  Gradient descent generalises naturally to Riemannian manifolds, and to
hyperbolic $n$-space, in particular. Namely, having calculated the gradient at
the point on the manifold representing the model parameters, the updated point
is obtained by travelling along the geodesic passing in the direction of the
gradient. Some recent works employing optimisation in hyperbolic space have not
attempted this procedure, however, employing instead various approximations to
avoid a calculation that was considered to be too complicated. In this
tutorial, we demonstrate that in the hyperboloid model of hyperbolic space, the
necessary calculations to perform gradient descent are in fact
straight-forward. The advantages of the approach are then both illustrated and
quantified for the optimisation problem of computing the Fr\'echet mean (i.e.
barycentre) of points in hyperbolic space.
