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 $\epsilon$ (and $\mu$ if it applies), up to log factors. Due to the geometric deformations, our rates have an extra factor, depending on the initial distance $R$ to a minimizer and the curvature $K$, with respect to Euclidean accelerated algorithms As a proxy for our solution, we solve a constrained non-convex Euclidean problem, under a condition between convexity and \emph{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. We also reduce global optimization to optimization over bounded balls where the effect of the curvature is reduced.

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法,优化双曲空间或球体子组定义的美元和大地曲线(g-convex)或$mu$强的G-convex功能。对于欧格利底亚加速算法以外的多元空间,这些是首个方法,作为我们解决方案的替代,我们在欧格西里底亚空间加速梯度下降率与美元和美元(如果适用,则为美元)相比,最高为日志因素。由于几何变形,我们的比率有一个额外因素,取决于最初距离美元至最小值和曲线子组子组的曲度;对于欧格利底亚加速算法来说,我们解决了一种受限制的非convex Euclidean 问题,在调和/emph{quarlon-consultion(如果适用的话,为美元)之间条件相同。由于几何体格调整,我们平整的平整和平整的平整度功能也提供了一个额外的额外因素。