Bounding the expectation of the supremum of empirical processes indexed by Hölder classes

In this note, we provide 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 in $$\mathbb{R}^{d}$$ . These results can alternatively be seen as non-asymptotic risk bounds, when the unknown distribution is estimated by its empirical counterpart, based on $$n$$ independent observations, and the error of estimation is quantified by integral probability metrics (IPM). In particular, IPM indexed by Hölder classes are considered and the corresponding rates are derived. These results interpolate between two well-known extreme cases: the rate $$n^{-1/d}$$ corresponding to the Wassertein-1 distance (the least smooth case) and the fast rate $$n^{-1/2}$$ corresponding to very smooth functions (for instance, functions from a RKHS defined by a bounded kernel).