论文标题
全球利曼（Riemannian）在双曲线和球形空间中的加速度
Global Riemannian Acceleration in Hyperbolic and Spherical Spaces
论文作者
论文摘要
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 $μ$-strongly g-convex functions defined on the hyperbolic space or a subset of the sphere.对于除欧几里得空间以外的歧管，这些是\ emph {全球}的第一种方法，与欧几里得空间中$ l $和$ε$（以及$μ$（如果适用）相同的梯度下降，则达到了相同的梯度下降。由于几何变形，我们的费率有一个额外的因素，具体取决于最小化器的初始距离$ r $，而曲率$ k $，相对于欧几里得加速算法 作为我们解决方案的代理，我们在独立利益的凸度和\ emph {quasar-convexity}之间解决了受约束的非convex欧几里得问题。此外，对于有界截面曲率的任何riemannian歧管，我们从优化方法中降低了平滑和G-Convex函数的优化方法，到了平滑且强烈的G-Convex函数的方法，反之亦然。我们还将全局优化减少到降低曲率效果的有限球上的优化。
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 $μ$-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 $ε$ (and $μ$ 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.
