We study the query complexity of geodesically convex (g-convex) optimization on a manifold. To isolate the effect of that manifold's curvature, we primarily focus on hyperbolic spaces. In a variety of settings (smooth or not; strongly g-convex or not; high- or low-dimensional), known upper bounds worsen with curvature. It is natural to ask whether this is warranted, or an artifact. For many such settings, we propose a first set of lower bounds which indeed confirm that (negative) curvature is detrimental to complexity. To do so, we build on recent lower bounds (Hamilton and Moitra, 2021; Criscitiello and Boumal, 2022) for the particular case of smooth, strongly g-convex optimization. Using a number of techniques, we also secure lower bounds which capture dependence on condition number and optimality gap, which was not previously the case. We suspect these bounds are not optimal. We conjecture optimal ones, and support them with a matching lower bound for a class of algorithms which includes subgradient descent, and a lower bound for a related game. Lastly, to pinpoint the difficulty of proving lower bounds, we study how negative curvature influences (and sometimes obstructs) interpolation with g-convex functions.

翻译：我们研究了流形上测地凸（g-convex）优化的查询复杂度。为隔离流形曲率的影响，我们主要聚焦于双曲空间。在多种设定下（光滑或非光滑；强测地凸或非强测地凸；高维或低维），已知上界随曲率增大而恶化。一个自然的问题是：这种恶化是必然现象还是人为假象？对于许多此类设定，我们提出了首批下界，证实（负）曲率确实会损害复杂度。为此，我们基于光滑强测地凸优化的特例下界（Hamilton 和 Moitra, 2021；Criscitiello 和 Boumal, 2022）展开研究。借助多种技术，我们还获得了捕捉条件数和最优性间隙依赖关系的下界，这是此前未能实现的。我们推测这些下界并非最优，并提出了猜想性最优下界，通过针对包含次梯度下降法的算法类匹配下界及相关博弈的下界加以支撑。最后，为明确证明下界的困难所在，我们研究了负曲率如何影响（有时阻碍）g-凸函数的插值问题。