Sufficient Conditions for Non-asymptotic Convergence of Riemannian Optimization Methods

Motivated by energy based analyses for descent methods in the Euclidean setting, we investigate a generalisation of such energy based 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, improving on work by Zhang and Sra [2016]. This analysis allows us to study acceleration of Riemannian gradient descent in the geodesically-convex setting, and improve on an existing result by Alimisis et al 2021]. 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.