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Global rates of convergence for nonconvex optimization on manifolds. (English) Zbl 07208096
IMA J. Numer. Anal. 39, No. 1, 1-33 (2019); erratum ibid. 40, No. 4, 2940 (2020).
Summary: 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 \(\epsilon \). 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 \(\epsilon\) 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 \(- \varepsilon\) 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 \(\varepsilon \) (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.

MSC:
65-XX Numerical analysis
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