On the Finite-Time Convergence for Intrinsic Decentralized Riemannian Optimization

We develop a finite-time convergence theory for intrinsic decentralized Riemannian optimization with deterministic and stochastic gradients. Using an exact weighted Fréchet-mean consensus step, we establish linear consensus contraction and $\mathcal{O}(1/T)$ network disagreement on manifolds with bounded sectional curvature, including positive curvature. For geodesically convex and smooth nonconvex objectives, we prove exact $\widetilde{\mathcal{O}}(T^{-1/2})$ convergence guarantees. Numerical experiments on decentralized PCA over the sphere support the theoretical results.