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Global rates of convergence for nonconvex optimization on manifolds
We consider the minimization of a cost function on a manifold using
Riemannian gradient descent and Riemannian trust regions (RTR). We focus on
satisfying necessary optimality conditions within a tolerance .
Specifically, we show that, under Lipschitz-type assumptions on the pullbacks
of to the tangent spaces of , both of these algorithms produce points
with Riemannian gradient smaller than in
iterations. Furthermore, RTR returns a point where also the Riemannian
Hessian's least eigenvalue is larger than in
iterations. There are no assumptions on initialization.
The rates match their (sharp) unconstrained counterparts as a function of the
accuracy (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 , under simpler assumptions.Comment: 33 pages, IMA Journal of Numerical Analysis, 201
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