We further research on the acceleration phenomenon on Riemannian manifolds by introducing the first global first-order method that achieves the same rates as accelerated gradient descent in the Euclidean space for the optimization of smooth and geodesically convex (g-convex) or strongly g-convex functions defined on the hyperbolic space or a subset of the sphere, up to constants and log factors. To the best of our knowledge, this is the first method that is proved to achieve these rates globally on functions defined on a Riemannian manifold $\mathcal{M}$ other than the Euclidean space. 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. As a proxy, we solve a non-convex Euclidean problem, which is between convexity and quasar-convexity in the constrained setting.

翻译：我们进一步研究里曼尼方块加速现象,方法是采用第一个全球第一阶方法,在欧几里德空间实现与加速梯度下降率相同的速度,以优化双曲空间或球子组界定的平滑和大地曲线(g-convex)或强烈的G-convex函数,直至恒定值和日志系数。根据我们所知,这是第一个在除欧几里德空间之外的里曼方块中定义的函数上实现这些速度的方法。此外,对于任何有线段弯曲的里曼方块,我们从光滑和格-convex函数优化方法到平滑和强凝结函数的方法,以及反之亦然。作为代理,我们解决了一种非culidean问题,这是在受限制的环境下的调和孔-conversion-conxity之间的问题。