Global rates of convergence for nonconvex optimization on manifolds

Abstract

Abstract We consider the minimization of a cost function f on a manifold $\mathcal{M}$ using Riemannian gradient descent and Riemannian trust regions (RTR). We focus on satisfying necessary optimality conditions within a tolerance ε. Specifically, we show that, under Lipschitz-type assumptions on the pullbacks of f to the tangent spaces of $\mathcal{M}$, both of these algorithms produce points with Riemannian gradient smaller than ε in $\mathcal{O}\big(1/\varepsilon ^{2}\big)$ iterations. Furthermore, RTR returns a point where also the Riemannian Hessian’s least eigenvalue is larger than −ε in $\mathcal{O} \big(1/\varepsilon ^{3}\big)$ iterations. There are no assumptions on initialization. The rates match their (sharp) unconstrained counterparts as a function of the accuracy ε (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 ${\mathbb{R}^{n}}$, under simpler assumptions.