Luby-Veličković-Wigderson再访：深度2电路的改进相关边界和伪随机发生器
Luby--Veličković--Wigderson revisited: Improved correlation bounds and pseudorandom generators for depth-two circuits
摘要：
我们研究深度为2的电路的相关界限和伪随机发生器，该电路由$\mathsf{SYM}$门（计算任意对称函数）或$\mathsf{THR}$门（计算任意线性阈值函数）组成，该门由$S$ $\mathsf{AND}$门提供。这种电路在Luby、Veličković和Wigderson [LVW 93]的早期有影响力的无条件去随机化工作中被考虑，他们给出了第一个种子长度为$2^{O(\sqrt{\log(S/\varepsilon)})}$的非平凡PRG，$\varepsilon$欺骗了这些电路。在这项工作中，我们得到了[LVW 93]的种子长度的第一个严格的改进：我们构建了一个PRG，$\varepsilon$ -傻瓜大小$S$ $\{\mathsf{SYM},\mathsf{THR}\} \circ\mathsf{AND}$电路在$\{0,1\}^n$与种子长度\[ 2^{O(\sqrt{\log S })} + \mathrm{polylog}(1/\varepsilon), \]的指数（和接近最佳）改进的$\varepsilon$ -依赖[LVW 93]。上面的PRG实际上是更一般的PRG的特殊情况，我们为包含多个$\mathsf{SYM}$或$\mathsf{THR}$门的恒定深度电路建立了PRG，包括作为特殊情况的$\{\mathsf{SYM},\mathsf{THR}\} \circ \mathsf{AC^0}$电路。这些更一般的结果加强了Viola [Vio 06]以前的结果，并基本上加强了Lovett和Srinivasan [LS 11]最近的结果。我们改进的PRG遵循改进的相关边界，通过Nisan-Wigderson“硬度与随机性”范式[NW 94]将其转换为PRG。我们改进相关边界的关键是使用了最近Håstad [Hås 14]提出的一个强大的\emph{multi-switching}引理。
英文摘要：
We study correlation bounds and pseudorandom generators for depth-two circuits that consist of a $\mathsf{SYM}$-gate (computing an arbitrary symmetric function) or $\mathsf{THR}$-gate (computing an arbitrary linear threshold function) that is fed by $S$ $\mathsf{AND}$ 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 $2^{O(\sqrt{\log(S/\varepsilon)})}$ that $\varepsilon$-fools these circuits. In this work we obtain the first strict improvement of [LVW93]'s seed length: we construct a PRG that $\varepsilon$-fools size-$S$ $\{\mathsf{SYM},\mathsf{THR}\} \circ\mathsf{AND}$ circuits over $\{0,1\}^n$ with seed length \[ 2^{O(\sqrt{\log S })} + \mathrm{polylog}(1/\varepsilon), \] an exponential (and near-optimal) improvement of the $\varepsilon$-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 $\mathsf{SYM}$ or $\mathsf{THR}$ gates, including as a special case $\{\mathsf{SYM},\mathsf{THR}\} \circ \mathsf{AC^0}$ 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 \emph{multi-switching} lemma due to Håstad [Hås14].
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