ABSTRACT

In this paper, we study the auxiliary problems that appear in p-order tensor methods for unconstrained minimization of convex functions with ν-Hölder continuous pth derivatives. This type of auxiliary problems corresponds to the minimization of a $(p+\nu )$-order regularization of the pth-order Taylor approximation of the objective. For the case p = 3, we consider the use of Gradient Methods with Bregman distance. When the regularization parameter is sufficiently large, we prove that the referred methods take at most $\mathcal{O}(\log ({ϵ}^{-1}\left)\right)$ iterations to find either a suitable approximate stationary point of the tensor model or an ε-approximate stationary point of the original objective function.

摘要

在本文中，我们研究了用ν- Hölder连续p次导数无约束最小化p阶张量方法中出现的辅助问题。这种类型的辅助问题对应于最小化$（p+\nu ）$目标的p阶泰勒近似的2阶正则化。对于p  = 3的情况，我们考虑使用Bregman距离的梯度方法。当正则化参数足够大时，我们证明所引用的方法最多$\mathcal{Ø}（\mathrm{日志}（{ϵ}^{-\mathrm{1个}}））$迭代以找到张量模型的合适的近似平稳点或原始目标函数的ε近似平稳点。