# Uniform Convergence Beyond Glivenko-Cantelli

arXiv: https://arxiv.org/abs/2510.21506 · HTML: https://mlsanity.com/paper/2510.21506/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 2 of its 10 references that are in the map.
- First posted on arXiv 24 Oct 2025.

## Builds on (most cited first, 2 of 2)

- [Density Estimation in Infinite Dimensional Exponential Families](https://mlsanity.com/paper/1312.3516/) (2013)
- [The Empirical Mean is Minimax Optimal for Local Glivenko-Cantelli](https://mlsanity.com/paper/2410.02835/) (2024)

## Built on by (most cited first, 1 of 1)

- [Lipschitzian SLLNs for random functions](https://mlsanity.com/paper/2607.20411/) (2026)

## Abstract (arXiv)

We characterize conditions under which collections of distributions on 0,1^mathbbN admit uniform estimation of their mean. Prior work from Vapnik and Chervonenkis (1971) has focused on uniform convergence using the empirical mean estimator, leading to the principle known as P- Glivenko-Cantelli. We extend this framework by moving beyond the empirical mean estimator and introducing Uniform Mean Estimability, also called UME-learnability, which captures when a collection permits uniform mean estimation by any arbitrary estimator. We work on the space created by the mean vectors of the collection of distributions. For each distribution, the mean vector records the expected value in each coordinate. We show that separability of the mean vectors is a sufficient condition for UME-learnability. However, we show that separability of the mean vectors is not necessary for UME-learnability by constructing a collection of distributions whose mean vectors are non-separable yet UME-learnable using techniques fundamentally different from those used in our separability-based analysis. Finally, we establish that countable unions of UME-learnable collections are also UME-learnable, solving the conjecture posed in Cohen et al. (2025).