Adversarial laws of large numbers and optimal regret in online classification

Noga Alon, Omri Ben-Eliezer, Yuval Dagan, Shay Moran, Moni Naor, Eylon Yogev

Research output: Chapter in Book/Report/Conference proceedingConference contribution

1 Scopus citations

Abstract

Laws of large numbers guarantee that given a large enough sample from some population, the measure of any fixed sub-population is well-estimated by its frequency in the sample. We study laws of large numbers in sampling processes that can affect the environment they are acting upon and interact with it. Specifically, we consider the sequential sampling model proposed by Ben-Eliezer and Yogev (2020), and characterize the classes which admit a uniform law of large numbers in this model: these are exactly the classes that are online learnable. Our characterization may be interpreted as an online analogue to the equivalence between learnability and uniform convergence in statistical (PAC) learning. The sample-complexity bounds we obtain are tight for many parameter regimes, and as an application, we determine the optimal regret bounds in online learning, stated in terms of Littlestone's dimension, thus resolving the main open question from Ben-David, Pál, and Shalev-Shwartz (2009), which was also posed by Rakhlin, Sridharan, and Tewari (2015).

Original languageEnglish (US)
Title of host publicationSTOC 2021 - Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing
EditorsSamir Khuller, Virginia Vassilevska Williams
PublisherAssociation for Computing Machinery
Pages447-455
Number of pages9
ISBN (Electronic)9781450380539
DOIs
StatePublished - Jun 15 2021
Event53rd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2021 - Virtual, Online, Italy
Duration: Jun 21 2021Jun 25 2021

Publication series

NameProceedings of the Annual ACM Symposium on Theory of Computing
ISSN (Print)0737-8017

Conference

Conference53rd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2021
Country/TerritoryItaly
CityVirtual, Online
Period6/21/216/25/21

All Science Journal Classification (ASJC) codes

  • Software

Keywords

  • Littlestone dimension
  • adversarial robustness
  • online learning
  • random sampling
  • robust sampling

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