In this paper, we consider the sequential decision problem where the goal is to minimize the general dynamic regret on a complete Riemannian manifold. The task of offline optimization on such a domain, also known as a geodesic metric space, has recently received significant attention. The online setting has received significantly less attention, and it has remained an open question whether the body of results that hold in the Euclidean setting can be transplanted into the land of Riemannian manifolds where new challenges (e.g., curvature) come into play. In this paper, we show how to get optimistic regret bound on manifolds with non-positive curvature whenever improper learning is allowed and propose an array of adaptive no-regret algorithms. To the best of our knowledge, this is the first work that considers general dynamic regret and develops "optimistic" online learning algorithms which can be employed on geodesic metric spaces.

翻译：在本文中，我们考虑在完全黎曼流形上最小化一般动态遗憾的序贯决策问题。此类域（也称为测地度量空间）上的离线优化任务近期受到广泛关注，而在线设置的研究则显著较少。一个悬而未决的问题是：欧几里得设置下成立的一系列结论能否移植到具有新挑战（如曲率）的黎曼流形领域？本文展示了在允许非正曲率流形上进行非正当学习时如何获得乐观遗憾界，并提出了一系列自适应无遗憾算法。据我们所知，这是首个考虑一般动态遗憾并开发可用于测地度量空间的"乐观"在线学习算法的工作。