  In this paper, the worst-case probability measure over the data is introduced
as a tool for characterizing the generalization capabilities of machine
learning algorithms. More specifically, the worst-case probability measure is a
Gibbs probability measure and the unique solution to the maximization of the
expected loss under a relative entropy constraint with respect to a reference
probability measure. Fundamental generalization metrics, such as the
sensitivity of the expected loss, the sensitivity of the empirical risk, and
the generalization gap are shown to have closed-form expressions involving the
worst-case data-generating probability measure. Existing results for the Gibbs
algorithm, such as characterizing the generalization gap as a sum of mutual
information and lautum information, up to a constant factor, are recovered. A
novel parallel is established between the worst-case data-generating
probability measure and the Gibbs algorithm. Specifically, the Gibbs
probability measure is identified as a fundamental commonality of the model
space and the data space for machine learning algorithms.
