  Numerous applications in machine learning and data analytics can be
formulated as equilibrium computation over Riemannian manifolds. Despite the
extensive investigation of their Euclidean counterparts, the performance of
Riemannian gradient-based algorithms remain opaque and poorly understood. We
revisit the original scheme of Riemannian gradient descent (RGD) and analyze it
under a geodesic monotonicity assumption, which includes the well-studied
geodesically convex-concave min-max optimization problem as a special case. Our
main contribution is to show that, despite the phenomenon of distance
distortion, the RGD scheme, with a step size that is agnostic to the manifold's
curvature, achieves a curvature-independent and linear last-iterate convergence
rate in the geodesically strongly monotone setting. To the best of our
knowledge, the possibility of curvature-independent rates and/or last-iterate
convergence in the Riemannian setting has not been considered before.

Optimization and Control;Computer Science and Game Theory;Machine Learning