PAC-Bayes Un-Expected Bernstein Inequality

Zakaria Mhammedi (1, 2) , Peter Grünwald (3, 4) , Benjamin Guedj (5, 6, 4, 7, 8)

1 ANU - Australian National University
2 CSIRO - Data61 [Canberra]
3 CWI - Centrum Wiskunde & Informatica
4 Inria-CWI - Inria-CWI
5 UCL - University College of London [London]
6 UCL-CS - Department of Computer science [University College of London]
7 MODAL - MOdel for Data Analysis and Learning
8 Inria-London - The Inria London Programme

We present a new PAC-Bayesian generalization bound. Standard bounds contain a $\sqrt{L_n \cdot \KL/n}$ 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 {\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 $X^2$ taken outside its expectation.

Domaines: Machine Learning [stat.ML]; Apprentissage [cs.LG]; Théorie [stat.TH]

Zakaria Mhammedi, Peter Grünwald, Benjamin Guedj. PAC-Bayes Un-Expected Bernstein Inequality. NeurIPS 2019, Dec 2019, Vancouver, Canada. ⟨hal-02401295⟩