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.

翻译：我们研究在可实现设置下使用一般度量损失进行学习时的强通用贝叶斯一致性，将经典刻画从$0$-$1$分类（Bousquet等人，2020；Hanneke等人，2021）和实值回归（Attias等人，2024）中推广出来。给定一个实例空间$(X,ρ)$、一个可能具有无界损失的标签空间$(Y,\ell)$以及一个假设类$H \subseteq Y^{X}$，我们解决了Tsir Cohen和Kontorovich（2022）提出的一个开放问题的可实现情形。具体而言，我们找到了假设类$H$上的充要条件，在此条件下存在一个与分布无关的学习规则，对于每一个可实现的数据生成分布，其风险几乎必然收敛到类内最佳风险（即零）。我们的主要贡献是以组合障碍的形式给出了这一精确刻画：类似于Attias等人（2024），我们引入了无限非递减$(\gamma_k)$-Littlestone树的概念，其中$\gamma_k \to \infty$。这将在Bousquet等人（2020）中使用的Littlestone树结构推广到了度量损失设置中。