Open Problem: Polynomial linearly-convergent method for geodesically convex optimization?

TL;DR

This paper investigates the existence of a polynomially efficient, geodesically convex optimization method on Riemannian manifolds, providing an ellipsoid-like algorithm for constant curvature spaces and discussing challenges for general manifolds.

Contribution

It introduces an ellipsoid-like algorithm with polynomial query complexity for constant curvature manifolds and explores the obstacles in extending it to general Riemannian manifolds.

Findings

Algorithm with $O(d^2 \, \log^2(\epsilon^{-1}))$ query complexity for constant curvature spaces

Per-query complexity of $O(d^2)$ in these spaces

Discussion of challenges in generalizing to arbitrary Riemannian manifolds

Abstract

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.