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].
@InProceedings{servedio_et_al:LIPIcs.APPROX-RANDOM.2018.56, author = {Servedio, Rocco A. and Tan, Li-Yang}, title = {{Luby-Velickovic-Wigderson Revisited: Improved Correlation Bounds and Pseudorandom Generators for Depth-Two Circuits}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)}, pages = {56:1--56:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-085-9}, ISSN = {1868-8969}, year = {2018}, volume = {116}, editor = {Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.56}, URN = {urn:nbn:de:0030-drops-94601}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2018.56}, annote = {Keywords: Pseudorandom generators, correlation bounds, constant-depth circuits} }
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