Implicit Riemannian Optimism with Applications to Min-Max Problems

We introduce a Riemannian optimistic online learning algorithm for Hadamard manifolds based on inexact implicit updates. Unlike prior work, our method can handle in-manifold constraints, and matches the best known regret bounds in the Euclidean setting, removing the dependence on geometric constants, like the minimum curvature. Building on this method, we develop multiple algorithms for g-convex, g-concave smooth minmax problems on Hadamard manifolds. Notably, one method nearly matches the gradient oracle complexity of the lower bound for Euclidean problems, for the first time. * Equal contribution (arbitrary order).