Near Optimal Methods for Minimizing Convex Functions with Lipschitz $p$-th Derivatives

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Alexander Gasnikov, Pavel Dvurechensky, Eduard Gorbunov, Evgeniya Vorontsova, Daniil Selikhanovych, César A. Uribe, Bo Jiang, Haoyue Wang, Shuzhong Zhang, Sébastien Bubeck, Qijia Jiang, Yin Tat Lee, Yuanzhi Li, Aaron Sidford ;
Proceedings of the Thirty-Second Conference on Learning Theory, PMLR 99:1392-1393, 2019.

Abstract

In this merged paper, we consider the problem of minimizing a convex function with Lipschitz-continuous $p$-th order derivatives. Given an oracle which when queried at a point returns the first $p$-derivatives of the function at that point we provide some methods which compute an $\e$ approximate minimizer in $O\left(\e^{-\frac{2}{3p+1}} \right)$ iterations. These methods match known lower bounds up to polylogarithmic factors for constant $p$.

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