Generalization Error Bounds Via Rényi-, $f$-Divergences and Maximal Leakage

Abstract

In this work, the probability of an event under some joint distribution is bounded by measuring it with the product of the marginals instead (which is typically easier to analyze) together with a measure of the dependence between the two random variables. These results find applications in adaptive data analysis, where multiple dependencies are introduced and in learning theory, where they can be employed to bound the generalization error of a learning algorithm. Bounds are given in terms of Sibson's Mutual Information, $\alpha -$Divergences, Hellinger Divergences, and $f-$Divergences. A case of particular interest is the Maximal Leakage (or Sibson's Mutual Information of order infinity), since this measure is robust to post-processing and composes adaptively. The corresponding bound can be seen as a generalization of classical bounds, such as Hoeffding's and McDiarmid's inequalities, to the case of dependent random variables.

arXiv admin note: text overlap with arXiv:1903.01777

References in corpus (6)

• Rényi Divergence and Kullback-Leibler Divergence
• $f$-divergence Inequalities
• Information-theoretic analysis of generalization capability of learning algorithms
• Generalization in Adaptive Data Analysis and Holdout Reuse
• Learners that Use Little Information
• Chaining Mutual Information and Tightening Generalization Bounds

Cited by in corpus (4)

• Generalization Bounds via Information Density and Conditional Information Density
• Tighter expected generalization error bounds via Wasserstein distance
• Characterizing the Generalization Error of Gibbs Algorithm with Symmetrized KL information
• On Conditional $\alpha$-Information and its Application to Side-Channel Analysis