Acceleration in Hyperbolic and Spherical Spaces release_gcnd423y3fatpdy727c7tgrcyu

by David Martínez-Rubio

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2020  

Abstract

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 geodesically 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 ℳ other than the Euclidean space. 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. As a proxy, we solve a non-convex Euclidean problem, which is between convexity and quasar-convexity in the constrained setting.
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Date   2020-12-07
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arXiv  2012.03618v1
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