An Empirical Bernstein Inequality for Dependent Data in Hilbert Spaces and Applications

Erfan Mirzaei, Andreas Maurer, Vladimir R Kostic, Massimiliano Pontil
Proceedings of The 28th International Conference on Artificial Intelligence and Statistics, PMLR 258:5158-5166, 2025.

Abstract

Learning from non-independent and non-identically distributed data poses a persistent challenge in statistical learning. In this study, we introduce data-dependent Bernstein inequalities tailored for vector-valued processes in Hilbert space. Our inequalities apply to both stationary and non-stationary processes and exploit the potential rapid decay of correlations between temporally separated variables to improve estimation. We demonstrate the utility of these bounds by applying them to covariance operator estimation in the Hilbert-Schmidt norm and to operator learning in dynamical systems, achieving novel risk bounds. Finally, we perform numerical experiments to illustrate the practical implications of these bounds in both contexts.

Cite this Paper

BibTeX
@InProceedings{pmlr-v258-mirzaei25a, title = {An Empirical Bernstein Inequality for Dependent Data in Hilbert Spaces and Applications}, author = {Mirzaei, Erfan and Maurer, Andreas and Kostic, Vladimir R and Pontil, Massimiliano}, booktitle = {Proceedings of The 28th International Conference on Artificial Intelligence and Statistics}, pages = {5158--5166}, year = {2025}, editor = {Li, Yingzhen and Mandt, Stephan and Agrawal, Shipra and Khan, Emtiyaz}, volume = {258}, series = {Proceedings of Machine Learning Research}, month = {03--05 May}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v258/main/assets/mirzaei25a/mirzaei25a.pdf}, url = {https://proceedings.mlr.press/v258/mirzaei25a.html}, abstract = {Learning from non-independent and non-identically distributed data poses a persistent challenge in statistical learning. In this study, we introduce data-dependent Bernstein inequalities tailored for vector-valued processes in Hilbert space. Our inequalities apply to both stationary and non-stationary processes and exploit the potential rapid decay of correlations between temporally separated variables to improve estimation. We demonstrate the utility of these bounds by applying them to covariance operator estimation in the Hilbert-Schmidt norm and to operator learning in dynamical systems, achieving novel risk bounds. Finally, we perform numerical experiments to illustrate the practical implications of these bounds in both contexts.} }
Endnote
%0 Conference Paper %T An Empirical Bernstein Inequality for Dependent Data in Hilbert Spaces and Applications %A Erfan Mirzaei %A Andreas Maurer %A Vladimir R Kostic %A Massimiliano Pontil %B Proceedings of The 28th International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2025 %E Yingzhen Li %E Stephan Mandt %E Shipra Agrawal %E Emtiyaz Khan %F pmlr-v258-mirzaei25a %I PMLR %P 5158--5166 %U https://proceedings.mlr.press/v258/mirzaei25a.html %V 258 %X Learning from non-independent and non-identically distributed data poses a persistent challenge in statistical learning. In this study, we introduce data-dependent Bernstein inequalities tailored for vector-valued processes in Hilbert space. Our inequalities apply to both stationary and non-stationary processes and exploit the potential rapid decay of correlations between temporally separated variables to improve estimation. We demonstrate the utility of these bounds by applying them to covariance operator estimation in the Hilbert-Schmidt norm and to operator learning in dynamical systems, achieving novel risk bounds. Finally, we perform numerical experiments to illustrate the practical implications of these bounds in both contexts.
APA
Mirzaei, E., Maurer, A., Kostic, V.R. & Pontil, M.. (2025). An Empirical Bernstein Inequality for Dependent Data in Hilbert Spaces and Applications. Proceedings of The 28th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 258:5158-5166 Available from https://proceedings.mlr.press/v258/mirzaei25a.html.

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