Distributional Generalization: Characterizing Classifiers Beyond Test Error

We present a new set of empirical properties of interpolating classifiers, including neural networks, kernel machines and decision trees. Informally, the output distribution of an interpolating classifier matches the distribution of true labels, when conditioned on certain subgroups of the input space. For example, if we mislabel 30% of images of dogs as cats in the train set of CIFAR-10, then a ResNet trained to interpolation will in fact mislabel roughly 30% of dogs as cats on the *test set* as well, while leaving other classes unaffected. These behaviors are not captured by classical generalization, which would only consider the average error over the inputs, and not *where* these errors occur. We introduce and experimentally validate a formal conjecture that specifies the subgroups for which we expect this distributional closeness. Further, we show that these properties can be seen as a new form of generalization, which advances our understanding of the implicit bias of interpolating methods.