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Luby-Velickovic-Wigderson Revisited: Improved Correlation Bounds and Pseudorandom Generators for Depth-Two Circuits

Authors Rocco A. Servedio, Li-Yang Tan

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Author Details

Rocco A. Servedio
  • Department of Computer Science, Columbia University, New York, NY, USA
Li-Yang Tan
  • Department of Computer Science, Stanford University, Stanford, California, USA

Funding

  • Servedio, Rocco A.: Supported by NSF grants CCF-1420349 and CCF-1563155.
  • Tan, Li-Yang: Supported by NSF grant CCF-1563122. Part of this research was done during a visit to Columbia University.

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Rocco A. Servedio and Li-Yang Tan. Luby-Velickovic-Wigderson Revisited: Improved Correlation Bounds and Pseudorandom Generators for Depth-Two Circuits. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 56:1-56:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2018.56

Abstract

We study correlation bounds and pseudorandom generators for depth-two circuits that consist of a SYM-gate (computing an arbitrary symmetric function) or THR-gate (computing an arbitrary linear threshold function) that is fed by S {AND} gates. Such circuits were considered in early influential work on unconditional derandomization of Luby, Velickovi{c}, and Wigderson [Michael Luby et al., 1993], who gave the first non-trivial PRG with seed length 2^{O(sqrt{log(S/epsilon)})} that epsilon-fools these circuits. In this work we obtain the first strict improvement of [Michael Luby et al., 1993]'s seed length: we construct a PRG that epsilon-fools size-S {SYM,THR} oAND circuits over {0,1}^n with seed length 2^{O(sqrt{log S})} + polylog(1/epsilon), an exponential (and near-optimal) improvement of the epsilon-dependence of [Michael Luby et al., 1993]. The above PRG is actually a special case of a more general PRG which we establish for constant-depth circuits containing multiple SYM or THR gates, including as a special case {SYM,THR} o AC^0 circuits. These more general results strengthen previous results of Viola [Viola, 2006] and essentially strengthen more recent results of Lovett and Srinivasan [Lovett and Srinivasan, 2011]. Our improved PRGs follow from improved correlation bounds, which are transformed into PRGs via the Nisan-Wigderson "hardness versus randomness" paradigm [Nisan and Wigderson, 1994]. The key to our improved correlation bounds is the use of a recent powerful multi-switching lemma due to Håstad [Johan Håstad, 2014].

Subject Classification

ACM Subject Classification
  • Theory of computation → Pseudorandomness and derandomization
Keywords
  • Pseudorandom generators
  • correlation bounds
  • constant-depth circuits

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