Nonconvex stochastic optimization on manifolds via Riemannian Frank-Wolfe methods

We study stochastic projection-free methods for constrained optimization of smooth functions on Riemannian manifolds (where there are additional constraints beyond the parameter domain being a manifold). Specifically, we introduce stochastic Riemannian Frank-Wolfe methods for both nonconvex and geodesically convex problems. We present algorithms for both stochastic optimization and finite-sum problems. For the latter, we develop variance reduced methods, including a Riemannian adaptation of the recently proposed Spider technique. For all settings, we recover convergence rates that are comparable to the best-known rates for their Euclidean counterparts. Finally, we discuss applications to two basic tasks: computation of the Karcher mean and Wasserstein barycenters for multivariate normal distributions. For both tasks, stochastic Fw methods yield state-of-the-art empirical performance.