Universal Online Learning with Bounded Loss: Reduction to Binary Classification

Moise Blanchard, Romain Cosson
Proceedings of Thirty Fifth Conference on Learning Theory, PMLR 178:479-495, 2022.

Abstract

We study universal consistency of non-i.i.d. processes in the context of online learning. A stochastic process is said to admit universal consistency if there exists a learner that achieves vanishing average loss for any measurable response function on this process. When the loss function is unbounded, [1] showed that the only processes admitting strong universal consistency are those taking a finite number of values almost surely. However, when the loss function is bounded, the class of processes admitting strong universal consistency is much richer and its characterization could be dependent on the response setting [2]. In this paper, we show that this class of processes is independent from the response setting thereby closing an open question of [3] (Open Problem 3). Specifically, we show that the class of processes that admit universal online learning is the same for binary classification as for multiclass classification with countable number of classes. Consequently, any output setting with bounded loss can be reduced to binary classification. Our reduction is constructive and practical. Indeed, we show that the nearest neighbor algorithm is transported by our construction. For binary classification on a process admitting strong universal learning, we prove that nearest neighbor successfully learns at least all finite unions of intervals.

Cite this Paper

BibTeX
@InProceedings{pmlr-v178-blanchard22a, title = {Universal Online Learning with Bounded Loss: Reduction to Binary Classification}, author = {Blanchard, Moise and Cosson, Romain}, booktitle = {Proceedings of Thirty Fifth Conference on Learning Theory}, pages = {479--495}, year = {2022}, editor = {Loh, Po-Ling and Raginsky, Maxim}, volume = {178}, series = {Proceedings of Machine Learning Research}, month = {02--05 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v178/blanchard22a/blanchard22a.pdf}, url = {https://proceedings.mlr.press/v178/blanchard22a.html}, abstract = {We study universal consistency of non-i.i.d. processes in the context of online learning. A stochastic process is said to admit universal consistency if there exists a learner that achieves vanishing average loss for any measurable response function on this process. When the loss function is unbounded, [1] showed that the only processes admitting strong universal consistency are those taking a finite number of values almost surely. However, when the loss function is bounded, the class of processes admitting strong universal consistency is much richer and its characterization could be dependent on the response setting [2]. In this paper, we show that this class of processes is independent from the response setting thereby closing an open question of [3] (Open Problem 3). Specifically, we show that the class of processes that admit universal online learning is the same for binary classification as for multiclass classification with countable number of classes. Consequently, any output setting with bounded loss can be reduced to binary classification. Our reduction is constructive and practical. Indeed, we show that the nearest neighbor algorithm is transported by our construction. For binary classification on a process admitting strong universal learning, we prove that nearest neighbor successfully learns at least all finite unions of intervals.} }
Endnote
%0 Conference Paper %T Universal Online Learning with Bounded Loss: Reduction to Binary Classification %A Moise Blanchard %A Romain Cosson %B Proceedings of Thirty Fifth Conference on Learning Theory %C Proceedings of Machine Learning Research %D 2022 %E Po-Ling Loh %E Maxim Raginsky %F pmlr-v178-blanchard22a %I PMLR %P 479--495 %U https://proceedings.mlr.press/v178/blanchard22a.html %V 178 %X We study universal consistency of non-i.i.d. processes in the context of online learning. A stochastic process is said to admit universal consistency if there exists a learner that achieves vanishing average loss for any measurable response function on this process. When the loss function is unbounded, [1] showed that the only processes admitting strong universal consistency are those taking a finite number of values almost surely. However, when the loss function is bounded, the class of processes admitting strong universal consistency is much richer and its characterization could be dependent on the response setting [2]. In this paper, we show that this class of processes is independent from the response setting thereby closing an open question of [3] (Open Problem 3). Specifically, we show that the class of processes that admit universal online learning is the same for binary classification as for multiclass classification with countable number of classes. Consequently, any output setting with bounded loss can be reduced to binary classification. Our reduction is constructive and practical. Indeed, we show that the nearest neighbor algorithm is transported by our construction. For binary classification on a process admitting strong universal learning, we prove that nearest neighbor successfully learns at least all finite unions of intervals.
APA
Blanchard, M. & Cosson, R.. (2022). Universal Online Learning with Bounded Loss: Reduction to Binary Classification. Proceedings of Thirty Fifth Conference on Learning Theory, in Proceedings of Machine Learning Research 178:479-495 Available from https://proceedings.mlr.press/v178/blanchard22a.html.

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