Learning-Related complexity of linear ranking functions

Authors:
Atsuyoshi Nakamura
Affiliations:
Graduate School of Information Science and Technology, Hokkaido University, Sapporo, Japan
Venue:
ALT'06 Proceedings of the 17th international conference on Algorithmic Learning Theory
Year:
2006

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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 Θ(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.