No-go Theorem for Acceleration in the Hyperbolic Plane
This resolves a key open question in optimization theory, showing that acceleration is fundamentally impossible in hyperbolic geometry, which is incremental as it builds on prior partial answers but provides a definitive no-go theorem.
The paper tackles the problem of whether accelerated gradient methods, akin to Nesterov's in Euclidean spaces, exist for geodesically convex functions on Riemannian manifolds, specifically proving that no such method exists for the hyperbolic plane even with exponentially small noise, due to rapid volume growth in negatively curved spaces.
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 dash these hopes. 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.