research-article

Average-case lower bounds and satisfiability algorithms for small threshold circuits

Authors:

Rahul Santhanam,

Srikanth SrinivasanAuthors Info & Claims

CCC '16: Proceedings of the 31st Conference on Computational Complexity

Article No.: 1, Pages 1 - 35

Published: 29 May 2016 Publication History

Abstract

We show average-case lower bounds for explicit Boolean functions against bounded-depth threshold circuits with a superlinear number of wires. We show that for each integer d > 1, there is ε_d > 0 such that Parity has correlation at most 1/n^Ω(1) with depth-d threshold circuits which have at most n^1+ε_d wires, and the Generalized Andreev Function has correlation at most 1/2^nΩ(1) with depth-d threshold circuits which have at most n^1+ε_d wires. Previously, only worst-case lower bounds in this setting were known [22].

We use our ideas to make progress on several related questions. We give satisfiability algorithms beating brute force search for depth-d threshold circuits with a superlinear number of wires. These are the first such algorithms for depth greater than 2. We also show that Parity cannot be computed by polynomial-size AC⁰ circuits with n^o(1) general threshold gates. Previously no lower bound for Parity in this setting could handle more than log(n) gates. This result also implies subexponential-time learning algorithms for AC⁰ with n^o(1) threshold gates under the uniform distribution. In addition, we give almost optimal bounds for the number of gates in a depth-d threshold circuit computing Parity on average, and show average-case lower bounds for threshold formulas of any depth.

Our techniques include adaptive random restrictions, anti-concentration and the structural theory of linear threshold functions, and bounded-read Chernoff bounds.

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Cited By

Chen LTell RCharikar MCohen E(2019)Bootstrapping results for threshold circuits “just beyond” known lower boundsProceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing10.1145/3313276.3316333(34-41)Online publication date: 23-Jun-2019
https://dl.acm.org/doi/10.1145/3313276.3316333
Williams RServedio R(2018)Limits on representing boolean functions by linear combinations of simple functionsProceedings of the 33rd Computational Complexity Conference10.5555/3235586.3235592(1-24)Online publication date: 22-Jun-2018
https://dl.acm.org/doi/10.5555/3235586.3235592
Chapman BCzumaj A(2018)The gotsman-linial conjecture is falseProceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms10.5555/3174304.3175315(692-699)Online publication date: 7-Jan-2018
https://dl.acm.org/doi/10.5555/3174304.3175315
Show More Cited By

Index Terms

Average-case lower bounds and satisfiability algorithms for small threshold circuits
1. Theory of computation
  1. Computational complexity and cryptography
    1. Complexity classes

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Published In

cover image Guide Proceedings

CCC '16: Proceedings of the 31st Conference on Computational Complexity

May 2016

843 pages

ISBN:9783959770088

Editor:
Ran Raz
Weizmann Institute of Science, Rehovot, Israel

Sponsors

Microsoft Reasearch: Microsoft Reasearch

Publisher

Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik

Dagstuhl, Germany

Publication History

Published: 29 May 2016

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Cited By

Chen LTell RCharikar MCohen E(2019)Bootstrapping results for threshold circuits “just beyond” known lower boundsProceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing10.1145/3313276.3316333(34-41)Online publication date: 23-Jun-2019
https://dl.acm.org/doi/10.1145/3313276.3316333
Williams RServedio R(2018)Limits on representing boolean functions by linear combinations of simple functionsProceedings of the 33rd Computational Complexity Conference10.5555/3235586.3235592(1-24)Online publication date: 22-Jun-2018
https://dl.acm.org/doi/10.5555/3235586.3235592
Chapman BCzumaj A(2018)The gotsman-linial conjecture is falseProceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms10.5555/3174304.3175315(692-699)Online publication date: 7-Jan-2018
https://dl.acm.org/doi/10.5555/3174304.3175315
Tell RDiakonikolas IKempe DHenzinger M(2018)Quantified derandomization of linear threshold circuitsProceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing10.1145/3188745.3188822(855-865)Online publication date: 20-Jun-2018
https://dl.acm.org/doi/10.1145/3188745.3188822
Kabanets VKane DLu ZHatami HMcKenzie PKing V(2017)A polynomial restriction lemma with applicationsProceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing10.1145/3055399.3055470(615-628)Online publication date: 19-Jun-2017
https://dl.acm.org/doi/10.1145/3055399.3055470

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