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Uniform Brackets, Containers, and Combinatorial Macbeath Regions

Authors Kunal Dutta , Arijit Ghosh, Shay Moran

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Author Details

Kunal Dutta
  • Department of Informatics, University of Warsaw, Poland
Arijit Ghosh
  • Indian Statistical Institute, Kolkata, India
Shay Moran
  • Technion – Israel Institute of Technology, Haifa, Israel
  • Google Research, Tel Aviv, Israel

Funding

  • Dutta, Kunal: Supported by the Polish NCN SONATA Grant no. 2019/35/D/ST6/04525.
  • Moran, Shay: Shay Moran is a Robert J. Shillman Fellow and his research is supported by the ISF, grant no. 1225/20, by an Azrieli Faculty Fellowship, and by BSF grant 2018385.

Cite As Get BibTex

Kunal Dutta, Arijit Ghosh, and Shay Moran. Uniform Brackets, Containers, and Combinatorial Macbeath Regions. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 59:1-59:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.ITCS.2022.59

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.

Subject Classification

ACM Subject Classification
  • Theory of computation
Keywords
  • communication complexity
  • distributed learning
  • emperical process theory
  • online algorithms
  • discrete geometry
  • computational geometry

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