Learning-related complexity of linear ranking functions

Abstract

In this paper, we study learning-related complexity of linear ranking functions from n-dimensional Euclidean space to {1, 2,..., k}. We show that their graph dimension, a kind of measure for PAC learning complexity in the multiclass classification setting, is (n+k). This graph dimension is significantly smaller than the graph dimension Ω(nk) of the class of {1, 2,..., k}-valued decision-list functions naturally defined using k - 1 linear discrimination functions. We also show a risk bound of learning linear ranking functions in the ordinal regression setting by a technique similar to that used in the proof of an upper bound of their graph dimension. © Springer-Verlag Berlin Heidelberg 2006.