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

Authors: Ruiwen Chen, Rahul Santhanam, and Srikanth Srinivasan

Published in: LIPIcs, Volume 50, 31st Conference on Computational Complexity (CCC 2016)


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.

Cite as

Ruiwen Chen, Rahul Santhanam, and Srikanth Srinivasan. Average-Case Lower Bounds and Satisfiability Algorithms for Small Threshold Circuits. In 31st Conference on Computational Complexity (CCC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 50, pp. 1:1-1:35, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)


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@InProceedings{chen_et_al:LIPIcs.CCC.2016.1,
  author =	{Chen, Ruiwen and Santhanam, Rahul and Srinivasan, Srikanth},
  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 =	{Raz, Ran},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2016.1},
  URN =		{urn:nbn:de:0030-drops-58447},
  doi =		{10.4230/LIPIcs.CCC.2016.1},
  annote =	{Keywords: threshold circuit, satisfiability algorithm, circuit lower bound}
}
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