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تسجيل مستخدم جديدFrom Nesterovs Estimate Sequence to Riemannian Acceleration
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We propose the first global accelerated gradient method for Riemannian manifolds. Toward establishing our result we revisit Nesterovs estimate sequence technique and develop an alternative analysis for it that may also be of independent interest. Then, we extend this analysis to the Riemannian setting, localizing the key difficulty due to non-Euclidean structure into a certain ``metric distortion. We control this distortion by developing a novel geometric inequality, which permits us to propose and analyze a Riemannian counterpart to Nesterovs accelerated gradient method.
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We further research on the acceleration phenomenon on Riemannian manifolds by introducing the first global first-order method that achieves the same rates as accelerated gradient descent in the Euclidean space for the optimization of smooth and geode
sically convex (g-convex) or strongly g-convex functions defined on the hyperbolic space or a subset of the sphere, up to constants and log factors. To the best of our knowledge, this is the first method that is proved to achieve these rates globally on functions defined on a Riemannian manifold $mathcal{M}$ other than the Euclidean space. As a proxy, we solve a constrained non-convex Euclidean problem, under a condition between convexity and quasar-convexity, of independent interest. Additionally, for any Riemannian manifold of bounded sectional curvature, we provide reductions from optimization methods for smooth and g-convex functions to methods for smooth and strongly g-convex functions and vice versa.
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