Schloss Dagstuhl - Leibniz-Zentrum für Informatik GmbH Schloss Dagstuhl - Leibniz-Zentrum für Informatik GmbH scholarly article en Dutta, Kunal; Ghosh, Arijit; Moran, Shay https://www.dagstuhl.de/lipics License: Creative Commons Attribution 4.0 license (CC BY 4.0)
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URN: urn:nbn:de:0030-drops-156551
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Uniform Brackets, Containers, and Combinatorial Macbeath Regions

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Abstract

We study the connections between three seemingly different combinatorial structures - uniform brackets in statistics and probability theory, containers in online and distributed learning theory, and combinatorial Macbeath regions, or Mnets in discrete and computational geometry. We show that these three concepts are manifestations of a single combinatorial property that can be expressed under a unified framework along the lines of Vapnik-Chervonenkis type theory for uniform convergence. These new connections help us to bring tools from discrete and computational geometry to prove improved bounds for these objects. Our improved bounds help to get an optimal algorithm for distributed learning of halfspaces, an improved algorithm for the distributed convex set disjointness problem, and improved regret bounds for online algorithms against σ-smoothed adversary for a large class of semi-algebraic threshold functions.

BibTeX - Entry

@InProceedings{dutta_et_al:LIPIcs.ITCS.2022.59,
  author =	{Dutta, Kunal and Ghosh, Arijit and Moran, Shay},
  title =	{{Uniform Brackets, Containers, and Combinatorial Macbeath Regions}},
  booktitle =	{13th Innovations in Theoretical Computer Science Conference (ITCS 2022)},
  pages =	{59:1--59:10},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-217-4},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{215},
  editor =	{Braverman, Mark},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2022/15655},
  URN =		{urn:nbn:de:0030-drops-156551},
  doi =		{10.4230/LIPIcs.ITCS.2022.59},
  annote =	{Keywords: communication complexity, distributed learning, emperical process theory, online algorithms, discrete geometry, computational geometry}
}

Keywords: communication complexity, distributed learning, emperical process theory, online algorithms, discrete geometry, computational geometry
Seminar: 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)
Issue date: 2022
Date of publication: 25.01.2022


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