  We study the complexity of producing $(\delta,\epsilon)$-stationary points of
Lipschitz objectives which are possibly neither smooth nor convex, using only
noisy function evaluations. Recent works proposed several stochastic zero-order
algorithms that solve this task, all of which suffer from a
dimension-dependence of $\Omega(d^{3/2})$ where $d$ is the dimension of the
problem, which was conjectured to be optimal. We refute this conjecture by
providing a faster algorithm that has complexity
$O(d\delta^{-1}\epsilon^{-3})$, which is optimal (up to numerical constants)
with respect to $d$ and also optimal with respect to the accuracy parameters
$\delta,\epsilon$, thus solving an open question due to Lin et al.
(NeurIPS'22). Moreover, the convergence rate achieved by our algorithm is also
optimal for smooth objectives, proving that in the nonconvex stochastic
zero-order setting, nonsmooth optimization is as easy as smooth optimization.
We provide algorithms that achieve the aforementioned convergence rate in
expectation as well as with high probability. Our analysis is based on a simple
yet powerful lemma regarding the Goldstein-subdifferential set, which allows
utilizing recent advancements in first-order nonsmooth nonconvex optimization.
