# Tensor Methods for Minimizing Convex Functions with Hölder Continuous Higher-Order Derivatives

In this paper we study p-order methods for unconstrained minimization of convex functions that are p-times differentiable with $\nu$-Hölder continuous pth 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 know, 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 for this problem class is also obtained.

## Citation

SIAM Journal on Optimization 30(4), 2750–2779 (2020)