论文标题
足够的条件，用于黎曼优化方法的非反应收敛性
Sufficient conditions for non-asymptotic convergence of Riemannian optimisation methods
论文作者
论文摘要
通过基于能量的欧几里得环境中下降方法的分析，我们研究了这种分析的概括，用于下降方法上的riemannian歧管。这样，我们发现可以为这种下降方法得出无曲率的保证。这也使我们能够为$ G $ -CONVEX函数的Riemannian立方规范算法提供第一个已知的保证，该算法扩展了Agarwal等人[2021]的保证，用于适应性的Riemannian立方体调查的Newton Newton算法，而不是一般的非convex函数。该分析使我们研究了$ G $ -CONVEX设置中的Riemannian梯度下降的加速度，并且我们在Alimisis等人[2021]的现有结果中提高了[2021]的改进，尽管依赖曲率依赖性速率。最后，扩展了Ahn和SRA [2020]的分析，我们试图为在强大的地球上凸设置中加速Riemannian下降方法提供一些足够的条件。
Motivated by energy based analyses for descent methods in the Euclidean setting, we investigate a generalisation of such analyses for descent methods over Riemannian manifolds. In doing so, we find that it is possible to derive curvature-free guarantees for such descent methods. This also enables us to give the first known guarantees for a Riemannian cubic-regularised Newton algorithm over $g$-convex functions, which extends the guarantees by Agarwal et al [2021] for an adaptive Riemannian cubic-regularised Newton algorithm over general non-convex functions. This analysis leads us to study acceleration of Riemannian gradient descent in the $g$-convex setting, and we improve on an existing result by Alimisis et al [2021], albeit with a curvature-dependent rate. Finally, extending the analysis by Ahn and Sra [2020], we attempt to provide some sufficient conditions for the acceleration of Riemannian descent methods in the strongly geodesically convex setting.
