Online optimization has witnessed a massive surge of research attention in recent years. In this paper, we propose online gradient descent and online bandit algorithms over Riemannian manifolds in full information and bandit feedback settings respectively, for both geodesically convex and strongly geodesically convex functions. We establish a series of upper bounds on the regrets for the proposed algorithms over Hadamard manifolds. We also find a universal lower bound for achievable regret on Hadamard manifolds. Our analysis shows how time horizon, dimension, and sectional curvature bounds have impact on the regret bounds. When the manifold permits positive sectional curvature, we prove similar regret bound can be established by handling non-constrictive project maps. In addition, numerical studies on problems defined on symmetric positive definite matrix manifold, hyperbolic spaces, and Grassmann manifolds are provided to validate our theoretical findings, using synthetic and real-world data.

We consider online optimization over Riemannian manifolds, where a learner attempts to minimize a sequence of time-varying loss functions defined on Riemannian manifolds. Though many Euclidean online convex optimization algorithms have been proven useful in a wide range of areas, less attention has been paid to their Riemannian counterparts. In this paper, we study Riemannian online gradient descent (R-OGD) on Hadamard manifolds for both geodesically convex and strongly geodesically convex loss functions, and Riemannian bandit algorithm (R-BAN) on Hadamard homogeneous manifolds for geodesically convex functions. We establish upper bounds on the regrets of the problem with respect to time horizon, manifold curvature, and manifold dimension. We also find a universal lower bound for the achievable regret by constructing an online convex optimization problem on Hadamard manifolds. All the obtained regret bounds match the corresponding results are provided in Euclidean spaces. Finally, some numerical experiments validate our theoretical results.