Fast-rate PAC-Bayes Generalization Bounds via Shifted Rademacher Processes
This work addresses a theoretical gap in machine learning generalization theory, offering incremental advancements in PAC-Bayesian analysis for researchers in statistical learning.
The paper tackles the problem of bridging Rademacher complexity and PAC-Bayesian theory by extending prior work to match fast-rate bounds using shifted Rademacher processes and deriving a new bound based on empirical risk surface flatness, establishing a new framework for fast-rate PAC-Bayes bounds.
The developments of Rademacher complexity and PAC-Bayesian theory have been largely independent. One exception is the PAC-Bayes theorem of Kakade, Sridharan, and Tewari (2008), which is established via Rademacher complexity theory by viewing Gibbs classifiers as linear operators. The goal of this paper is to extend this bridge between Rademacher complexity and state-of-the-art PAC-Bayesian theory. We first demonstrate that one can match the fast rate of Catoni's PAC-Bayes bounds (Catoni, 2007) using shifted Rademacher processes (Wegkamp, 2003; Lecué and Mitchell, 2012; Zhivotovskiy and Hanneke, 2018). We then derive a new fast-rate PAC-Bayes bound in terms of the "flatness" of the empirical risk surface on which the posterior concentrates. Our analysis establishes a new framework for deriving fast-rate PAC-Bayes bounds and yields new insights on PAC-Bayesian theory.