  Machine learning theory has mostly focused on generalization to samples from
the same distribution as the training data. Whereas a better understanding of
generalization beyond the training distribution where the observed distribution
changes is also fundamentally important to achieve a more powerful form of
generalization. In this paper, we attempt to study through the lens of
information measures how a particular architecture behaves when the true
probability law of the samples is potentially different at training and testing
times. Our main result is that the testing gap between the empirical
cross-entropy and its statistical expectation (measured with respect to the
testing probability law) can be bounded with high probability by the mutual
information between the input testing samples and the corresponding
representations, generated by the encoder obtained at training time. These
results of theoretical nature are supported by numerical simulations showing
that the mentioned mutual information is representative of the testing gap,
capturing qualitatively the dynamic in terms of the hyperparameters of the
network.
