Anthony, Martin and Ratsaby, Joel
(2018)
Largewidth bounds for learning halfspaces on distance spaces.
Discrete Applied Mathematics.
ISSN 0166218X
Abstract
A halfspace over a distance space is a generalization of a halfspace in a vector space. An important advantage of a distance space over a metric space is that the triangle inequality need not be satisfied, which makes our results potentially very useful in practice. Given two points in a set, a halfspace is defined by them, as the set of all points closer to the first point than to the second. In this paper we consider the problem of learning halfspaces in any finite distance space, that is, any finite set equipped with a distance function. We make use of a notion of ‘width’ of a halfspace at a given point: this is defined as the difference between the distances of the point to the two points that define the halfspace. We obtain probabilistic bounds on the generalization error when learning halfspaces from samples. These bounds depend on the empirical error (the fraction of sample points on which the halfspace does not achieve a large width) and on the VCdimension of the effective class of halfspaces that have a large sample width. Unlike some previous work on learning classification over metric spaces, the bound does not involve the covering number of the space, and can therefore be tighter
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