Luby--Veli\v{c}kovi\'c--Wigderson revisited: Improved correlation bounds and pseudorandom generators for depth-two circuits

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 √ log(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 √ logS) + 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 circuits. These more general results strengthen previous results of Viola [Vio07] 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 H̊astad [H̊as14]. Supported by NSF grants CCF-1420349 and CCF-1563155. [email protected] Supported by NSF grant CCF-1563122. Part of this research was done during a visit to Columbia University. [email protected]