The Complexity of Finding Stationary Points in Nonsmooth Nonconvex Optimization

We prove that first-order algorithms require $Ω(δ^{-1}ε^{-3})$ gradient queries (in the worst case) to find a $(δ,ε)$-Goldstein stationary point of a Lipschitz function, at which there is a convex combination of gradients within distance $δ$ whose norm is at most $ε$. This lower bound is tight, matc