Escaping from saddle points on Riemannian manifolds

We consider minimizing a nonconvex, smooth function $f$ on a Riemannian manifold $\mathcal{M}$. We show that a perturbed version of the gradient descent algorithm converges to a second-order stationary point for this problem (and hence is able to escape saddle points on the manifold). While the unconstrained problem is well-studied, our result is the first to prove such a rate for nonconvex, manifold-constrained problems. The rate of convergence depends as $1/\epsilon^2$ on the accuracy $\epsilon$, which matches a rate known only for unconstrained smooth minimization. The convergence rate also has a polynomial dependence on the parameters denoting the curvature of the manifold and the smoothness of the function.