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

Rocco A. Servedio, Li-Yang Tan, Michael Wagner
2018 International Workshop on Approximation Algorithms for Combinatorial Optimization  
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, Veličković, and Wigderson [31], who gave the first non-trivial PRG with seed length 2 O( √ log(S/ε)) that ε-fools these circuits. In this work we obtain the
more » ... t strict improvement of [31]'s seed length: we construct a PRG that ε-fools size-S {SYM, THR} • AND circuits over {0, 1} n with seed length an exponential (and near-optimal) improvement of the ε-dependence of [31] . 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} • AC 0 circuits. These more general results strengthen previous results of Viola [47] and essentially strengthen more recent results of Lovett and Srinivasan [30] . Our improved PRGs follow from improved correlation bounds, which are transformed into PRGs via the Nisan-Wigderson "hardness versus randomness" paradigm [37] . The key to our improved correlation bounds is the use of a recent powerful multi-switching lemma due to Håstad [21] .
doi:10.4230/lipics.approx-random.2018.56 dblp:conf/approx/ServedioT18 fatcat:knjpbzpf3zg2zpel4wrheeaqe4