NDLI: Efficient deterministic approximate counting for low-degree polynomial threshold functions

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Efficient deterministic approximate counting for low-degree polynomial threshold functions

Content Provider	ACM Digital Library
Author	De, Anindya Servedio, Rocco A.
Abstract	We give a deterministic algorithm for approximately counting satisfying assignments of a degree-d polynomial threshold function (PTF). Given a degree-d input polynomial p(x) over $R^{n}$ and a parameter ε > 0, our algorithm approximates Pr [EQUATION] to within an additive ±ε in time $O_{d,ε}(1)$ · $poly(n^{d}).$ (Since it is NP-hard to determine whether the above probability is nonzero, any sort of efficient multiplicative approximation is almost certainly impossible even for randomized algorithms.) Note that the running time of our algorithm (as a function of $n^{d},$ the number of coefficients of a degree-d PTF) is a fixed polynomial. The fastest previous algorithm for this problem [Kan12b], based on constructions of unconditional pseudorandom generators for degree-d PTFs, runs in time [EQUATION] for all c > 0. The key novel technical contributions of this work are • A new multivariate central limit theorem, proved using tools from Malliavin calculus and Stein's Method. This new CLT shows that any collection of Gaussian polynomials with small eigenvalues must have a joint distribution which is very close to a multidimensional Gaussian distribution. • A new decomposition of low-degree multilinear polynomials over Gaussian inputs. Roughly speaking we show that (up to some small error) any such polynomial can be decomposed into a bounded number of multilinear polynomials all of which have extremely small eigenvalues. We use these new ingredients to give a deterministic algorithm for a Gaussian-space version of the approximate counting problem, and then employ standard techniques for working with low-degree PTFs (invariance principles and regularity lemmas) to reduce the original approximate counting problem over the Boolean hypercube to the Gaussian version. As an application of our result, we give the first deterministic fixed-parameter tractable algorithm for the following moment approximation problem: given a degree-d polynomial $p(x_{1},.,$ $x_{n})$ over {--1, $1\}^{n},$ a positive integer k and an error parameter ε, output a (1±ε)-multiplicatively accurate estimate to [EQUATION]. Our algorithm runs in time $O_{d,ε,k}(1)$ · $poly(n_{d}).$
Starting Page	832
Ending Page	841
Page Count	10
File Format	PDF MP4
ISBN	9781450327107
DOI	10.1145/2591796.2591800
Language	English
Publisher	Association for Computing Machinery (ACM)
Publisher Date	2014-05-31
Publisher Place	New York
Access Restriction	Subscribed
Subject Keyword	Polynomial threshold function Derandomization Approximate counting
Content Type	Audio Text
Resource Type	Article

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