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Numerous applications in machine learning and data analytics can beformulated as equilibrium computation over Riemannian manifolds. Despite theextensive investigation of their Euclidean counterparts, the performance ofRiemannian gradient-based algorithms remain opaque and poorly understood. Werevisit the original scheme of Riemannian gradient descent (RGD) and analyze itunder a geodesic monotonicity assumption, which includes the well-studiedgeodesically convex-concave min-max optimization problem as a special case. Ourmain contribution is to show that, despite the phenomenon of distancedistortion, the RGD scheme, with a step size that is agnostic to the manifold'scurvature, achieves a curvature-independent and linear last-iterate convergencerate in the geodesically strongly monotone setting. To the best of ourknowledge, the possibility of curvature-independent rates and/or last-iterateconvergence in the Riemannian setting has not been considered before.

Curvature-Independent Last-Iterate Convergence for Games on Riemannian Manifolds