Tensor methods for finding approximate stationary points of convex functions
(2022) Optimization Methods and Software — Vol. 37, n° 2, p. 605-638 (2022)
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- 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.
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Nunes Grapiglia, G., & Nesterov, Y. (2022). Tensor methods for finding approximate stationary points of convex functions. Optimization Methods and Software, 37(2), 605-638. https://doi.org/10.1080/10556788.2020.1818082 (Original work published 2022)