Sufficient conditions for non-asymptotic convergence of Riemannian optimisation methods

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.