Decentralized Online Riemannian Optimization Beyond Hadamard Manifolds
This work addresses optimization challenges in non-Euclidean spaces for applications like robotics or machine learning, representing an incremental advance by extending methods beyond Hadamard manifolds.
The paper tackles decentralized online optimization on Riemannian manifolds with positive curvature, where traditional consensus methods fail due to non-convex structures, and achieves a linear convergence for consensus and an O(√T) regret bound for gradient descent and bandit feedback setups.
We study decentralized online Riemannian optimization over manifolds with possibly positive curvature, going beyond the Hadamard manifold setting. Decentralized optimization techniques rely on a consensus step that is well understood in Euclidean spaces because of their linearity. However, in positively curved Riemannian spaces, a main technical challenge is that geodesic distances may not induce a globally convex structure. In this work, we first analyze a curvature-aware Riemannian consensus step that enables a linear convergence beyond Hadamard manifolds. Building on this step, we establish a $O(\sqrt{T})$ regret bound for the decentralized online Riemannian gradient descent algorithm. Then, we investigate the two-point bandit feedback setup, where we employ computationally efficient gradient estimators using smoothing techniques, and we demonstrate the same $O(\sqrt{T})$ regret bound through the subconvexity analysis of smoothed objectives.