NDLI: An invariance principle for polytopes

Content Provider	ACM Digital Library
Author	Harsha, Prahladh Klivans, Adam Meka, Raghu
Copyright Year	2013
Abstract	Let $\textit{X}$ be randomly chosen from ${-1,1}^{n},$ and let $\textit{Y}$ be randomly chosen from the standard spherical Gaussian on $ℝ^{n}.$ For any (possibly unbounded) polytope $\textit{P}$ formed by the intersection of $\textit{k}$ halfspaces, we prove that $\|Pr[\textit{X}$ ∈ $\textit{P}]$ - $Pr[\textit{Y}$ ∈ $\textit{P}]\|$ ≤ $log^{8/5}k$ ṡ 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 $\textit{k}.$ Previously, only bounds that were at least linear in $\textit{k}$ 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 $log^{O(1)}k$ ṡ $ε^{1/6}$ on the Boolean noise sensitivity of intersections of $\textit{k}$ “regular” halfspaces (previous work gave bounds linear in $\textit{k}).$ 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 $\textit{k}$ faces within error Δ and seed-length $\textit{O}(log$ $\textit{n}$ poly(log $\textit{k},1/Δ)).$ 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.
Starting Page	1
Ending Page	25
Page Count	25
File Format	PDF
ISSN	00045411
e-ISSN	1557735X
DOI	10.1145/2395116.2395118
Journal	Journal of the ACM (JACM)
Volume Number	59
Issue Number	6
Language	English
Publisher	Association for Computing Machinery (ACM)
Publisher Date	2013-01-09
Publisher Place	New York
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Invariance Agnostic learning Intersection of halfspaces Noise sensitivity Polytopes Pseudorandomness
Content Type	Text
Resource Type	Article
Subject	Hardware and Architecture Information Systems Control and Systems Engineering Artificial Intelligence Software

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1	Ministry of Education (GoI), Department of Higher Education	Sanctioning Authority	https://www.education.gov.in/ict-initiatives
2	Indian Institute of Technology Kharagpur	Host Institute of the Project: The host institute of the project is responsible for providing infrastructure support and hosting the project	https://www.iitkgp.ac.in
3	National Digital Library of India Office, Indian Institute of Technology Kharagpur	The administrative and infrastructural headquarters of the project	Dr. B. Sutradhar bsutra@ndl.gov.in
4	Project PI / Joint PI	Principal Investigator and Joint Principal Investigators of the project	Dr. B. Sutradhar bsutra@ndl.gov.in Prof. Saswat Chakrabarti will be added soon
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