Improved Pseudorandom Generators from Peudorandom Multi-switching Lemmas

We give the best known pseudorandom generators for two touchstone classes in unconditional derandomization: small-depth circuits and sparse F-2 polynomials. Our main results are an epsilon-PRG for the class of size-M depth-d AC(0) circuits with seed length log(M)(d+O(1)) . log(1/epsilon), and an epsilon-PRG for the class of S-sparse F-2 polynomials with seed length 2(O(root logS)) . log (1 / epsilon ) These results bring the state of the art for unconditional deran- domization of these classes into sharp alignment with the state of the art for computational hardness for all parameter settings: substantially improving on the seed lengths of either PRG would require a breakthrough on longstanding and notorious circuit lower bound problems. 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 Hastad'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 (SODA'12) and Hastad (SICOMP 2014). This optimality of our derandomization translates into the optimality (given current circuit lower bounds) of our PRGs for AC(0) and sparse F-2 polynomials.