Let $f \colon \mathcal{M} \to \mathbb{R}$ be a Lipschitz and geodesically convex function defined on a $d$-dimensional Riemannian manifold $\mathcal{M}$. Does there exist a first-order deterministic algorithm which (a) uses at most $O(\mathrm{poly}(d) \log(\epsilon^{-1}))$ subgradient queries to find a point with target accuracy $\epsilon$, and (b) requires only $O(\mathrm{poly}(d))$ arithmetic operations per query? In convex optimization, the classical ellipsoid method achieves this. After detailing related work, we provide an ellipsoid-like algorithm with query complexity $O(d^2 \log^2(\epsilon^{-1}))$ and per-query complexity $O(d^2)$ for the limited case where $\mathcal{M}$ has constant curvature (hemisphere or hyperbolic space). We then detail possible approaches and corresponding obstacles for designing an ellipsoid-like method for general Riemannian manifolds.

翻译：设 $f \colon \mathcal{M} \to \mathbb{R}$ 是定义在 $d$ 维黎曼流形 $\mathcal{M}$ 上的利普希茨且测地凸函数。是否存在一种一阶确定性算法，满足：(a) 在达到目标精度 $\epsilon$ 的情况下，最多使用 $O(\mathrm{poly}(d) \log(\epsilon^{-1}))$ 次次梯度查询；(b) 每次查询仅需 $O(\mathrm{poly}(d))$ 次算术运算？在凸优化中，经典椭球法实现了这一目标。在梳理相关研究后，我们针对常曲率流形（半球面或双曲空间）这一限定情形，提出了一种类椭球算法，其查询复杂度为 $O(d^2 \log^2(\epsilon^{-1}))$，每次查询复杂度为 $O(d^2)$。随后，我们详细探讨了为一般黎曼流形设计类椭球方法的可能途径及相应困难。