  We introduce a new notion of generalization -- Distributional Generalization
-- which roughly states that outputs of a classifier at train and test time are
close *as distributions*, as opposed to close in just their average error. For
example, if we mislabel 30% 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. This
behavior is not captured by classical generalization, which would only consider
the average error and not the distribution of errors over the input domain. Our
formal conjectures, which are much more general than this example, characterize
the form of distributional generalization that can be expected in terms of
problem parameters: model architecture, training procedure, number of samples,
and data distribution. We give empirical evidence for these conjectures across
a variety of domains in machine learning, including neural networks, kernel
machines, and decision trees. Our results thus advance our empirical
understanding of interpolating classifiers.
