PAC-Bayes Generalisation Bounds for Heavy-Tailed Losses through Supermartingales

Maxime Haddouche, Benjamin Guedj

Abstract: While PAC-Bayes is now an established learning framework for light-tailed losses (\emph{e.g.}, subgaussian or subexponential), its extension to the case of heavy-tailed losses remains largely uncharted and has attracted a growing interest in recent years. We contribute PAC-Bayes generalisation bounds for heavy-tailed losses under the sole assumption of bounded variance of the loss function. Under that assumption, we extend previous results from \citet{kuzborskij2019efron}. Our key technical contribution is exploiting an extention of Markov's inequality for supermartingales. Our proof technique unifies and extends different PAC-Bayesian frameworks by providing bounds for unbounded martingales as well as bounds for batch and online learning with heavy-tailed losses.

Submission Length: Regular submission (no more than 12 pages of main content)

Changes Since Last Submission: We updated the final version of our manuscript.

Assigned Action Editor: ~Pierre_Alquier1

License: Creative Commons Attribution 4.0 International (CC BY 4.0)

Submission Number: 853