English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Let $V_* : \mathbb{R}^d \to \mathbb{R}$ be some (possibly non-convex)potential function, and consider the probability measure $\pi \proptoe^{-V_*}$. When $\pi$ exhibits multiple modes, it is known that samplingtechniques based on Wasserstein gradient flows of the Kullback-Leibler (KL)divergence (e.g. Langevin Monte Carlo) suffer poorly in the rate ofconvergence, where the dynamics are unable to easily traverse between modes. Instark contrast, the work of Lu et al. (2019; 2022) has shown that the gradientflow of the KL with respect to the Fisher-Rao (FR) geometry exhibits aconvergence rate to $\pi$ is that \textit{independent} of the potentialfunction. In this short note, we complement these existing results in theliterature 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 theburn-in time is guaranteed to be finite. Our proof is based on observing asimilarity between FR gradient flows and simulated annealing with linearscaling, and facts about cumulant generating functions. We conclude with simplesynthetic experiments that demonstrate our theoretical findings are indeedtight. Based on our numerics, we conjecture that the asymptotic rates ofconvergence for Wasserstein-Fisher-Rao gradient flows are possibly related tothis expansion in some cases.

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