Uniform Brackets, Containers, and Combinatorial Macbeath Regions

TL;DR

This paper unifies three combinatorial structures—uniform brackets, containers, and Macbeath regions—under a single framework, enabling the transfer of geometric tools to improve algorithms in learning, geometry, and online optimization.

Contribution

It introduces a unified combinatorial framework connecting brackets, containers, and Macbeath regions, leading to improved bounds and algorithms across multiple fields.

Findings

Derived improved bounds for combinatorial objects.

Developed optimal algorithms for distributed learning of halfspaces.

Achieved better regret bounds for online algorithms against smoothed adversaries.

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 a smoothed adversary for a large class of semi-algebraic threshold functions.