Horoballs and the subgradient method
Adrian S. Lewis, Genaro Lopez-Acedo, Adriana Nicolae
TL;DR
This work develops a tangent-free subgradient-style method for convex optimization in Hadamard spaces by exploiting horospherical convexity of level sets, thereby avoiding tangent-space machinery. Using a projection oracle and a support oracle grounded in horoballs, the authors prove a convergence rate of $O\left(\tfrac{1}{\sqrt{n}}\right)$ that depends only on the Lipschitz constant $L$ and the feasible diameter $D$, with no lower curvature bound required. They introduce horoballs and horospherical convexity, analyze the algorithm in general Hadamard spaces, and apply it to the minimal enclosing ball and circumcenter problems, including non-manifold CAT(0) complexes and BHV tree spaces. The results broaden the applicability of first-order methods to non-Euclidean, non-manifold settings, offering practical tools for problems in phylogenetic geometry and robotics. Overall, the paper presents a geometrically natural subgradient framework that extends non-Euclidean optimization beyond manifolds while preserving familiar $1/\sqrt{n}$ complexity behavior.
Abstract
To explore convex optimization on Hadamard spaces, we consider an iteration in the style of a subgradient algorithm. Traditionally, such methods assume that the underlying spaces are manifolds and that the objectives are geodesically convex: the methods are described using tangent spaces and exponential maps. By contrast, our iteration applies in a general Hadamard space, is framed in the underlying space itself, and relies instead on horospherical convexity of the objective level sets. For this restricted class of objectives, we prove a complexity result of the usual form. Notably, the complexity does not depend on a lower bound on the space curvature. We illustrate our subgradient algorithm on the minimal enclosing ball problem in Hadamard spaces.