Shame on me! I somehow managed to miss the ``recent'' developments in this area. (This paper has appeared in the 37th CCC, 2022.)
The issue at hand is the seed length used by (unconditional) PRGs that fool AC0 circuits. Specifically, for depth $d$ and size $m$, the current paper gives a seed-length that is $\tildeO(\log m)^d$ (or rather $O((\log m)^d\cdot\log\log m)$).
Actually, the result is better for very small deviation parameter $\eps$ (i.e., $\eps \ll 1/poly(m)$). In that the seed length is $O((\log m)^{d-1}\cdot\log(m/\eps)\cdot\log\log m)$).
As stated in the introducttion, whereas Trevisan and Xue [TX13] achieved seed length $\log^{d+O(1)}(m/\eps)$, more recent (incomparable) improvements achieved seed lengths $\log^{d+O(1)}m\cdot\log(1/\eps)$ (Servedio and Tan [ST19]) and $\log^{d}(m/\eps)\log n$ (where $n$ is the number of inputs, Kelley [Kel21]).
We show a new PRG construction fooling depth-, size- AC circuits within error , which has seed length . Our PRG improves on previous work (Trevisan and Xue 2013, Servedio and Tan 2019, Kelley 2021) from various aspects. It has optimal dependence on and is only one ``'' away from the lower bound barrier. For the case of , 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 AC, 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?stad 2014).
See arXiv 2301.10102.