The Large-Error Approximate Degree of AC(0)

We prove two new results about the inability of low-degree polynomials to uniformly approximate constant-depth circuits, even to slightly-better-than-trivial error. First, we prove a tight (Omega) over tilde (n(1/2)) lower bound on the threshold degree of the SURJECTIVITY function on n variables. This matches the best known threshold degree bound for any AC(0) function, previously exhibited by a much more complicated circuit of larger depth (Sherstov, FOCS'15). Our result also extends to a 2((Omega) over tilde (n1/2)) lower bound on the sign-rank of an AC(0) function, improving on the previous best bound of 2(Omega(n2/5)) (Bun and Thaler, ICALP'16). Second, for any delta > 0, we exhibit a function f : {-1,1}(n) -> {-1,1} that is computed by a circuit of depth O(1/delta) and is hard to approximate by polynomials in the following sense: f cannot be uniformly approximated to error epsilon = 1-2(-Omega(n1-delta)), even by polynomials of degree n(1-delta). In our FOCS'17 paper we proved a similar lower bound, but which held only for error epsilon = 1/3. Our result implies 2(Omega(n1-delta)) lower bounds on the complexity of AC(0) under a variety of measures, including discrepancy, margin complexity, and threshold weight, that are central to communication complexity and learning theory. This nearly matches the trivial upper bound of 2(O(n)) that holds for every function. The previous best lower bound on AC(0) for these measures was 2(Omega(n1/2)) (Sherstov, FOCS'15). Additional applications in learning theory, communication complexity, and cryptography are described.