. Goodwin Avenue, Urbana, IL 61801, USA
{sagarwal,danr}@cs.uiuc.edu
Abstract. The problem of ranking, in which the goal is to learn a realvalued
ranking function that induces a ranking or ordering over an instance space, has
recently gained attention in machine learning. We define a model of learnability
for ranking functions in a particular setting of the ranking problem known as the
bipartite ranking problem, and derive a number of results in this model. Our first
main result provides a sufficient condition for the learnability of a class of ranking
functions F: we show that F is learnable if its bipartite rankshatter coefficients,
which measure the richness of a ranking function class in the same way as do the
standard VCdimension related shatter coefficients (growth function) for classes
of classification functions, do not grow too quickly. Our second main result gives
a necessary condition for learnability: we define a new combinatorial parameter
for a class of ranking functions F that we term the rank dimension of F, and
show that F is learnable only if its rank dimension is finite. Finally, we investigate
