AI 中文总结
该研究在Hadamard流形上分析高阶优化的曲率影响，发现h凸性下负曲率可优化速率，而g凸性下负曲率会带来信息论障碍，明确两类凸性的根本差异。
AI 中文摘要
我们研究Hadamard流形上确定性黎曼p阶预言机复杂度（p≥2），分别在强球心（h）凸性与强测地（g）凸性条件下展开分析。这两个概念在欧氏空间中完全一致；而在弯曲的Hadamard流形上，h凸性是比g凸性更强的概念，可提供全局球心信息。记p阶条件参数为Qₚ=LₚRᵖ⁻¹/μ，我们在所有Hadamard流形上针对强h凸目标函数得到了欧氏最优速率Qₚ^(2/(3p+1))，并给出了匹配的固定曲率下界。在双曲空间中，所得球支持可实现曲率尺度的定位，该定位需承担对数级代价，且条件参数Qₚ会被替换为Qₚ·min{1,4/(κR)}ᵖ⁻¹。因此，不断增大的负曲率（κR→∞）可在h凸性下进一步优化欧氏最优速率。对于强g凸目标，当κR=O(1)时，匹配的上下界会得到与欧氏空间相同的指数；相反，当κR不断增大时，我们在双曲空间上构造了一个难解族，其满足Qₚ≍ₚ(1+κR)ᵖ，需要Ω̃ₚ(Qₚ^(1/p))次查询。这揭示了一个根本差异：使球定位更精准的双曲发散，却对完整的g凸类构成了信息论层面的障碍。
英文摘要
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 $Q_p=L_pR^{p-1}/μ$, 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 $Q_p \min\{1, 4/(κR) \}^{p-1}$, subject to a logarithmic localization cost. Thus growing negative curvature ($κR \rightarrow \infty$) 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 $κR=O(1)$. In contrast, with growing $κR$, we construct a hard family on the hyperbolic space with $Q_p\asymp_p(1+κR)^p$ that requires $\widetildeΩ_p(Q_p^{1/p})$ 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.
