The exponentially weighted average forecaster in geodesic spaces of non-positive curvature

The problem of prediction with expert advice [ Cesa-Bianchi and Lugosi, 2006 ] is a by now standard model of online learning. Traditionally studied for outcom es taking values in a vector space, less seems to be known when the outcome space is a more general metr ic space. This paper partly addresses the problem by focusing on the case of NPC spaces, i .e., geodesic metric spaces with non-positive curvature in the sense of Alexandrov. The class of NPC spaces includes many metric spaces of partic ular interest in the data sciences. Apart from Hilbert spaces, interesting examples are hyperb olic spaces [ Nickel and Kiela, 2017 ], the space of real symmetric positive-definite matrices with Log -Euclidean [ Arsigny et al., 2007 ] or Log-Cholesky [ Lin, 2019 ] Riemannian metrics and more generally all complete and sim ply connected Riemannian manifolds with non-positive sectional curvatu re.