In this paper, we consider the problem of finding ε-approximate stationary points of convex functions that are p-times differentiable with ν-Hölder continuous pth derivatives. We present tensor methods with and without acceleration. Specifically, we show that the non-accelerated schemes take at most $\mathcal{O}\left({ϵ}^{-1/(p+\nu -1)}\right)$ iterations to reduce the norm of the gradient of the objective below given $ϵ\in (0,1)$. For accelerated tensor schemes, we establish improved complexity bounds of $\mathcal{O}\left({ϵ}^{-(p+\nu )/\left[\right(p+\nu -1\left)\right(p+\nu +1\left)\right]}\right)$ and $\mathcal{O}\left(|\log \left(ϵ\right)|{ϵ}^{-1/(p+\nu )}\right)$, when the Hölder parameter $\nu \in [0,1]$ is known. For the case in which ν is unknown, we obtain a bound of $\mathcal{O}\left({ϵ}^{-(p+1)/\left[\right(p+\nu -1\left)\right(p+2\left)\right]}\right)$ for a universal accelerated scheme. Finally, we also obtain a lower complexity bound of $\mathcal{O}\left({ϵ}^{-2/\left[3\right(p+\nu )-2]}\right)$ for finding ε-approximate stationary points using p-order tensor methods.

在本文中，我们考虑寻找ε - 近似的凸函数驻点的问题，这些驻点是可与ν -Hölder 连续p th 导数进行p次微分的。我们提出了有加速和没有加速的张量方法。具体来说，我们表明非加速方案最多需要$\mathcal{○}\left({\epsilon }^{-1/(p+\nu -1)}\right)$迭代以减少下面给出的目标的梯度范数$\epsilon \in (0,1)$. 对于加速张量方案，我们建立了改进的复杂性界限$\mathcal{○}\left({\epsilon }^{-(p+\nu )/\left[\right(p+\nu -1\left)\right(p+\nu +1\left)\right]}\right)$和$\mathcal{○}\left(|\mathrm{日志}\left(\epsilon \right)|{\epsilon }^{-1/(p+\nu )}\right)$, 当 Hölder 参数$\nu \in [0,1]$是已知的。对于ν未知的情况，我们得到一个界限$\mathcal{○}\left({\epsilon }^{-(p+1)/\left[\right(p+\nu -1\left)\right(p+2\left)\right]}\right)$对于通用加速方案。最后，我们还获得了一个较低的复杂度界限$\mathcal{○}\left({\epsilon }^{-2/\left[3\right(p+\nu )-2]}\right)$用于使用p阶张量方法找到ε近似静止点。