Robust Generalization via $α$-Mutual Information
This work addresses the problem of robust generalization for machine learning practitioners, offering theoretical tools for adaptive data analysis, but it is incremental as it builds on existing divergence and information measures.
The paper tackles the problem of bounding generalization error in learning algorithms by connecting probability measures using Rényi α-Divergences and Sibson's α-Mutual Information, extending results from Maximal Leakage to general alphabets and providing bounds for both high-probability events and expected generalization error.
The aim of this work is to provide bounds connecting two probability measures of the same event using Rényi $α$-Divergences and Sibson's $α$-Mutual Information, a generalization of respectively the Kullback-Leibler Divergence and Shannon's Mutual Information. A particular case of interest can be found when the two probability measures considered are a joint distribution and the corresponding product of marginals (representing the statistically independent scenario). In this case, a bound using Sibson's $α-$Mutual Information is retrieved, extending a result involving Maximal Leakage to general alphabets. These results have broad applications, from bounding the generalization error of learning algorithms to the more general framework of adaptive data analysis, provided that the divergences and/or information measures used are amenable to such an analysis ({\it i.e.,} are robust to post-processing and compose adaptively). The generalization error bounds are derived with respect to high-probability events but a corresponding bound on expected generalization error is also retrieved.