2018/01/10 by Servedio, Rocco A., Tan, Li-Yang
#Computational Complexity (cs.CC) #FOS: Computer and information sciences
paper · doi:10.48550/arxiv.1801.03590
We give the best known pseudorandom generators for two touchstone classes in unconditional derandomization: an ε-PRG for the class of size-M depth-d AC0 circuits with seed length log(M)d+O(1)⋅ log(1/ε), and an ε-PRG for the class of S-sparse \mathbbF2 polynomials with seed length 2O(√(log S))⋅ log(1/ε). These results bring the state of the art for unconditional derandomization of these classes into sharp alignment with the state of the art for computational hardness for all parameter settings: improving on the seed lengths of either PRG would require breakthrough progress on longstanding and notorious circuit lower bounds. The key enabling ingredient in our approach is a new pseudorandom multi-switching lemma. We derandomize recently-developed multi-switching lemmas, which are powerful generalizations of Håstad's switching lemma that deal with families of depth-two circuits. Our pseudorandom multi-switching lemma---a randomness-efficient algorithm for sampling restrictions that simultaneously simplify all circuits in a family---achieves the parameters obtained by the (full randomness) multi-switching lemmas of Impagliazzo, Matthews, and Paturi [IMP12] and Håstad [Hås14]. This optimality of our derandomization translates into the optimality (given current circuit lower bounds) of our PRGs for AC0 and sparse \mathbbF2 polynomials.