Bounding the expectation of the supremum of empirical processes indexed by Hölder classes
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We obtain upper bounds on the expectation of the supremum of empirical processes indexed by Hölder classes of any smoothness and for any distribution supported on a bounded set. Another way to see it is from the point of view of integral probability metrics (IPM), a class of metrics on the space of probability measures: our rates quantify how quickly the empirical measure obtained from n independent samples from a probability measure P approaches P with respect to the IPM indexed by Hölder classes. As an extremal case we recover the known rates for the Wassertein-1 distance.
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