Prediction from Partial Information and Hindsight, with Application to Circuit Lower Bounds

Consider a random sequence of n bits that has entropy at least n−k, where k≪ n. A commonly used observation is that an average coordinate of this random sequence is close to being uniformly distributed, that is, the coordinate “looks random.” In this work, we prove a stronger result that says, roughly, that the average coordinate looks random to an adversary that is allowed to query ≈nk other coordinates of the sequence, even if the adversary is non-deterministic. This implies corresponding results for decision trees and certificates for Boolean functions. As an application of this result, we prove a new result on depth-3 circuits, which recovers as a direct corollary the known lower bounds for the parity and majority functions, as well as a lower bound on sensitive functions due to Boppana (Circuits Inf Process Lett 63(5):257–261, 1997). An interesting feature of this proof is that it works in the framework of Karchmer and Wigderson (SIAM J Discrete Math 3(2):255–265, 1990), and, in particular, it is a “top-down” proof (Håstad et al. in Computat Complex 5(2):99–112, 1995). Finally, it yields a new kind of a random restriction lemma for non-product distributions, which may be of independent interest.