Learnability and the doubling dimension
Given a set F of classifiers and a probability distribution over their domain, one can define a metric by taking the distance between a pair of classifiers to be the probability that they classify a random item differently. We prove bounds on the sample complexity of PAC learning in terms of the doubling dimension of this metric. These bounds imply known bounds on the sample complexity of learning halfspaces with respect to the uniform distribution that are optimal up to a constant factor. We prove a bound that holds for any algorithm that outputs a classifier with zero error whenever this is possible; this bound is in terms of the maximum of the doubling dimension and the VC-dimension of F, and strengthens the best known bound in terms of the VC-dimension alone. We show that there is no bound on the doubling dimension in terms of the VC-dimension of F (in contrast with the metric dimension).