  Let $V_* : \mathbb{R}^d \to \mathbb{R}$ be some (possibly non-convex)
potential function, and consider the probability measure $\pi \propto
e^{-V_*}$. When $\pi$ exhibits multiple modes, it is known that sampling
techniques based on Wasserstein gradient flows of the Kullback-Leibler (KL)
divergence (e.g. Langevin Monte Carlo) suffer poorly in the rate of
convergence, where the dynamics are unable to easily traverse between modes. In
stark contrast, the work of Lu et al. (2019; 2022) has shown that the gradient
flow of the KL with respect to the Fisher-Rao (FR) geometry exhibits a
convergence rate to $\pi$ is that \textit{independent} of the potential
function. In this short note, we complement these existing results in the
literature by providing an explicit expansion of
$\text{KL}(\rho_t^{\text{FR}}\|\pi)$ in terms of $e^{-t}$, where
$(\rho_t^{\text{FR}})_{t\geq 0}$ is the FR gradient flow of the KL divergence.
In turn, we are able to provide a clean asymptotic convergence rate, where the
burn-in time is guaranteed to be finite. Our proof is based on observing a
similarity between FR gradient flows and simulated annealing with linear
scaling, and facts about cumulant generating functions. We conclude with simple
synthetic experiments that demonstrate our theoretical findings are indeed
tight. Based on our numerics, we conjecture that the asymptotic rates of
convergence for Wasserstein-Fisher-Rao gradient flows are possibly related to
this expansion in some cases.
