Realizable Bayes-Consistency for General Metric Losses

Problem
This paper addresses the gap in understanding strong universal Bayes-consistency in the realizable setting for learning with general metric losses. Previous works primarily focused on binary classification and real-valued regression, leaving a lack of comprehensive characterizations for more general loss functions. The authors tackle an open problem posed by Cohen et al. (2022) regarding the conditions under which a distribution-free learning rule can achieve risk convergence to the best-in-class risk (zero) for any realizable data-generating distribution. This work is presented as a preprint and has not yet undergone peer review.

Method
The authors introduce a novel combinatorial framework to characterize the necessary and sufficient conditions for Bayes-consistency in the realizable case. They extend the concept of Littlestone trees to the context of general metric losses by defining an infinite non-decreasing $(\gamma_k)$-Littlestone tree, where $\gamma_k \to \infty$. This structure allows for a more nuanced understanding of the hypothesis class $\mathcal{H}$ and its relationship to the risk of learning rules. The paper does not disclose specific training compute or datasets, focusing instead on theoretical contributions.

Results
The authors demonstrate that under their proposed framework, a learning rule can achieve almost sure convergence of risk to zero for every realizable distribution if the hypothesis class $\mathcal{H}$ satisfies certain combinatorial properties. They provide a sharp characterization of these properties, which significantly extends previous results in the literature. While specific numerical results or comparisons against named baselines are not provided, the theoretical implications suggest a robust framework for understanding learning dynamics in more complex settings.

Limitations
The authors acknowledge that their results are confined to the realizable case, which may not generalize to the non-realizable setting. Additionally, the reliance on combinatorial structures may limit applicability in practical scenarios where such structures are difficult to ascertain. The paper does not address potential computational complexities associated with implementing the proposed learning rules or the scalability of the methods to large datasets.

Why it matters
This work has significant implications for the theoretical foundations of machine learning, particularly in understanding the conditions under which learning algorithms can achieve optimal performance across a broader range of loss functions. By extending the concept of Bayes-consistency to general metric losses, the findings may influence future research on robust learning algorithms and contribute to the development of more effective learning frameworks in both theoretical and applied contexts. The introduction of the $(\gamma_k)$-Littlestone tree could inspire further exploration into combinatorial structures in machine learning, potentially leading to new insights in generalization and risk minimization.

Authors: Dan Tsir Cohen, Steve Hanneke, Aryeh Kontorovich
Source: arXiv:2605.03823
URL: https://arxiv.org/abs/2605.03823v1