On the Consistency of Kernel Methods with Dependent Observations

Pierre-François Massiani, Sebastian Trimpe, Friedrich Solowjow
Proceedings of the 41st International Conference on Machine Learning, PMLR 235:35013-35038, 2024.

Abstract

The consistency of a learning method is usually established under the assumption that the observations are a realization of an independent and identically distributed (i.i.d.) or mixing process. Yet, kernel methods such as support vector machines (SVMs), Gaussian processes, or conditional kernel mean embeddings (CKMEs) all give excellent performance under sampling schemes that are obviously non-i.i.d., such as when data comes from a dynamical system. We propose the new notion of empirical weak convergence (EWC) as a general assumption explaining such phenomena for kernel methods. It assumes the existence of a random asymptotic data distribution and is a strict weakening of previous assumptions in the field. Our main results then establish consistency of SVMs, kernel mean embeddings, and general Hilbert-space valued empirical expectations with EWC data. Our analysis holds for both finite- and infinite-dimensional outputs, as we extend classical results of statistical learning to the latter case. In particular, it is also applicable to CKMEs. Overall, our results open new classes of processes to statistical learning and can serve as a foundation for a theory of learning beyond i.i.d. and mixing.

Cite this Paper

BibTeX
@InProceedings{pmlr-v235-massiani24a, title = {On the Consistency of Kernel Methods with Dependent Observations}, author = {Massiani, Pierre-Fran\c{c}ois and Trimpe, Sebastian and Solowjow, Friedrich}, booktitle = {Proceedings of the 41st International Conference on Machine Learning}, pages = {35013--35038}, year = {2024}, editor = {Salakhutdinov, Ruslan and Kolter, Zico and Heller, Katherine and Weller, Adrian and Oliver, Nuria and Scarlett, Jonathan and Berkenkamp, Felix}, volume = {235}, series = {Proceedings of Machine Learning Research}, month = {21--27 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v235/main/assets/massiani24a/massiani24a.pdf}, url = {https://proceedings.mlr.press/v235/massiani24a.html}, abstract = {The consistency of a learning method is usually established under the assumption that the observations are a realization of an independent and identically distributed (i.i.d.) or mixing process. Yet, kernel methods such as support vector machines (SVMs), Gaussian processes, or conditional kernel mean embeddings (CKMEs) all give excellent performance under sampling schemes that are obviously non-i.i.d., such as when data comes from a dynamical system. We propose the new notion of empirical weak convergence (EWC) as a general assumption explaining such phenomena for kernel methods. It assumes the existence of a random asymptotic data distribution and is a strict weakening of previous assumptions in the field. Our main results then establish consistency of SVMs, kernel mean embeddings, and general Hilbert-space valued empirical expectations with EWC data. Our analysis holds for both finite- and infinite-dimensional outputs, as we extend classical results of statistical learning to the latter case. In particular, it is also applicable to CKMEs. Overall, our results open new classes of processes to statistical learning and can serve as a foundation for a theory of learning beyond i.i.d. and mixing.} }
Endnote
%0 Conference Paper %T On the Consistency of Kernel Methods with Dependent Observations %A Pierre-François Massiani %A Sebastian Trimpe %A Friedrich Solowjow %B Proceedings of the 41st International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2024 %E Ruslan Salakhutdinov %E Zico Kolter %E Katherine Heller %E Adrian Weller %E Nuria Oliver %E Jonathan Scarlett %E Felix Berkenkamp %F pmlr-v235-massiani24a %I PMLR %P 35013--35038 %U https://proceedings.mlr.press/v235/massiani24a.html %V 235 %X The consistency of a learning method is usually established under the assumption that the observations are a realization of an independent and identically distributed (i.i.d.) or mixing process. Yet, kernel methods such as support vector machines (SVMs), Gaussian processes, or conditional kernel mean embeddings (CKMEs) all give excellent performance under sampling schemes that are obviously non-i.i.d., such as when data comes from a dynamical system. We propose the new notion of empirical weak convergence (EWC) as a general assumption explaining such phenomena for kernel methods. It assumes the existence of a random asymptotic data distribution and is a strict weakening of previous assumptions in the field. Our main results then establish consistency of SVMs, kernel mean embeddings, and general Hilbert-space valued empirical expectations with EWC data. Our analysis holds for both finite- and infinite-dimensional outputs, as we extend classical results of statistical learning to the latter case. In particular, it is also applicable to CKMEs. Overall, our results open new classes of processes to statistical learning and can serve as a foundation for a theory of learning beyond i.i.d. and mixing.
APA
Massiani, P., Trimpe, S. & Solowjow, F.. (2024). On the Consistency of Kernel Methods with Dependent Observations. Proceedings of the 41st International Conference on Machine Learning, in Proceedings of Machine Learning Research 235:35013-35038 Available from https://proceedings.mlr.press/v235/massiani24a.html.

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