Realizable Bayes-Consistency for General Metric Losses

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., 2020; 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. (2024), 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. (2020) to the metric loss setting.

14 pages. Accepted to ICML 2026; v2: fixed abstract metadata rendering; v3: strengthened lower-bound theorem statement to almost-sure infinite risk (proof unchanged)

Machine Learning (cs.LG); Information Theory (cs.IT); Statistics Theory (math.ST)

I.2.6; F.2.0; G.3

arXiv:2605.03823 [cs.LG]

(or arXiv:2605.03823v3 [cs.LG] for this version)