Luby--Veličković--Wigderson revisited: Improved correlation bounds and pseudorandom generators for depth-two circuits
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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 [LVW93], who gave the first non-trivial PRG with seed length 2^O(√((S/ε))) that ε-fools these circuits. In this work we obtain the first strict improvement of [LVW93]'s seed length: we construct a PRG that ε-fools size-S {SYM,THR}∘AND circuits over {0,1}^n with seed length 2^O(√( S )) + polylog(1/ε), an exponential (and near-optimal) improvement of the ε-dependence of [LVW93]. 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 [Vio06] and essentially strengthen more recent results of Lovett and Srinivasan [LS11]. Our improved PRGs follow from improved correlation bounds, which are transformed into PRGs via the Nisan--Wigderson "hardness versus randomness" paradigm [NW94]. The key to our improved correlation bounds is the use of a recent powerful multi-switching lemma due to Håstad [Hås14].
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