Brief

Researchers have developed a new decentralized online Riemannian optimization algorithm capable of operating beyond the limitations of Hadamard manifolds, extending its applicability to spaces with positive curvature. The algorithm incorporates a curvature-aware consensus step that facilitates linear convergence even in these more complex geometric settings. This advancement leads to a $O(\sqrt{T})$ regret bound for the decentralized online Riemannian gradient descent method, with similar bounds achieved in a two-point bandit feedback scenario using efficient gradient estimators. AI