发表机构
Peking University（北京大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对双曲空间测地凸优化，建立匹配Zhang和Sra（2016）上界的确定性一阶优化下界，消除早期限制，明确查询复杂度与几何势消耗的关系。
AI 中文摘要
我们针对双曲空间上全局Lipschitz测地凸函数的确定性一阶优化建立了匹配下界。全局极小值点位于距已知点距离r范围内，每次查询返回精确函数值与选定黎曼次梯度，目标函数间隙为εLr，其中L为Lipschitz常数。对于截面曲率-k²<0，令ρ=kr，ζ=ρ/tanhρ。最优最坏情况查询复杂度Q*（在维度上均匀取值）满足：对所有ρ>0及0<ε≤1/64，Q*(ρ,ε)=Θ(ζε⁻²)。该新下界覆盖任意自适应确定性查询与任意输出，匹配Zhang和Sra（2016）的投影次梯度上界速率，消除了早期匹配下界的测地跨度与历史半空间限制。构造从带边界的凸载体扩展径向源，小残差切割选择极小值候选，而精确双曲持久性不等式确保所有历史接触在未来载体增长下保持有效。因此，一个固定全局凸目标与一个固定次梯度选择可同时实现所有精确回复，每次查询消耗几何势的O(ε²)，其可用预算为Θ(ζ)，在查询视界维度线性的情况下得到乘积下界。
英文摘要
We establish matching lower bounds for deterministic first-order optimization of globally Lipschitz geodesically convex functions on hyperbolic space. A global minimizer lies within distance $r$ of a known point, each query returns an exact function value and a selected Riemannian subgradient, and the target objective gap is $\varepsilon Lr$, where $L$ is the Lipschitz constant. For sectional curvature $-k^2<0$, let $ρ=kr$ and $ζ=ρ/\tanhρ$. The optimal worst-case query complexity $Q^\star$, taken uniformly over dimensions, satisfies $Q^\star(ρ,\varepsilon)=Θ(ζ\varepsilon^{-2})$ for every $ρ>0$ and $0<\varepsilon\le1/64$. The new lower bound covers arbitrary adaptive deterministic queries and arbitrary outputs, matching the projected-subgradient upper rate of Zhang and Sra (2016). It removes the geodesic-span and historical-halfspace restrictions of earlier matching lower bounds. The construction extends a radial source from a convex carrier with boundary. A small residual cut selects minimizer candidates while an exact hyperbolic persistence inequality keeps every historical contact valid under future carrier growth. One fixed globally convex objective and one fixed subgradient selection consequently realize all exact replies simultaneously. Each query consumes $O(\varepsilon^2)$ of a geometric potential whose available budget is $Θ(ζ)$, yielding the product lower bound in a dimension linear in the query horizon.
