Convex Set Disjointness, Distributed Learning of Halfspaces, and LP Feasibility

We study the Convex Set Disjointness (CSD) problem, where two players have input sets taken from an arbitrary fixed domain~Usubseteq ℝd of size lvert Urvert = n. Their mutual goal is to decide using minimum communication whether the convex hulls of their sets intersect (equivalently, whether their sets can be separated by a hyperplane). Different forms of this problem naturally arise in distributed learning and optimization: it is equivalent to em Distributed Linear Program (LP) Feasibility -- a basic task in distributed optimization, and it is tightly linked to it Distributed Learning of Halfdpaces in ℝd. In communication complexity theory, CSD can be viewed as a geometric interpolation between the classical problems of Set Disjointness (when~d≥ n-1) and Greater-Than (when d=1). We establish a nearly tight bound of tilde Theta(dlog n) on the communication complexity of learning halfspaces in ℝd. For Convex Set Disjointness (and the equivalent task of distributed LP feasibility) we derive upper and lower bounds of tilde O(d2log n) and~Omega(dlog n). These results improve upon several previous works in distributed learning and optimization. Unlike typical works in communication complexity, the main technical contribution of this work lies in the upper bounds. In particular, our protocols are based on a it Container Lemma for Halfspaces and on two variants of it Carath'eodory's Theorem, which may be of independent interest. These geometric statements are used by our protocols to provide a compressed summary of the players' input.