Improved Pseudorandom Generators from Pseudorandom Multi-Switching Lemmas

We give the best known pseudorandom generators for two touchstone classes in unconditional derandomization: an $\varepsilon$-PRG for the class of size-$M$ depth-$d$ $\mathsf{AC}^0$ circuits with seed length $\log(M)^{d+O(1)}\cdot \log(1/\varepsilon)$, and an $\varepsilon$-PRG for the class of $S$-sparse $\mathbb{F}_2$ polynomials with seed length $2^{O(\sqrt{\log S})}\cdot \log(1/\varepsilon)$. 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 \emph{pseudorandom multi-switching lemma}. We derandomize recently-developed \emph{multi}-switching lemmas, which are powerful generalizations of H{\aa}stad's switching lemma that deal with \emph{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 H{\aa}stad [H{\aa}s14]. This optimality of our derandomization translates into the optimality (given current circuit lower bounds) of our PRGs for $\mathsf{AC}^0$ and sparse $\mathbb{F}_2$ polynomials

The chow parameters problem

In the 2nd Annual FOCS (1961), C. K. Chow proved that every Boolean threshold function is uniquely determined by its degree-0 and degree-1 Fourier coefficients. These numbers became known as the Chow Parameters. Providing an algorithmic version of Chow’s theorem — i.e., efficiently construct-ing a representation of a threshold function given its Chow Parameters — has remained open ever since. This problem has received significant study in the fields of circuit complexity [Elg60, Cho61, Der65, Win71], game theory and the design of voting systems [DS79, Lee03, TT06, APL07], and learning theory [BDJ+98, Gol06]. In this paper we effectively solve the problem, giving a randomized PTAS with the following behav-ior: Theorem: Given the Chow Parameters of a Boolean threshold function f over n bits and any con-stant > 0, the algorithm runs in time O(n2 log2 n) and with high probability outputs a representation of a threshold function f ′ which is -close to f. Along the way we prove several new results of independent interest about Boolean threshold func-tions. In addition to various structural results, these include the following new algorithmic results i