# Bounding the expectation of the supremum of empirical processes indexed by H"older classes

arXiv: https://arxiv.org/abs/2003.13530 · HTML: https://mlsanity.com/paper/2003.13530/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 30 Mar 2020.

## Built on by (most cited first, 8 of 8)

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- [Near-optimal learning with average H"older smoothness](https://mlsanity.com/paper/2302.06005/) (2023)
- [Non-Asymptotic Error Bounds for Bidirectional GANs](https://mlsanity.com/paper/2110.12319/) (2021)
- [A Wasserstein perspective of Vanilla GANs](https://mlsanity.com/paper/2403.15312/) (2024)
- [Distributional Shrinkage II: Higher-Order Scores Encode Brenier Map](https://mlsanity.com/paper/2512.09295/) (2025)

## Abstract (arXiv)

In this note, we provide upper bounds on the expectation of the supremum of empirical processes indexed by H"older classes of any smoothness and for any distribution supported on a bounded set in mathbb Rd. These results can be alternatively 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 the integral probability metrics (IPM). In particular, the IPM indexed by a H"older class is considered and the corresponding rates are derived. These results interpolate between the 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 an RKHS defined by a bounded kernel).