  In recent years, interest in gradient-based optimization over Riemannian
manifolds has surged. However, a significant challenge lies in the reliance on
hyperparameters, especially the learning rate, which requires meticulous tuning
by practitioners to ensure convergence at a suitable rate. In this work, we
introduce innovative learning-rate-free algorithms for stochastic optimization
over Riemannian manifolds, eliminating the need for hand-tuning and providing a
more robust and user-friendly approach. We establish high probability
convergence guarantees that are optimal, up to logarithmic factors, compared to
the best-known optimally tuned rate in the deterministic setting. Our approach
is validated through numerical experiments, demonstrating competitive
performance against learning-rate-dependent algorithms.
