  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.
