Tensor methods for minimizing convex functions with holder continuous higher-order derivatives

Abstract

In this paper, we study p-order methods for unconstrained minimization of convex functions that are p-times differentiable (p ≥2) with ν -H older continuous pth derivatives. We propose tensor schemes with and without acceleration. For the schemes without acceleration, we establish iteration complexity bounds of O (ϵ-1/(p+ν) ) for reducing the functional residual below a given in (0, 1). Assuming that ν is known, we obtain an improved complexity bound of O(ϵ-1/(p+ν) ) for the corresponding accelerated scheme. For the case in which ν is unknown, we present a universal accelerated tensor scheme with iteration complexity of O ( p/[(p+1)(p+ν 1)] ) . A lower complexity bound of O (ϵ-2/[3(p+ν) 2] ) is also obtained for this problem class.