Abstract
We study correlation bounds and pseudorandom generators for depthtwo circuits that consist of a SYMgate (computing an arbitrary symmetric function) or THRgate (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 nontrivial PRG with seed length 2^{O(sqrt{log(S/epsilon)})} that epsilonfools 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 epsilonfools sizeS {SYM,THR} oAND circuits over {0,1}^n with seed length 2^{O(sqrt{log S})} + polylog(1/epsilon), an exponential (and nearoptimal) improvement of the epsilondependence of [Michael Luby et al., 1993]. The above PRG is actually a special case of a more general PRG which we establish for constantdepth 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 NisanWigderson "hardness versus randomness" paradigm [Nisan and Wigderson, 1994]. The key to our improved correlation bounds is the use of a recent powerful multiswitching lemma due to Håstad [Johan Håstad, 2014].
BibTeX  Entry
@InProceedings{servedio_et_al:LIPIcs:2018:9460,
author = {Rocco A. Servedio and LiYang Tan},
title = {{LubyVelickovicWigderson Revisited: Improved Correlation Bounds and Pseudorandom Generators for DepthTwo Circuits}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)},
pages = {56:156:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770859},
ISSN = {18688969},
year = {2018},
volume = {116},
editor = {Eric Blais and Klaus Jansen and Jos{\'e} D. P. Rolim and David Steurer},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2018/9460},
URN = {urn:nbn:de:0030drops94601},
doi = {10.4230/LIPIcs.APPROXRANDOM.2018.56},
annote = {Keywords: Pseudorandom generators, correlation bounds, constantdepth circuits}
}
Keywords: 

Pseudorandom generators, correlation bounds, constantdepth circuits 
Seminar: 

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018) 
Issue Date: 

2018 
Date of publication: 

02.08.2018 