Learning half-spaces on general infinite spaces equipped with a distance function

For a general infinite distance space chi, with no assumptions about the distance function, which need not satisfy the metric axioms, it is not clear what the VC-dimension of the class Hof half-spaces in chi may be and if there are generalization error bounds for learning H. We define a combinatorial dimension of chi to be the independence number of the class of balls in chi. We compute it for Euclidean space and for several non-metric distance spaces. Using this dimension, we are able to provide a generalization error bound for learning Hover any infinite distance space chi. (c) 2023 Elsevier Inc. All rights reserved.