The Blessing and Curse of Curvature in Higher-Order Optimization: Horospherical versus Geodesic Convexity

Abstract

We study deterministic Riemannian $p$-th-order oracle complexity (for $p\ge2$) on Hadamard manifolds, under strong horospherical ($h$)-convexity and strong geodesic ($g$)-convexity. The two notions agree in the Euclidean space. On a curved Hadamard manifold, $h$-convexity is a stronger notion than $g$-convexity and supplies global horospherical information. Writing the $p$-th-order condition parameter $Q_p=L_pR^{p-1}/\mu$, we obtain the Euclidean-optimal rate $Q_p^{2/(3p+1)}$ for strongly $h$-convex objectives on every Hadamard manifold, with a matching fixed-curvature lower bound. On hyperbolic space, the resulting horoball supports enable localization to the curvature scale. This replaces the condition parameter $Q_p$ by $Q_p \min\{1, 4/(\kappa R) \}^{p-1}$, subject to a logarithmic localization cost. Thus growing negative curvature ($\kappa R \rightarrow \infty$) can further improve the optimal Euclidean rate under $h$-convexity. For strongly $g$-convex objectives, matching upper and lower bounds recover the same Euclidean exponent when $\kappa R=O(1)$. In contrast, with growing $\kappa R$, we construct a hard family on the hyperbolic space with $Q_p\asymp_p(1+\kappa R)^p$ that requires $\widetilde{\Omega}_p(Q_p^{1/p})$ queries. This reveals a fundamental separation: the same hyperbolic divergence that sharpens horoball localization yields an information-theoretic obstruction for the full $g$-convex class.