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DOI: 10.4230/LIPIcs.CCC.2016.1
URN: urn:nbn:de:0030-drops-58447
URL: http://drops.dagstuhl.de/opus/volltexte/2016/5844/
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Chen, Ruiwen ; Santhanam, Rahul ; Srinivasan, Srikanth

Average-Case Lower Bounds and Satisfiability Algorithms for Small Threshold Circuits

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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 epsilon_d > 0 such that Parity has correlation at most 1/n^{Omega(1)} with depth-d threshold circuits which have at most n^{1+epsilon_d} wires, and the Generalized Andreev Function has correlation at most 1/2^{n^{Omega(1)}} with depth-d threshold circuits which have at most n^{1+epsilon_d} wires. Previously, only worst-case lower bounds in this setting were known [Impagliazzo/Paturi/Saks, SIAM J. Comp., 1997]. 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 AC^0 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 AC^0 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 ofany depth. Our techniques include adaptive random restrictions, anti-concentration and the structural theory of linear threshold functions, and bounded-read Chernoff bounds.

BibTeX - Entry

@InProceedings{chen_et_al:LIPIcs:2016:5844,
  author =	{Ruiwen Chen and Rahul Santhanam and Srikanth Srinivasan},
  title =	{{Average-Case Lower Bounds and Satisfiability Algorithms for Small Threshold Circuits}},
  booktitle =	{31st Conference on Computational Complexity (CCC 2016)},
  pages =	{1:1--1:35},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-008-8},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{50},
  editor =	{Ran Raz},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2016/5844},
  URN =		{urn:nbn:de:0030-drops-58447},
  doi =		{10.4230/LIPIcs.CCC.2016.1},
  annote =	{Keywords: threshold circuit, satisfiability algorithm, circuit lower bound}
}

Keywords: threshold circuit, satisfiability algorithm, circuit lower bound
Seminar: 31st Conference on Computational Complexity (CCC 2016)
Issue Date: 2016
Date of publication: 18.05.2016


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