A No-go Theorem for Robust Acceleration in the Hyperbolic PlaneDownload PDF

21 May 2021, 20:48 (modified: 01 Feb 2022, 04:10)NeurIPS 2021 PosterReaders: Everyone
Keywords: geodesic convexity, acceleration, lower bounds
TL;DR: We prove that in a noisy setting, there is no analogue of accelerated gradient descent for geodesically convex functions on the hyperbolic plane.
Abstract: In recent years there has been significant effort to adapt the key tools and ideas in convex optimization to the Riemannian setting. One key challenge has remained: Is there a Nesterov-like accelerated gradient method for geodesically convex functions on a Riemannian manifold? Recent work has given partial answers and the hope was that this ought to be possible. Here we prove that in a noisy setting, there is no analogue of accelerated gradient descent for geodesically convex functions on the hyperbolic plane. Our results apply even when the noise is exponentially small. The key intuition behind our proof is short and simple: In negatively curved spaces, the volume of a ball grows so fast that information about the past gradients is not useful in the future.
Code Of Conduct: I certify that all co-authors of this work have read and commit to adhering to the NeurIPS Statement on Ethics, Fairness, Inclusivity, and Code of Conduct.
14 Replies

Loading