Improved Average-Case Lower Bounds for De Morgan Formula Size: Matching Worst-Case Lower Bound

We give a function h : {0, 1} → {0, 1} such that every deMorgan formula of size n3−o(1)/r2 agrees with h on at most a fraction of 1 2 + 2 −Ω(r) of the inputs. This improves the previous average-case lower bound of Komargodski and Raz (STOC, 2013). Our technical contributions include a theorem that shows that the “expected shrinkage” result of H̊astad (SIAM J. Comput., 1998) actually holds with very high probability (where the restrictions are chosen from a certain distribution that takes into account the structure of the formula), combining ideas of both Impagliazzo, Meka and Zuckerman (FOCS, 2012) and Komargodski and Raz. In addition, using a bit-fixing extractor in the construction of h allows us to simplify a major part of the analysis of Komargodski and Raz.