Uniform Brackets, Containers, and Combinatorial Macbeath Regions
This work provides a unified theoretical framework connecting disparate combinatorial concepts, with applications in distributed learning, computational geometry, and online algorithms.
The paper shows that three combinatorial structures—uniform brackets, containers, and combinatorial Macbeath regions—are manifestations of a single property under a unified VC-type framework, enabling improved bounds. These bounds lead to an optimal algorithm for distributed learning of halfspaces, an improved algorithm for distributed convex set disjointness, and improved regret bounds for online algorithms against a smoothed adversary for semi-algebraic threshold functions.
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.