An invariance principle for polytopes

  • Authors:

  • Prahladh Harsha;Adam Klivans;Raghu Meka

  • Affiliations:

  • Tata Institute of Fundamanetal Researach, Mumbai, India;University of Texas, Austin, TX, USA;University of Texas, Austin, TX, USA

  • Venue:
  • Proceedings of the forty-second ACM symposium on Theory of computing
  • Year:
  • 2010

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Abstract

Let X be randomly chosen from {-1,1}n, and let Y 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]| ≤ log8/5k • 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. We give two important applications of our main result: A bound of logO(1)k • ε1/6 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. A pseudorandom generator (PRG) with seed length O(log n, poly(log k,1/Δ)) that Δ-fools all polytopes with k faces with respect to the Gaussian distribution. 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.