Hierarchical Domain Generalization

We study hierarchical domain generalization as a problem of extrapolation from finite observed regions to an entire instance space, replacing i.i.d. sampling with arbitrary domain hierarchies. We show that the central obstruction is not only the complexity of the hypothesis class, but the train/test domain partition through which evidence is revealed. In particular, on an infinite domain, for any class with more than two hypotheses and any training size, some partition makes generalization fail for some target. These results suggest that modern generalization theory must treat domain structure as a first-class object.