- Tensor Methods for Finding Approximate Stationary Points of Convex Functions Geovani Grapiglia(grapigliaufpr.br) Yurii Nesterov(yurii.nesterovuclouvain.be) Abstract: In this paper we consider the problem of finding \epsilon-approximate stationary points of convex functions that are p-times differentiable with \nu-Hölder continuous pth derivatives. We present tensor methods with and without acceleration. Specifically, we show that the non-accelerated schemes take at most O(\epsilon^{-1/(p+\nu-1)}) iterations to reduce the norm of the gradient of the objective below a given \epsilon\in (0,1). For accelerated tensor schemes we establish improved complexity bounds of O(\epsilon^{-(p+\nu)/[(p+\nu-1)(p+\nu+1)]}) and O(|\log(\epsilon)|\epsilon^{-1/(p+\nu)}), when the Hölder parameter \nu\in [0,1] is known. For the case in which \nu is unknown, we obtain a bound of O(\epsilon^{-(p+1)/[(p+\nu-1)(p+2)]}) for a universal accelerated scheme. Finally, we also obtain a lower complexity bound of O(\epsilon^{-2/[3(p+\nu)-2]}) for finding \epsilon-approximate stationary points using p-order tensor methods. Keywords: unconstrained minimization, high-order methods, tensor methods, Hölder condition, worst-case complexity Category 1: Convex and Nonsmooth Optimization (Convex Optimization ) Citation: Download: [PDF]Entry Submitted: 07/13/2019Entry Accepted: 07/13/2019Entry Last Modified: 07/13/2019Modify/Update this entry Visitors Authors More about us Links Subscribe, Unsubscribe Digest Archive Search, Browse the Repository Submit Update Policies Coordinator's Board Classification Scheme Credits Give us feedback Optimization Journals, Sites, Societies Optimization Online is supported by the Mathematical Optmization Society.