Escaping from saddle points on Riemannian manifolds
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We consider minimizing a nonconvex, smooth function f on a Riemannian manifold M. We show that a perturbed version of Riemannian gradient descent algorithm converges to a second-order stationary point (and hence is able to escape saddle points on the manifold). The rate of convergence depends as 1/ϵ^2 on the accuracy ϵ, which matches a rate known only for unconstrained smooth minimization. The convergence rate depends polylogarithmically on the manifold dimension d, hence is almost dimension-free. The rate also has a polynomial dependence on the parameters describing the curvature of the manifold and the smoothness of the function. While the unconstrained problem (Euclidean setting) is well-studied, our result is the first to prove such a rate for nonconvex, manifold-constrained problems.
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