$\require{mathtools}
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% Greek
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\let\fi\phi % because it looks like an f
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% .. over and under
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% .. relations
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% .. miscer
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% (I needed to create my own because the MathJax version of \DeclarePairedDelimiter doesn't have \mathopen{} and that messes up the spacing)
% .. one-part
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% .... (use phantom to force at least the standard height of double bars)
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% .. operators
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% .. analysis
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% Complexity classes
% .. classical
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% .. probabilistic
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% .. circuits
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% .. resources
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Say we want to make pseudorandom generators for $\AC^0$. How big does the seed length $r$ need to be as a function of the size $S$ and the depth $d$?
Dream: match random sample
The gold standard would be to match the performance of a randomly chosen sample of $2^r$ strings from $\zo^n$, which requires
\[
r = \Theta(\log \log |\CC| + \log(1/\eps)).
\]
for a complexity class $\CC$.
What’s $|\CC|$ for $\AC^0$ of size $S$ and depth $d$? Well, depth doesn’t really matter here: basically all you need to know is which wire goes from where to where, which gives $2^{\poly(S)}$ possibilities. So a random $g$ would require seed length
\[r = \Theta(\log \log 2^{\poly(S)} + \log(1/\eps)) = \Theta(\log (S/\eps)).\]
This is in fact tight! Indeed, for any sample of $2^r$ points, we can adversarially build a CNF of with $2\eps \cdot 2^r$ clauses that are unsatisfied on $2\eps \cdot 2^r$ of those points, but is otherwise always satisfied. Such a CNF would have sample mean $1-2\eps$ but true mean $\approx 1$, which means it isn’t $\eps$-fooled. So we need to make sure that
\[2\eps \cdot 2^r > S \Rightarrow r \geq \Omega(\log(S/\eps)).\]
What to hope for, barring breakthrough
On the other hand, it turns out that PRGs imply lower bounds for the corresponding circuit class. And it seems very unlikely that we’d prove a new circuit lower bound by crafting a PRG for it: PRGs are pretty complicated to construct, and historically, PRGs have always been found later than the circuit lower bound they imply.
For now (and since the ’80s), lower bounds against $\AC^0$ are stuck at $S = 2^{O(n^{1/(d-1)})}$. So we should probably not try to go for PRGs that would imply stronger lower bounds on size.
Fooling the class
Why do PRGs imply circuit lower bounds? Well, the image of a PRG is supposed to look random to every function in the class, so in particular it cannot be in the class. Formally, let $g:\zo^r \to \zo^n$ be a PRG $\eps$-fooling a class $\CC$ for some $\eps < 1/2$, and consider the “image indicator” of $g$:
\[\Img g(x) \ce \One[\exists s \in \zo^r, g(s)=x].\]
Suppose $\Img g$ is in $\CC$. By construction, the “sample mean” of $\Img g$ is $1$, so its “true mean” should be at least $1-\eps$. But it returns $1$ for only $2^r \le 2^n/2$ points, so that’s impossible.
So for depth-$d$ $\AC^0$, this means we shouldn’t expect to find a PRG with $r = o((\log S)^{d-1})$ for constant $\eps$. Indeed, otherwise we could set $S \ce 2^{\omega(n^{1/(d-1)})}$ and still get $r < n$, which means $\Img g$ wouldn’t have size $2^{O(n^{1/(d-1)})}$ circuits.
Fooling ORs of the class
Actually, you can even say even more if $\eps$ is small enough. Indeed, if you squint at it, the above argument shows not only that is $\Img g$ not in $\CC$, but also that no function in $\CC$ can cover a bigger than $\approx\eps$ fraction of it without also satisfying points outside it: indeed, the image of $g$ is a tiny fraction of $\zo^n$, so a function that’s completely contained in it would have “true mean” basically $0$.
Now, say you want to represent $\Img g$ with an OR $f_1 \or \cdots \or f_m$ of functions in $\CC$. The functions $f_1, \ldots, f_m$ must all be be “contained within” $\Img g$, so none of them can cover more than an $\eps$ fraction of it. But they must collectively cover all of $\Img g$! So you would need $m \geq 1/\eps$ of them. In other words, a PRG that $\eps$-fools a class $\CC$ gives a $1/\eps$ size lower bound against “OR-$\CC$”.
When we set $\CC$ to be $\AC^0$ circuits of depth $d$ whose top gate is an AND, then “OR-$\CC$” are depth-$(d+1)$ circuits: that means a PRG with low error gives lower bounds for higher depth! And the lower bounds we have for depth $d+1$ are even worse than for depth $d$, so this limits our expectations even more.
Concretely, depth-$(d+1)$ circuits of size $S$ (with an OR at the top) are ORs of $\leq S$ depth-$d$ circuits of size $\leq S$, so it’s enough to set $\eps \ce 1/S$. This means we shouldn’t expect to find a PRG for depth-$d$ $\AC^0$ with $r = o((\log S)^{d-1}\log(1/\eps))$. Indeed, setting $\eps \ce 1/S$ and $S \ce 2^{\omega(n^{1/d})}$ would give $r<n$, so it would give $2^{\omega(n^{1/d})}$ lower bounds against depth-$(d+1)$, which is beyond what is currently known.
To summarize, the breakthrough-free dream seed-length is
\[r = O((\log S)^{d-1}\log(1/\eps)).\]
Not quite there
All PRGs for $\AC^0$ I know of are based on various derandomizations of Switching lemma and his multi-switching lemma. As far as I can tell, the state of the art for now is
\[r = O((\log S)^{d-1}\log(S/\eps)\log \log S).\]
This is pretty good when the error is small: for $\eps\leq 1/\poly(S)$, this is $O((\log S)^{d-1} \log(1/\eps) \log \log S)$, only one $\log \log S$ factor from the “right dependence” $O((\log S)^{d-1} \log(1/\eps))$. However, for constant $\eps$, it’s one full $\log S$ factor off.
The special case of DNFs (depth $d=2$) and constant $\eps$ is especially interesting. The state of the art is $r = O((\log S)^2\log \log S)$, and has been this way since 2010. It would be glorious to improve this to the optimal $O(\log S)$.
TODO: describe what’s known for related notions like hitting sets