Curvature-Dependant Global Convergence Rates for Optimization on Manifolds of Bounded Geometry

TL;DR

This paper establishes curvature-dependent convergence rates for optimization algorithms on manifolds of bounded geometry, providing tighter bounds on the Hessian norm and explicit calculations for common manifolds.

Contribution

It introduces improved bounds on the Hessian norm of the exponential map and derives curvature-dependent convergence rates for Riemannian gradient descent and trivialization algorithms.

Findings

Derived explicit curvature-dependent convergence rates.

Provided tighter bounds on the Hessian norm of the exponential map.

Applied results to manifolds like the special orthogonal group and Grassmannian.

Abstract

We give curvature-dependant convergence rates for the optimization of weakly convex functions defined on a manifold of 1-bounded geometry via Riemannian gradient descent and via the dynamic trivialization algorithm. In order to do this, we give a tighter bound on the norm of the Hessian of the Riemannian exponential than the previously known. We compute these bounds explicitly for some manifolds commonly used in the optimization literature such as the special orthogonal group and the real Grassmannian. Along the way, we present self-contained proofs of fully general bounds on the norm of the differential of the exponential map and certain cosine inequalities on manifolds, which are commonly used in optimization on manifolds.