Tensor Methods for Finding Approximate Stationary Points of Convex Functions

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