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• W2765150911 abstract "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 $epsilon^{-1/15}logfrac{1}{epsilon}$ 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 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 the rate $epsilon^{-1}logfrac{1}{epsilon}$, showing that finding stationary points is easier given convexity." @default.
• W2765150911 created "2017-11-10" @default.
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• W2765150911 date "2017-11-02" @default.
• W2765150911 modified "2023-09-27" @default.
• W2765150911 title "Lower Bounds for Finding Stationary Points II: First-Order Methods" @default.
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