We present a new PAC-Bayesian generalization bound. Standard bounds contain a $\sqrt{L_n \cdot \KL/n}$ complexity term which dominates unless Ln, the empirical error of the learning algorithm's randomized predictions, vanishes. We manage to replace Ln by a term which vanishes in many more situations, essentially whenever the employed learning algorithm is sufficiently stable on the dataset at hand. Our new bound consistently beats state-of-the-art bounds both on a toy example and on UCI datasets (with large enough n). Theoretically, unlike existing bounds, our new bound can be expected to converge to 0 faster whenever a Bernstein/Tsybakov condition holds, thus connecting PAC-Bayesian generalization and {\em excess risk\/} bounds---for the latter it has long been known that faster convergence can be obtained under Bernstein conditions. Our main technical tool is a new concentration inequality which is like Bernstein's but with X2 taken outside its expectation.

Safe Bayesian Inference: A Theory of Misspecification based on Statistical Learning
Machine Learning

Mhammedi, Z., Grünwald, P., & Guedj, B. (2019). PAC-Bayes unexpected Bernstein inequality. In Proceedings NeurIPS (Annual Conference on Neural Information Processing Systems).