  We study the convergence rate of Bregman gradient methods for convex
optimization in the space of measures on a $d$-dimensional manifold. Under
basic regularity assumptions, we show that the suboptimality gap at iteration
$k$ is in $O(log(k)k^{--1})$ for multiplicative updates, while it is in
$O(k^{--q/(d+q)})$ for additive updates for some $q \in {1, 2, 4}$ determined
by the structure of the objective function. Our flexible proof strategy, based
on approximation arguments, allows us to painlessly cover all Bregman Proximal
Gradient Methods (PGM) and their acceleration (APGM) under various geometries
such as the hyperbolic entropy and $L^p$ divergences. We also prove the
tightness of our analysis with matching lower bounds and confirm the
theoretical results with numerical experiments on low dimensional problems.
Note that all these optimization methods must additionally pay the
computational cost of discretization, which can be exponential in $d$.
