Generalization Bounds for Out-of-distribution Generalization

Out-of-distribution (OOD) generalization has attracted increasing research attention in recent years, owing to its promising empirical results in real-world applications. However, theoretical studies on OOD generalization remain limited, particularly with respect to lower bounds on the generalization error. To better understand how source data contributes to improved OOD generalization performance, we take an initial step toward establishing a lower bound on the OOD generalization error, and subsequently investigate upper bounds from the perspective of statistical learning theory. Interestingly, we find that under some conditions, simply minimizing the average empirical risk over the source domains can yield a nearly optimal error rate (up to a logarithmic factor) \textit{without} requiring knowledge or estimation of distributional parameters or the discrepancy between source and target domains. This finding offers an explanation for the surprising phenomenon observed in DomainBed, where carefully designed OOD generalization algorithms fail to outperform the simple empirical risk minimization (ERM) algorithm. Our results also imply a no-free-lunch theorem and provide an optimistic bound for OOD generalization.