Local Riemannian Stochastic Gradient Descent: Learning and Uncertainty Quantification

We study statistical inference with infrequent communication for the full-participation Riemannian Federated Averaging Gradient Streams (RFedAGS) recursion with growing communication intervals, where client drift and geometric error must be controlled at the scale of all local observations. A moment-sensitive synchronized-state bound makes these errors negligible and yields a central limit theorem (CLT) for a computable fixed-anchor average in a regular local basin. The covariance separates the objective Hessian, client noise and communication schedule. With a fixed burn-in, admissible slowly varying intervals preserve first-order precision while communication frequency vanishes. Fresh pooled scores and exact or sample-based Hessians consistently estimate this covariance, giving feasible Wald regions. Gaussian and sphere experiments at matched total oracle budgets show that a prespecified half-tail attains near-nominal coverage with substantially lower communication cost than a fixed-interval baseline, at the cost of a larger uncertainty scale. A four-site principal-subspace application illustrates the construction.