PDF e-Pub

## Section: New Results

### Axis 2: PAC-Bayes Un-Expected Bernstein Inequality

Participant: Benjamin Guedj

We present a new PAC-Bayesian generalization bound. Standard bounds contain a $\sqrt{{L}_{n}·KL/n}$ complexity term which dominates unless ${L}_{n}$, the empirical error of the learning algorithm's randomized predictions, vanishes. We manage to replace ${L}_{n}$ 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 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 ${X}^{2}$ taken outside its expectation.

Joint work with Peter Grünwald (CWI), Zakaria Mhammedi (Australian National University).

This work has been accepted at NeurIPS 2019, will be presented as a poster in the main conference and as a oral in the workshop “Machine Learning with guarantees”, and is included in the proceedings of NeurIPS 2019.

Published: [37]