Leveraging PAC-Bayes Theory and Gibbs Distributions for Generalization Bounds with Complexity Measures
This work addresses a foundational problem in statistical learning theory by enabling more flexible generalization bounds, though it appears incremental as it builds on existing PAC-Bayes frameworks.
The paper tackles the limitation of generalization bounds being tied to specific complexity measures by deriving a general bound using disintegrated PAC-Bayes theory and Gibbs distributions, which can be instantiated with arbitrary complexity measures and adapts to both the hypothesis class and task.
In statistical learning theory, a generalization bound usually involves a complexity measure imposed by the considered theoretical framework. This limits the scope of such bounds, as other forms of capacity measures or regularizations are used in algorithms. In this paper, we leverage the framework of disintegrated PAC-Bayes bounds to derive a general generalization bound instantiable with arbitrary complexity measures. One trick to prove such a result involves considering a commonly used family of distributions: the Gibbs distributions. Our bound stands in probability jointly over the hypothesis and the learning sample, which allows the complexity to be adapted to the generalization gap as it can be customized to fit both the hypothesis class and the task.