A note on the unique properties of the Kullback--Leibler divergence for sampling via gradient flows

We consider the problem of sampling from a probability distribution $\pi$ which admits a density w.r.t. a dominating measure. It is well known that this can be written as an optimisation problem over the space of probability distributions in which we aim to minimise a divergence from $\pi$. The optimisation problem is normally solved through gradient flows in the space of probability distributions with an appropriate metric. We show that the Kullback–Leibler divergence is the only divergence in the family of Bregman divergences whose gradient flow w.r.t. many popular metrics does not require knowledge of the normalising constant of $\pi$.