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 $\eps$-PRG for the class of size-$M$ depth-$d$ $\acz$ circuits with seed length $\log(M)^{d+O(1)}\cdot \log(1/\eps)$, and an $\eps$-PRG for the class of $S$-sparse $\F_2$ polynomials with seed length $2^{O(\sqrt{\log S})}\cdot \log(1/\eps)$. 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~\cite{IMP12} and H{\aa}stad~\cite{Has14}. This optimality of our derandomization translates into the optimality (given current circuit lower bounds) of our PRGs for $\acz$ and sparse $\F_2$ polynomials.