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：我们研究流形上测地凸（g-凸）优化的查询复杂度。为隔离流形曲率的影响，我们主要关注双曲空间。在多种设定下（光滑或非光滑；强g-凸与否；高维或低维），已知的上界随曲率增大而恶化。我们自然要问：这种恶化是必然的，还是人为假象？针对多种此类设定，我们提出首批下界结果，确实验证（负）曲率对复杂度具有不利影响。为此，我们基于近期关于光滑强g-凸优化的下界结果（Hamilton 和 Moitra, 2021；Criscitiello 和 Boumal, 2022）展开研究。通过多种技术手段，我们还获得了反映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，并猜想最优下界的形式——通过为包含次梯度下降法的算法类别匹配下界，以及为相关博弈过程建立下界来佐证。最后，为厘清证明下界的难点，我们研究了负曲率如何影响（有时甚至阻碍）g-凸函数的内插性质。