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The information-theoretic framework of Russo and J. Zou (2016) and Xu andRaginsky (2017) provides bounds on the generalization error of a learningalgorithm in terms of the mutual information between the algorithm's output andthe training sample. In this work, we study the proposal, by Steinke andZakynthinou (2020), to reason about the generalization error of a learningalgorithm by introducing a super sample that contains the training sample as arandom subset and computing mutual information conditional on the super sample.We first show that these new bounds based on the conditional mutual informationare tighter than those based on the unconditional mutual information. We thenintroduce 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 boundsto the study of Langevin dynamics algorithm, showing that conditioning on thesuper sample allows us to exploit information in the optimization trajectory toobtain tighter bounds based on hypothesis tests.

Sharpened Generalization Bounds based on Conditional Mutual Information and an Application to Noisy, Iterative Algorithms