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We establish lower bounds on the complexity of finding ϵ-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 ϵ better than ϵ^-8/5, which is within ϵ^-1/151/ϵ of the best known rate for such methods. Moreover, for functions with Lipschitz first and second derivatives, we prove no deterministic first-order method can achievearXiv:1711.00841v1 fatcat:pih7vinkybcxtggmbmzqwibxle