提出无需依赖曲率的在线优化方法，提升复杂流形上的学习效率。
Online Optimization on Hadamard Manifolds: Curvature Independent Regret Bounds on Horospherically Convex Objectives
- 基于双曲凸性设计在线梯度下降算法
- 实现与欧氏空间相同的 $O(\sqrt{T})$ 与 $O(\log T)$ 误差界
- 适用于对称正定矩阵流形，如协方差估计
研究在哈达玛德流形上基于双曲凸性（h-convexity）的在线黎曼优化。以往工作多依赖测地凸性（g-convexity），导致误差界随流形曲率恶化。本文分析了针对h-凸和强h-凸函数的黎曼在线梯度下降，分别建立了$O(\sqrt{T})$和$O(\log T)$的误差保证，且不依赖曲率，与欧氏情形一致。通过在对称正定（SPD）矩阵流形（采用仿射不变度量）上的实验验证，研究了在线Tyler's $M$-估计和在线Fréchet均值计算，展示了h-凸性在实际中的应用价值。
原文摘要 · Abstract (English)
We study online Riemannian optimization on Hadamard manifolds under the framework of horospherical convexity (h-convexity). Prior work mostly relies on the geodesic convexity (g-convexity), leading to regret bounds scaling poorly with the manifold curvature. To address this limitation, we analyze Riemannian online gradient descent for h-convex and strongly h-convex functions and establish $O(\sqrt{T})$ and $O(\log(T))$ regret guarantees, respectively. These bounds are curvature-independent and match the results in the Euclidean setting. We validate our approach with experiments on the manifold of symmetric positive definite (SPD) matrices equipped with the affine-invariant metric. In particular, we investigate online Tyler's $M$-estimation and online Fréchet mean computation, showing the application of h-convexity in practice.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。