# Sufficient conditions for non-asymptotic convergence of Riemannian optimisation methods

arXiv: https://arxiv.org/abs/2212.05972 · HTML: https://mlsanity.com/paper/2212.05972/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 Dec 2022.

## Built on by (most cited first, 1 of 1)

- [Mirror Descent on Riemannian Manifolds](https://mlsanity.com/paper/2603.17527/) (2026)

## Abstract (arXiv)

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.