We further research on the accelerated optimization phenomenon on Riemannian manifolds by introducing accelerated global first-order methods for the optimization of $L$-smooth and geodesically convex (g-convex) or $\mu$-strongly g-convex functions defined on the hyperbolic space or a subset of the sphere. For a manifold other than the Euclidean space, these are the first methods to \emph{globally} achieve the same rates as accelerated gradient descent in the Euclidean space with respect to $L$ and $\varepsilon$ (and $\mu$ if it applies), up to log factors. Previous results with these accelerated rates only worked, given strong g-convexity, in a generally small neighborhood (initial distance $R$ to a minimizer being $R = O((\mu/L)^{3/4})$). Our rates have a polynomial factor on $1/\cos(R)$ (spherical case) or $\cosh(R)$ (hyperbolic case). Thus, we completely match the Euclidean case for a constant initial distance, and for larger $R$ we incur greater constants due to the geometry. As a proxy for our solution, we solve a constrained non-convex Euclidean problem, under a condition between convexity and \textit{quasar-convexity}, of independent interest. Additionally, for any Riemannian manifold of bounded sectional curvature, we provide reductions from optimization methods for smooth and g-convex functions to methods for smooth and strongly g-convex functions and vice versa.

翻译：我们进一步研究里格曼多元的加速优化现象,方法是采用加速全球一阶方法,优化超球空间或球区子集中定义的美元和地平线(g-convex)或$\muauty g-convex功能。对于欧格里多尼亚空间以外的多个区域,这是第一个达到 empph{global} 速度与欧格里多尼亚空间加速梯度下降率相同,在美元和瓦里普斯朗(如果适用的话,则美元)方面采用加速的全球一阶方法,在日志因素上优化美元和地平方平线(g- convex)或美元(yperecepslon),然后在一般小的街区中,这些加速率效果只能起作用(最短的距离为$至最低的R=O(( mu/L) 3/4}}美元),这是第一个方法。我们的汇率在1美元/colfrecial(R)或美元(ropercol)之间,对于我们最初的平比方平方平方格(rox),一个持续的递法则完全符合Ecolx的递的递减法。因此一个持续的递解-rox,对于我们的货币-rocol-rocol-rox的递减法则提供了一个持续的周期-rol-rocol-rol-ro-ro-ro-rocol-rocol-rox),对于我们比。