How Many Domains Suffice for Domain Generalization? A Tight Characterization via the Domain Shattering Dimension

2025-06-20

AI Summary

This paper addresses the fundamental sample complexity problem in domain generalization: how many source domains must be randomly drawn from a family of data distributions—potentially including unseen domains—to guarantee robust model performance across all domains in the family? To this end, we introduce *domain shattering dimension*, a novel combinatorial complexity measure that tightly characterizes domain sample complexity for the first time, and establish its precise quantitative relationship with the VC dimension. Within the PAC learning framework, integrating combinatorial analysis and statistical learning theory, we develop the first learnability theory for domain generalization: we prove that any hypothesis class PAC-learnable under standard assumptions remains learnable in this setting, and we derive necessary and sufficient conditions on the number of source domains required—achieving theoretically optimal characterization of domain complexity.

Technology Category

• Machine Learning: Transfer, Domain Adaptation, Multi-Task Learning
• Constraint Satisfaction and Optimization: Constraint Learning and Acquisition
• Search and Optimization: Learning to Search

Application Category

• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Search and Retrieval-Augmented AI: Vertical and domain-specific search
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

We study a fundamental question of domain generalization: given a family of domains (i.e., data distributions), how many randomly sampled domains do we need to collect data from in order to learn a model that performs reasonably well on every seen and unseen domain in the family? We model this problem in the PAC framework and introduce a new combinatorial measure, which we call the domain shattering dimension. We show that this dimension characterizes the domain sample complexity. Furthermore, we establish a tight quantitative relationship between the domain shattering dimension and the classic VC dimension, demonstrating that every hypothesis class that is learnable in the standard PAC setting is also learnable in our setting.

Problem

• Characterize domain sample complexity for generalization
• Introduce domain shattering dimension as new measure
• Relate domain shattering dimension to VC dimension

Innovation

• Introduces domain shattering dimension measure
• Links domain sample complexity to VC dimension
• PAC framework for domain generalization