设$V_* : \mathbb{R}^d \to \mathbb{R}$是某个（可能是非凸的）势函数，并考虑概率测度$π\propto e^{-V_*}$。当$π$表现出多个模式时，已知基于Kullback-Leibler（KL）发散的Wasserstein梯度流的采样技术（例如Langevin Monte Carlo）在收敛速率方面受到不良影响，其中动态不能容易地在模式之间穿越。与此形成鲜明对比的是，Lu等人（2019; 2022）的工作表明，相对于Fisher-Rao（FR）几何的KL的梯度流显示出$π$的收敛速度是势函数的\textit{independent}。在这个简短的说明中，我们补充这些现有的文献中的结果，提供了一个明确的扩展$\text{KL}(ρ_t^{\text{FR}}\|π)$的$e^{-t}$，其中$(ρ_t^{\text{FR}})_{t\geq 0}$是FR梯度流的KL散度。反过来，我们能够提供一个干净的渐近收敛率，其中的老化时间是有限的保证。我们的证明是基于观察FR梯度流和模拟退火与线性缩放之间的相似性，以及关于累积量生成函数的事实。最后，我们用简单的合成实验来证明我们的理论发现确实是严密的。基于我们的数值计算，我们推测Wasserstein-Fisher-Rao梯度流的渐近收敛速度在某些情况下可能与这种展开有关。
英文摘要：
Let $V_* : \mathbb{R}^d \to \mathbb{R}$ be some (possibly non-convex) potential function, and consider the probability measure $π\propto e^{-V_*}$. When $π$ 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 $π$ 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 $\text{KL}(ρ_t^{\text{FR}}\|π)$ in terms of $e^{-t}$, where $(ρ_t^{\text{FR}})_{t\geq 0}$ 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.
On The Linear Convergence of Bregman Proximal Gradient Methods with Applications to Kullback--Leibler regression
University of Toulouse, Toulouse INP, CNRS, IRIT（图卢兹大学）；MaLGa Center, DIBRIS, University of Genoa（热那亚大学）；Laboratoire J. A. Dieudonné, Université Côte d’Azur（蔚蓝海岸大学）
