  We study the oracle complexity of producing $(\delta,\epsilon)$-stationary
points of Lipschitz functions, in the sense proposed by Zhang et al. [2020].
While there exist dimension-free randomized algorithms for producing such
points within $\widetilde{O}(1/\delta\epsilon^3)$ first-order oracle calls, we
show that no dimension-free rate can be achieved by a deterministic algorithm.
On the other hand, we point out that this rate can be derandomized for smooth
functions with merely a logarithmic dependence on the smoothness parameter.
Moreover, we establish several lower bounds for this task which hold for any
randomized algorithm, with or without convexity. Finally, we show how the
convergence rate of finding $(\delta,\epsilon)$-stationary points can be
improved in case the function is convex, a setting which we motivate by proving
that in general no finite time algorithm can produce points with small
subgradients even for convex functions.
