An invariance principle for polytopes

Abstract

Let X be randomly chosen from {-1,1}ⁿ, and let γ be randomly chosen from the standard spherical Gaussian on ℝⁿ. For any (possibly unbounded) polytope P formed by the intersection of k halfspaces, we prove that |Pr[X ∈ P] - Pr[Y ∈ P]| ≤ log^8/5k ⋅ Δ, where Δ is a parameter that is small for polytopes formed by the intersection of “regular” halfspaces (i.e., halfspaces with low influence). The novelty of our invariance principle is the polylogarithmic dependence on k. Previously, only bounds that were at least linear in k were known. The proof of the invariance principle is based on a generalization of the Lindeberg method for proving central limit theorems and could be of use elsewhere. We give two important applications of our invariance principle, one from learning theory and the other from pseudorandomness. (1) A bound of log^O(1)k ⋅ ε^1/6 on the Boolean noise sensitivity of intersections of k “regular” halfspaces (previous work gave bounds linear in k). This gives a corresponding agnostic learning algorithm for intersections of regular halfspaces. (2) A pseudorandom generator (PRG) for estimating the Gaussian volume of polytopes with k faces within error δ and seed-length O(log n poly(log k,1/δ)). We also obtain PRGs with similar parameters that fool polytopes formed by intersection of regular halfspaces over the hypercube. Using our PRG constructions, we obtain the first deterministic quasi-polynomial time algorithms for approximately counting the number of solutions to a broad class of integer programs, including dense covering problems and contingency tables.