Salvato in:
| Autori principali: | , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/2510.21506 |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866912827008090112 |
|---|---|
| author | Devale, Tanmay Devulapalli, Pramith Hanneke, Steve |
| author_facet | Devale, Tanmay Devulapalli, Pramith Hanneke, Steve |
| contents | 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). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_21506 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Uniform Convergence Beyond Glivenko-Cantelli Devale, Tanmay Devulapalli, Pramith Hanneke, Steve Machine Learning 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). |
| title | Uniform Convergence Beyond Glivenko-Cantelli |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2510.21506 |