Realizable Bayes-Consistency for General Metric Losses

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,ρ), 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 (γ_k)-Littlestone tree, where γ_k \to \infty. This extends the Littlestone tree structure used in Bousquet et al. (2020) to the metric loss setting.