Generalization Error Bounds Via Rényi-, $f$-Divergences and Maximal Leakage
This work addresses the challenge of analyzing generalization error in adaptive data analysis and learning theory, where dependencies are common, but it is incremental as it extends existing bounds to more complex scenarios.
The paper tackles the problem of bounding generalization error in learning algorithms by developing bounds that account for dependencies between random variables, using measures like Sibson's Mutual Information and Maximal Leakage, which generalize classical inequalities like Hoeffding's to dependent cases.
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, $α-$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.