# Tensor Methods for Minimizing Convex Functions with H\"{o}lder Continuous Higher-Order Derivatives

## TL;DR

This paper develops tensor-based optimization methods for convex functions with higher-order derivatives that are Hölder continuous, providing complexity bounds for both accelerated and universal schemes, advancing the theoretical understanding of such optimization problems.

## Contribution

It introduces new tensor schemes with and without acceleration for convex minimization, establishing their iteration complexity bounds and a universal scheme for unknown Hölder parameters.

## Key findings

- Accelerated tensor schemes achieve improved complexity bounds.
- Universal scheme works without knowing Hölder continuity parameter.
- Lower bounds match the proposed schemes' complexity.

## Abstract

In this paper we study $p$-order methods for unconstrained minimization of convex functions that are $p$-times differentiable ($p\geq 2$) with $\nu$-H\"{o}lder continuous $p$th derivatives. We propose tensor schemes with and without acceleration. For the schemes without acceleration, we establish iteration complexity bounds of $\mathcal{O}\left(\epsilon^{-1/(p+\nu-1)}\right)$ for reducing the functional residual below a given $\epsilon\in (0,1)$. Assuming that $\nu$ is known, we obtain an improved complexity bound of $\mathcal{O}\left(\epsilon^{-1/(p+\nu)}\right)$ for the corresponding accelerated scheme. For the case in which $\nu$ is unknown, we present a universal accelerated tensor scheme with iteration complexity of $\mathcal{O}\left(\epsilon^{-p/[(p+1)(p+\nu-1)]}\right)$. A lower complexity bound of $\mathcal{O}\left(\epsilon^{-2/[3(p+\nu)-2]}\right)$ is also obtained for this problem class.