  It is well-known that given a smooth, bounded-from-below, and possibly
nonconvex function, standard gradient-based methods can find
$\epsilon$-stationary points (with gradient norm less than $\epsilon$) in
$\mathcal{O}(1/\epsilon^2)$ iterations. However, many important nonconvex
optimization problems, such as those associated with training modern neural
networks, are inherently not smooth, making these results inapplicable. In this
paper, we study nonsmooth nonconvex optimization from an oracle complexity
viewpoint, where the algorithm is assumed to be given access only to local
information about the function at various points. We provide two main results:
First, we consider the problem of getting near $\epsilon$-stationary points.
This is perhaps the most natural relaxation of finding $\epsilon$-stationary
points, which is impossible in the nonsmooth nonconvex case. We prove that this
relaxed goal cannot be achieved efficiently, for any distance and $\epsilon$
smaller than some constants. Our second result deals with the possibility of
tackling nonsmooth nonconvex optimization by reduction to smooth optimization:
Namely, applying smooth optimization methods on a smooth approximation of the
objective function. For this approach, we prove under a mild assumption an
inherent trade-off between oracle complexity and smoothness: On the one hand,
smoothing a nonsmooth nonconvex function can be done very efficiently (e.g., by
randomized smoothing), but with dimension-dependent factors in the smoothness
parameter, which can strongly affect iteration complexity when plugging into
standard smooth optimization methods. On the other hand, these dimension
factors can be eliminated with suitable smoothing methods, but only by making
the oracle complexity of the smoothing process exponentially large.
