Curvature-Independent Last-Iterate Convergence for Games on Riemannian
  Manifolds
Yang Cai Michael I. Jordan Tianyi Lin Argyris Oikonomou Emmanouil-Vasileios Vlatakis-Gkaragkounis
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.