Global rates of convergence for nonconvex optimization on manifolds

We consider the minimization of a cost function f on a manifold M using Riemannian gradient descent and Riemannian trust regions (RTR). We focus on satisfying necessary optimality conditions within a tolerance epsilon. Specifically, we show that, under Lipschitz-type assumptions on the pullbacks of f to the tangent spaces of M, both of these algorithms produce points with Riemannian gradient smaller than epsilon in O(1/epsilon(2)) iterations. Furthermore, RTR returns a point where also the Riemannian Hessian's least eigenvalue is larger than -epsilon in O(1/epsilon(3)) iterations. There are no assumptions on initialization. The rates match their (sharp) unconstrained counterparts as a function of the accuracy epsilon (up to constants) and hence are sharp in that sense. These are the first deterministic results for global rates of convergence to approximate first- and second-order Karush-Kuhn-Tucker points on manifolds. They apply in particular for optimization constrained to compact submanifolds of R-n, under simpler assumptions.