We study strong universal Bayes-consistency in the realizable setting for learning with general metric losses, extending classical characterizations beyond $0$-$1$ classification \citep{bousquet_theory_2021, hanneke2021universalbayesconsistencymetric} and real-valued regression \citep{attias_universal_2024}. Given an instance space $(\mathcal X,ρ)$, a label space $(\mathcal Y,\ell)$ with possibly unbounded loss, and a hypothesis class $\mathcal H \subseteq \mathcal Y^{\mathcal X}$, we resolve the realizable case of an open problem presented in \citet{pmlr-v178-cohen22a}. Specifically, we find the necessary and sufficient conditions on the hypothesis class $\mathcal H$ 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 \citet{attias2024optimallearnersrealizableregression}, 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 \citet{bousquet_theory_2021} to the metric loss setting.

翻译：我们研究在可实现设定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 \citep{bousquet_theory_2021, hanneke2021universalbayesconsistencymetric} 和实值回归 \citep{attias_universal_2024} 进行了推广。给定一个实例空间 $(\mathcal X,ρ)$、一个可能具有无界损失的标签空间 $(\mathcal Y,\ell)$，以及一个假设类 $\mathcal H \subseteq \mathcal Y^{\mathcal X}$，我们解决了 \citet{pmlr-v178-cohen22a} 中提出的一个公开问题的可实现情形。具体地，我们找到了假设类 $\mathcal H$ 的必要充分条件，使得存在一个无分布学习规则，其对每个可实现数据生成分布的风险几乎必然收敛到类内最优风险（即零）。我们的主要贡献是这一基于组合障碍的精确刻画：与 \citet{attias2024optimallearnersrealizableregression} 类似，我们引入了无限非递减 $(γ_k)$-Littlestone 树的概念，其中 $γ_k \to \infty$。这将在 \citet{bousquet_theory_2021} 中使用的 Littlestone 树结构扩展到了度量损失设定。