Uniform Convergence Beyond Glivenko-Cantelli

Tanmay Devale, Pramith Devulapalli, Steve Hanneke
Proceedings of The 37th International Conference on Algorithmic Learning Theory, PMLR 313:1-21, 2026.

Abstract

We characterize conditions under which collections of distributions on $\{0,1\}^\mathbb{N}$ 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).

Cite this Paper

BibTeX
@InProceedings{pmlr-v313-devale26a, title = {Uniform Convergence Beyond Glivenko-Cantelli}, author = {Devale, Tanmay and Devulapalli, Pramith and Hanneke, Steve}, booktitle = {Proceedings of The 37th International Conference on Algorithmic Learning Theory}, pages = {1--21}, year = {2026}, editor = {Telgarsky, Matus and Ullman, Jonathan}, volume = {313}, series = {Proceedings of Machine Learning Research}, month = {23--26 Feb}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v313/main/assets/devale26a/devale26a.pdf}, url = {https://proceedings.mlr.press/v313/devale26a.html}, abstract = {We characterize conditions under which collections of distributions on $\{0,1\}^\mathbb{N}$ 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).} }
Endnote
%0 Conference Paper %T Uniform Convergence Beyond Glivenko-Cantelli %A Tanmay Devale %A Pramith Devulapalli %A Steve Hanneke %B Proceedings of The 37th International Conference on Algorithmic Learning Theory %C Proceedings of Machine Learning Research %D 2026 %E Matus Telgarsky %E Jonathan Ullman %F pmlr-v313-devale26a %I PMLR %P 1--21 %U https://proceedings.mlr.press/v313/devale26a.html %V 313 %X We characterize conditions under which collections of distributions on $\{0,1\}^\mathbb{N}$ 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).
APA
Devale, T., Devulapalli, P. & Hanneke, S.. (2026). Uniform Convergence Beyond Glivenko-Cantelli. Proceedings of The 37th International Conference on Algorithmic Learning Theory, in Proceedings of Machine Learning Research 313:1-21 Available from https://proceedings.mlr.press/v313/devale26a.html.

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