Acceleration in Hyperbolic and Spherical Spaces
release_gcnd423y3fatpdy727c7tgrcyu
by
David Martínez-Rubio
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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