  The information-theoretic framework of Russo and J. Zou (2016) and Xu and
Raginsky (2017) provides bounds on the generalization error of a learning
algorithm in terms of the mutual information between the algorithm's output and
the training sample. In this work, we study the proposal, by Steinke and
Zakynthinou (2020), to reason about the generalization error of a learning
algorithm by introducing a super sample that contains the training sample as a
random subset and computing mutual information conditional on the super sample.
We first show that these new bounds based on the conditional mutual information
are tighter than those based on the unconditional mutual information. We then
introduce yet tighter bounds, building on the "individual sample" idea of Bu,
S. Zou, and Veeravalli (2019) and the "data dependent" ideas of Negrea et al.
(2019), using disintegrated mutual information. Finally, we apply these bounds
to the study of Langevin dynamics algorithm, showing that conditioning on the
super sample allows us to exploit information in the optimization trajectory to
obtain tighter bounds based on hypothesis tests.
