Lower bounds for finding stationary points II: first-order methods

We establish lower bounds on the complexity of finding $$\epsilon $$ -stationary points of smooth, non-convex high-dimensional functions using first-order methods. We prove that deterministic first-order methods, even applied to arbitrarily smooth functions, cannot achieve convergence rates in $$\epsilon $$ better than $$\epsilon ^{-8/5}$$ , which is within $$\epsilon ^{-1/15}\log \frac{1}{\epsilon }$$ of the best known rate for such methods. Moreover, for functions with Lipschitz first and second derivatives, we prove that no deterministic first-order method can achieve convergence rates better than $$\epsilon ^{-12/7}$$ , while $$\epsilon ^{-2}$$ is a lower bound for functions with only Lipschitz gradient. For convex functions with Lipschitz gradient, accelerated gradient descent achieves a better rate, showing that finding stationary points is easier given convexity.