在流形上做优化比较，解决传统方法无法处理的推荐与机器人问题。
Riemannian Dueling Optimization
- 基于流形结构设计比较优化算法，突破欧氏空间限制。
- 提出RDNGD和RDFW算法，理论证明收敛速度与光滑性相关。
- 适合需流形约束且无法投影的场景，如矩阵优化与姿态估计。
比较优化仅通过目标函数的比较查询进行优化，广泛应用于推荐系统和机器人等领域。现有方法主要针对欧氏空间中的无约束问题。本文研究在黎曼流形上的比较优化，涵盖现有算法无法解决的重要应用。我们提出黎曼对偶归一化梯度下降（RDNGD）方法，并在目标函数具有测地线L-光滑或测地线（强）凸性时建立其迭代复杂度。还提出无需投影的黎曼对偶Frank-Wolfe（RDFW）算法，适用于禁止投影的情形，并建立了其迭代与查询复杂度。通过在合成数据和真实应用上的数值实验验证了所提算法的有效性。
原文摘要 · Abstract (English)
Dueling optimization considers optimizing an objective with access to only a comparison oracle of the objective function. It finds important applications in emerging fields such as recommendation systems and robotics. Existing works on dueling optimization mainly focused on unconstrained problems in the Euclidean space. In this work, we study dueling optimization over Riemannian manifolds, which covers important applications that cannot be solved by existing dueling optimization algorithms. In particular, we propose a Riemannian Dueling Normalized Gradient Descent (RDNGD) method and establish its iteration complexity when the objective function is geodesically L-smooth or geodesically (strongly) convex. We also propose a projection-free algorithm, named Riemannian Dueling Frank-Wolfe (RDFW) method, to deal with the situation where projection is prohibited. We establish the iteration and oracle complexities for RDFW. We illustrate the effectiveness of the proposed algorithms through numerical experiments on both synthetic and real applications.
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