We wanted to jot down a clarification in response to both your comments. In brief, we believe that an average-case depth hierarchy theorem rules out the possibility of a converse to Hastad-Boppana-LMN when viewed as a statement about the total influence of *constant*-depth circuits. However, while the Inf(f) <= (O(\log(S)))^{d-1} bound is often applied in the setting where d is constant, it in fact holds for all values of d. It would interesting to explore the implications of our result in regimes where d is allowed to be super-constant (indeed, we are discussing this with Gil now), but for this comment let me focus on the constant d regime (which is the regime the paper addresses). Let me elaborate…

Fix \eps = 0.01. Conjecture 1 in our paper says that there exist universal constants d, K_1, K_2 such that every monotone f is 0.99-approximated by a depth-d circuit of size \exp((K_1 * \Inf(f))^{K_2}). Note that this conjecture captures Hamed's question (Problem 4.6.3 of http://cs.mcgill.ca/~hatami/comp760-2014/lectures.pdf) for functions of influence <= \log(n). (A major thrust of the O'Donnell-Wimmer paper was to rule out the possibility of having d=2 and K_2 = 1 in Conjecture 1.)

We disprove Conjecture 1, and actually disprove the following even weaker conjecture:

Conjecture 1': For every family {f_n} of n-variable monotone functions there exist *constants* d, K_1, and K_2 (all depending on the family but not n) such that f_n is 0.99-approximated by a depth-d circuit of size \exp((K_1 * \Inf(f))^{K_2}).

Conjecture 1' captures the possibilities that are raised in (4) and (5) of http://cstheory.stackexchange.com/questions/12769/are-all-the-functions-whose-fourier-weight-is-concentrated-on-the-small-sized-se/14926#14926.

Our counterexample to Conjecture 1' is the family of Sipser functions {S_n} of depth say (\log(n))^{0.49}. Our main result implies that for all fixed *constants* d, there is no depth-d circuit of size \exp(\poly(\Inf(f))) that even 51%-approximates S_n.

In fact, our main result implies that the same statement holds for values of d up to (\log(n))^{0.49} – 1. It does not imply the statement for d = (\log(n))^{0.49} (note that S_n is itself a depth (\log(n))^{0.49} formula of linear size!) Therefore, our result does not rule out the possibility of a converse to Hastad-Boppana-LMN as stated in Gil's post above where d can be arbitrary (and need not be constant). Specifically, it is still possible that there exists a function \alpha such that the following holds: for every family {f_n} of monotone functions there exists a function d(n) and a constant K such that f_n is 0.99-approximated by a depth-d(n) circuit of size \exp((K * \Inf(f))^{1/\alpha(d(n)-1)}). (The [OW] result shows that this does not hold with \alpha being the identity function.)

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Hopefully what is written above is clear, but if not the following is one way to view our result within the context of the original BKS conjecture, O'Donnell-Wimmer's counterexample to the original BKS conjecture, and Hamed and Gil's subsequent questions. Focusing on the case of \log(n) influence functions,

– The original BKS conjecture allowed for the possibility that every \log(n) influence function is 0.99-approximated by a \poly(n)-size depth-2 circuit.

– O'Donnell-Wimmer gave a counterexample to this in their paper. Specifically, they proved the existence of a \log(n) influence function such that any 0.51-approximating depth-2 circuit must have size 2^{\Omega(n/\log(n))}.

– However, [OW]'s hard function is a linear-size depth-3 circuit. Therefore, it is natural to wonder (as Hamed did in his lecture notes and Gil did in the CS theory StackExchange question) whether every \log(n) influence function is 0.99-approximated by a function in AC0.

– We show that even this is not possible: there are \log(n) influence functions that cannot be 0.51-approximated by constant-depth circuits of quasi-polynomial size (indeed, even by \sqrt{\log\log(n)}-depth circuits of size \exp(\exp(\tilde{\Omega}(2^{\sqrt{\log\log(n)}})})).)

Thanks again, Gil and Boaz, for your insightful comments and for pointing out these subtleties! We will be revising this section of the paper to clarify these issues.

]]>Maybe (like in the case of recursive majority) the influence is smaller than BB. But this is needed to be shown. I suppose that the influence can be directly computed without referring to Boppana’s bound.

]]>I don’t know if a direct computation of influence will give another value ]]>

There is still something which I am confused about. I thought that moving to a small number of variables is just to make the total influence polylog (n) so it will address one version of our conjecture perhaps by Ryan. But this is not needed to the way we think about the conjecture. The way I think about our conjecture if your function depends on a small number of variables just ignore the other variables. In our thinking and formulation the number of variables is irrelevant.

Now, when you consider the Sipser function on m = exp exp sqrt log log n variables and the depth that RST uses (what is the depth precisely?) my impression is that a direct calculation of influence gives you something which is larger than m^{K depth} for every constant K. (If the influence is m^depth or so it is *not* a counter example to our conjecture – for our formulation you can just regard n to be m and there is no need to force the influence to be polylog (n) ).

In any case, I think that the Sipser function on n variables with the correct parameters has influence which is substantially smaller than the log size ^ depth, and that the main difficulty is to show that you cannot approximate it by another circuit where (log s)^K depth explains the low influence witnessed.

]]>However, perhaps recursive majority can give a different example.

]]>The influence is n^beta for some power. log size ^ depth gives n^log n so it is not relevant. Our conjecture in a very weak form would say that recursive majority on threes can be approximated by size s depth d circuit where (log s)^d is poly (n) and if we believe that RST’s result extends to this example this would be false. I am not so happy with the term “a new reason” because the influence computation is easy here and so is, I suppose, for the Sipser function. It is that we do not have a unified reason for low influence in terms of circuit size and depth. It would be nice to state a general reason which explain why Sipser’s function’s influence or recursive majority are what they are. (So maybe Instead of writing “RST show a genuinely new reason for small influence!” I should say “RST shows that we do not know or even have a guess for a general reason for small influence!” ]]>

Does RST’s ‘Sipser function” have low influence for a truly new reason? After all, I thought that the reason it has low influence is because it has a small depth circuit, no?

BTW I gave a guest lecture on this result in Dana Moshkovitz’s class – the lecture notes (which are likely full of mistakes) can be found here http://www.boazbarak.org/Courses/avg_case_depth.pdf

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