Guy Kornowski

The Complexity of Finding Stationary Points in Nonsmooth Nonconvex Optimization

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