Sufficient conditions for non-asymptotic convergence of Riemannian optimization methodsDownload PDF

Published: 23 Nov 2022, Last Modified: 05 May 2023OPT 2022 OralReaders: Everyone
Abstract: 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 motivates 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.
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