An Explicit Expansion of the Kullback-Leibler Divergence along its Fisher-Rao Gradient Flow

Published: 25 Jun 2023, Last Modified: 25 Jun 2023Accepted by TMLREveryoneRevisionsBibTeX
Abstract: 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 numerical findings, we conjecture that the asymptotic rates of convergence for Wasserstein-Fisher-Rao gradient flows are possibly related to this expansion in some cases.
License: Creative Commons Attribution 4.0 International (CC BY 4.0)
Submission Length: Regular submission (no more than 12 pages of main content)
Changes Since Last Submission: Please see the supplementary file for a diff version of the PDF
Supplementary Material: pdf
Assigned Action Editor: ~Murat_A_Erdogdu1
Submission Number: 894