How difficult is it to prove new lower bounds?

Two recent papers address the challenge of proving new correlation bounds for low-degree polynomials, which as depicted in a previous post are also necessary for a number of other long-standing problems, such as lower bounds for number-on-forehead communication protocols and depth-3 Majority circuits.

Nonclassical polynomials as a barrier to polynomial lower bounds, by Bhowmick and Lovett brings non-classical polynomials into the picture and shows that those polynomials of degree only log(n) are capable of various things that classical polynomials we know or conjecture are not. Consider for example the correlation between the mod 3 function on n boolean variables, and polynomials of degree d modulo 2. It has been natural to conjecture that this correlation is say super-polynomially small (less than 1/nc for every c) for degrees d up to d = n0.1, but despite substantial effort we cannot even show correlation at most 1/n for degree log(n). However, we can show exponentially small correlation bounds for degrees at most 0.1 log(n), and correlation bounds of 1/n0.1 for degrees up to n0.1, see this survey.

Bhowmick and Lovett construct a non-classical polynomial of degree O(log n) that achieves correlation 99% with the mod 3 function. What is the trick? First, suppose that my polynomial was defined modulo 1, i.e., over the torus, and that I was allowed to divide by 3. Then I could consider the polynomial p(x1, x2, …, xn) = (x1 + x2 + … + xn)/3, which has degree 1, and obtain maximum correlation 1. You can’t quite do that, but close. Non-classical polynomials are indeed defined over the torus, and allow division by powers of 2 of their integer coefficients. Basically, you can arrange things so that with polynomials of degree d you can divide by 3(1+1/2d). Setting d = O(log n) gets you close enough to a division by 3 that you obtain correlation 99%.

They also exhibit degree-O(log n) non-classical polynomials that correlate well with the majority function, and that weak-represent Or. In all cases, the polynomial is (x1 + x2 + … + xn) divided by a suitable number — the square root of n in the case of majority.

It is not clear to me how serious an obstacle this is, since as mentioned above and in their paper we can still prove vanishing correlation bounds for classical polynomials of degree O(log n), so we do have techniques that separate classical from non-classical polynomials in certain regimes. But it is refreshing to have this new viewpoint.

You can see their title and abstract following the link above. Mine would have been something like this: The power of non-classical polynomials. We show that non-classical polynomials of logarithmic degree are capable of several feats that we know or conjecture classical polynomials are not.

Anti-concentration for random polynomials by Nguyen and Vu proves that real polynomials (as opposed to polynomials modulo 2) have correlation zero (not small, but exactly zero) with the parity function, up to degree log(n)/loglog(n), improving on a degree bound of loglog(n) in this paper with Razborov. Here the polynomial is supposed to compute the parity function exactly: any non-boolean output counts as a mistake, thus making proving correlation bounds supposedly easier. Again, we can’t prove bounds of 1/n on the correlation for degree log(n), a problem which as also depicted in the previous post appears even more fundamental than correlation bounds for polynomials modulo 2 (and formally is a necessary first step).

Nguyen and Vu obtain their exponential improvement by a corresponding improvement in the anti-concentration bound in our paper, which was in turn a slight improvement over a special case of a previous result by Costello, Tao, and Vu. (Our improvement was only for the probability of hitting one element, as opposed to landing in an interval, but that was sufficient for the application.) Nguyen and Vu simultaneously improve on both results.

Whereas our proof is more or less from first principles, Nguyen and Vu use a lot of machinery that was developed in theoretical computer science, including invariance principles and regularity lemmas, and it is very cool to see how everything fits together.

Having discussed these two papers in a sequence, a natural question is whether non-classical polynomials help for exact computation as considered in the second paper. In fact, this question is asked in the paper by Bhowmick and Lovett, who conjecture that the answer is negative: for exact computation, non-classical polynomials should not do better than classical.