Realizable Bayes-Consistency for General Metric Losses

Dan Tsir Cohen, Steve Hanneke, Aryeh Kontorovich
Proceedings of the 43rd International Conference on Machine Learning, PMLR 306:21088-21101, 2026.

Abstract

We study strong universal Bayes-consistency in the realizable setting for learning with general metric losses, extending classical characterizations beyond $0$-$1$ classification (Bousquet et al., 2021; Hanneke et al., 2021) and real-valued regression (Attias et al., 2024). Given an instance space $(X,\rho)$, a label space $(Y,\ell)$ with possibly unbounded loss, and a hypothesis class $H \subseteq Y^{X}$, we resolve the realizable case of an open problem presented in Tsir Cohen and Kontorovich (2022). Specifically, we find the necessary and sufficient conditions on the hypothesis class $H$ under which there exists a distribution-free learning rule whose risk converges almost surely to the best-in-class risk (which is zero) for every realizable data-generating distribution. Our main contribution is this sharp characterization in terms of a combinatorial obstruction: Similarly to Attias et al. (2023), we introduce the notion of an infinite non-decreasing $(\gamma_k)$-Littlestone tree, where $\gamma_k \to \infty$. This extends the Littlestone tree structure used in Bousquet et al. (2021) to the metric loss setting.

Cite this Paper

BibTeX
@InProceedings{pmlr-v306-cohen26b, title = {Realizable {B}ayes-Consistency for General Metric Losses}, author = {Cohen, Dan Tsir and Hanneke, Steve and Kontorovich, Aryeh}, booktitle = {Proceedings of the 43rd International Conference on Machine Learning}, pages = {21088--21101}, year = {2026}, editor = {Zhang, Tong and Dudik, Miroslav and Jaggi, Martin and Agarwal, Alekh and Li, Sharon and Schuurmans, Dale and Zhu, Jerry and Berkenkamp, Felix and Dong, Hanze and Bietti, Alberto}, volume = {306}, series = {Proceedings of Machine Learning Research}, month = {06--11 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v306/main/assets/cohen26b/cohen26b.pdf}, url = {https://proceedings.mlr.press/v306/cohen26b.html}, abstract = {We study strong universal Bayes-consistency in the realizable setting for learning with general metric losses, extending classical characterizations beyond $0$-$1$ classification (Bousquet et al., 2021; Hanneke et al., 2021) and real-valued regression (Attias et al., 2024). Given an instance space $(X,\rho)$, a label space $(Y,\ell)$ with possibly unbounded loss, and a hypothesis class $H \subseteq Y^{X}$, we resolve the realizable case of an open problem presented in Tsir Cohen and Kontorovich (2022). Specifically, we find the necessary and sufficient conditions on the hypothesis class $H$ under which there exists a distribution-free learning rule whose risk converges almost surely to the best-in-class risk (which is zero) for every realizable data-generating distribution. Our main contribution is this sharp characterization in terms of a combinatorial obstruction: Similarly to Attias et al. (2023), we introduce the notion of an infinite non-decreasing $(\gamma_k)$-Littlestone tree, where $\gamma_k \to \infty$. This extends the Littlestone tree structure used in Bousquet et al. (2021) to the metric loss setting.} }
Endnote
%0 Conference Paper %T Realizable Bayes-Consistency for General Metric Losses %A Dan Tsir Cohen %A Steve Hanneke %A Aryeh Kontorovich %B Proceedings of the 43rd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2026 %E Tong Zhang %E Miroslav Dudik %E Martin Jaggi %E Alekh Agarwal %E Sharon Li %E Dale Schuurmans %E Jerry Zhu %E Felix Berkenkamp %E Hanze Dong %E Alberto Bietti %F pmlr-v306-cohen26b %I PMLR %P 21088--21101 %U https://proceedings.mlr.press/v306/cohen26b.html %V 306 %X We study strong universal Bayes-consistency in the realizable setting for learning with general metric losses, extending classical characterizations beyond $0$-$1$ classification (Bousquet et al., 2021; Hanneke et al., 2021) and real-valued regression (Attias et al., 2024). Given an instance space $(X,\rho)$, a label space $(Y,\ell)$ with possibly unbounded loss, and a hypothesis class $H \subseteq Y^{X}$, we resolve the realizable case of an open problem presented in Tsir Cohen and Kontorovich (2022). Specifically, we find the necessary and sufficient conditions on the hypothesis class $H$ under which there exists a distribution-free learning rule whose risk converges almost surely to the best-in-class risk (which is zero) for every realizable data-generating distribution. Our main contribution is this sharp characterization in terms of a combinatorial obstruction: Similarly to Attias et al. (2023), we introduce the notion of an infinite non-decreasing $(\gamma_k)$-Littlestone tree, where $\gamma_k \to \infty$. This extends the Littlestone tree structure used in Bousquet et al. (2021) to the metric loss setting.
APA
Cohen, D.T., Hanneke, S. & Kontorovich, A.. (2026). Realizable Bayes-Consistency for General Metric Losses. Proceedings of the 43rd International Conference on Machine Learning, in Proceedings of Machine Learning Research 306:21088-21101 Available from https://proceedings.mlr.press/v306/cohen26b.html.

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