Polynomial Uniform Convergence of Relative Frequencies to Probabilities

We define the concept of polynomial uniform convergence of relative frequencies to probabilities in the distribution-dependent context. Let Xn = {O, l}n, let Pn be a probability distribution on Xn and let Fn C 2X ,. be a family of events. The family {(Xn, Pn, Fn)}n~l has the property of polynomial uniform convergence if the probability that the maximum difference (over Fn) between the relative frequency and the probability of an event exceed a given positive e be at most 6 (0 < 6 < 1), when the sample on which the frequency is evaluated has size polynomial in n,l/e,l/b. Given at-sample (Xl, ... ,Xt), let C~t)(XI, ... ,Xt) be the Vapnik-Chervonenkis dimension of the family {{x}, ... ,xtl n f I f E Fn} and M(n, t) the expectation E(C~t) It). We show that {(Xn, Pn, Fn)}n~l has the property of polynomial uniform convergence iff there exists f3 > 0 such that M(n, t) = O(n/t!3). Applications to distribution-dependent PAC learning are discussed.