Deterministic Approximate Counting for Juntas of Degree-2 Polynomial Threshold Functions

Anindya De, Ilias Diakonikolas, Rocco A Servedio

Research output: Chapter in Book/Report/Conference proceedingConference contribution


Let g: -1, 1^k to -1, 1 be any Boolean function and q_1, dots, q_k be any degree-2 polynomials over -1, 1^n. We give a deterministic algorithm which, given as input explicit descriptions of g, q_1, dots, q_k and an accuracy parameter eps>0, approximates [ Pr_x sim -1, 1^n[g(sign(q_ 1(x)), dots, sign(q_k(x)))=1] ] to within an additive pm eps. For any constant eps > 0 and k geq 1 the running time of our algorithm is a fixed polynomial in n (in fact this is true even for some not-too-small eps = o_n(1) and not-too-large k = omega_n(1)). This is the first fixed polynomial-time algorithm that can deterministically approximately count satisfying assignments of a natural class of depth-3 Boolean circuits. Our algorithm extends a recent result DDS13:deg2count which gave a deterministic approximate counting algorithm for a single degree-2 polynomial threshold function sign(q(x)), corresponding to the k=1 case of our result. Note that even in the k=1 case it is NP-hard to determine whether Pr_x sim -1, 1^n[sign(q(x))=1] is nonzero, so any sort of multiplicative approximation is almost certainly impossible even for efficient randomized algorithms. Our algorithm and analysis requires several novel technical ingredients that go significantly beyond the tools required to handle the k=1 case in citeDDS13:deg2count. One of these is a new multidimensional central limit theorem for degree-2 polynomials in Gaussian random variables which builds on recent Malliavin-calculus-based results from probability theory. We use this CLT as the basis of a new decomposition technique for k-tuples of degree-2 Gaussian polynomials and thus obtain an efficient deterministic approximate counting algorithm for the Gaussian distribution, i.e., an algorithm for estimating [ Pr_x sim N(0, 1)^n[g(sign(q_1(x)), dots, sign(q_k(x)))=1]. ] Finally, a third new ingredient is a "regularity lemma" for k-tuples of degree-d polynomial threshold functions. This ge- eralizes both the regularity lemmas of DSTW:10, HKM:09 (which apply to a single degree-d polynomial threshold function) and the regularity lemma of Gopalan et al GOWZ10 (which applies to a k-tuples of linear threshold functions, i.e., the case d=1). Our new regularity lemma lets us extend our deterministic approximate counting results from the Gaussian to the Boolean domain.
Original languageEnglish
Title of host publicationComputational Complexity (CCC), 2014 IEEE 29th Conference on
PublisherInstitute of Electrical and Electronics Engineers (IEEE)
Number of pages12
Publication statusPublished - Jun 2014


  • Approximation algorithms
  • Approximation methods
  • Calculus
  • Covariance matrices
  • Eigenvalues and eigenfunctions
  • Polynomials
  • Random variables
  • Approximate counting
  • derandomization
  • polynomial threshold function


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