Extending Rademacher Complexity-Based Generalization Theory to Domain Generalization

Under the independent and identically distributed (i.i.d.) assumption, Rademacher complexity-based statistical learning theory characterizes the model's generalization through a Rademacher complexity term and its high-probability confidence guarantee. However, this framework becomes inadequate in the domain generalization scenario where distribution shifts commonly occur, precisely because distribution shift itself gives rise to an additional, distribution-induced model complexity, whereas classical Rademacher complexity is conditioned solely on training samples and thus fails to capture the complexity variations induced by distribution shifts. To overcome this limitation, we propose to treat domain distributions as the fundamental objects for defining distribution-level complexity. Specifically, we introduce a risk functional class on the Wasserstein space and quantify the model complexity with respect to domain distributions via its Rademacher term. Furthermore, by exploiting the $\mathrm{CD}(K,\infty )$ curvature structure of the Wasserstein space, we modify the McDiarmid-type concentration inequality to obtain a Wasserstein-geometric confidence guarantee for the proposed complexity measure. Based on this complexity measure, we derive a domain generalization error bound that separates a distribution-level Rademacher complexity term, which captures the complexity caused by distribution shift, and that establishes a connection between the curvature of the sample space and domain generalization performance through its confidence term. This reveals a new principle of domain generalization: learning representation space with higher curvature facilitates better cross-domain generalization. Finally, we experimentally validate this conclusion.