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

Published: 25 Apr 2023, Last Modified: 25 Apr 2023Accepted by TMLREveryoneRevisionsBibTeX
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.
Supplementary Material: pdf
Assigned Action Editor: ~Pierre_Alquier1
License: Creative Commons Attribution 4.0 International (CC BY 4.0)
Submission Number: 853