Improved Pseudorandom Generators for $\mathsf{AC}^0$ Circuits

TL;DR

This paper presents a new pseudorandom generator for small-depth circuits with improved seed length, leveraging novel restriction lemmas and derandomization techniques, advancing the efficiency of derandomization in computational complexity.

Contribution

It introduces a PRG with near-optimal seed length for $ ext{AC}^0$ circuits, using a partitioning approach and derandomized multi-switching lemma, improving previous bounds.

Findings

Seed length is optimal up to a $ ext{log} ext{log}(m)$ factor.

Tightly matches the best PRG for CNFs when $d=2$.

Introduces new restriction lemmas and derandomization methods.

Abstract

We show a new PRG construction fooling depth-$d$, size-$m$ $\mathsf{AC}^0$ circuits within error $\varepsilon$, which has seed length $O(\log^{d-1}(m)\log(m/\varepsilon)\log\log(m))$. Our PRG improves on previous work (Trevisan and Xue 2013, Servedio and Tan 2019, Kelley 2021) from various aspects. It has optimal dependence on $\frac{1}{\varepsilon}$ and is only one ``$\log\log(m)$'' away from the lower bound barrier. For the case of $d=2$, the seed length tightly matches the best-known PRG for CNFs (De et al. 2010, Tal 2017). There are two technical ingredients behind our new result; both of them might be of independent interest. First, we use a partitioning-based approach to construct PRGs based on restriction lemmas for $\mathsf{AC}^0$, which follows and extends the seminal work of (Ajtai and Wigderson 1989). Second, improving and extending prior works (Trevisan and Xue 2013, Servedio and Tan 2019, Kelley 2021), we prove a full derandomization of the powerful multi-switching lemma for a family of DNFs (H{\aa}stad 2014).