Title: The Sufficiency of Domain Counts in Domain Generalization: A Precise Characterization Through Domain Shattering Dimension

Abstract: This paper addresses a core inquiry in domain generalization: when working with a specific family of domains, defined as data distributions, what is the necessary number of randomly sampled domains required to gather data for training a model that maintains robust performance across both observed and novel domains within that family? We formulate this challenge within the PAC learning framework and propose a novel combinatorial metric termed the "domain shattering dimension." Our analysis demonstrates that this dimension precisely defines the sample complexity for domains. Additionally, we derive a rigorous quantitative link between the domain shattering dimension and the traditional VC dimension, proving that any hypothesis class capable of learning in the standard PAC context remains learnable under our proposed framework.

Source: arXiv Generated at: 2026-06-02 00:00:00 UTC