Optimal design of the Barker proposal and other locally balanced Metropolis–Hastings algorithms

Summary We study the class of first-order locally balanced Metropolis–Hastings algorithms introduced in Livingstone & Zanella (2022). To choose a specific algorithm within the class, the user must select a balancing function $g:{\mathbb{R}}_+ \to {\mathbb{R}}_+$ satisfying $g(t) = tg(1/t)$ and a noise distribution for the proposal increment. Popular choices within the class are the Metropolis-adjusted Langevin algorithm and the recently introduced Barker proposal. We first establish a general limiting optimal acceptance rate of 57 $\%$ and scaling of $n^{-1/3}$ , as the dimension $n$ tends to infinity among all members of the class under mild smoothness assumptions on $g$ and when the target distribution for the algorithm is of product form. In particular, we obtain an explicit expression for the asymptotic efficiency of an arbitrary algorithm in the class, as measured by expected squared jumping distance. We then consider how to optimize this expression under various constraints. We derive an optimal choice of noise distribution for the Barker proposal, an optimal choice of balancing function under a Gaussian noise distribution, and an optimal choice of first-order locally balanced algorithm among the entire class, which turns out to depend on the specific target distribution. Numerical simulations confirm our theoretical findings, and in particular, show that a bimodal choice of noise distribution in the Barker proposal gives rise to a practical algorithm that is consistently more efficient than the original Gaussian version.Summary We study the class of first-order locally balanced Metropolis–Hastings algorithms introduced in Livingstone & Zanella (2022). To choose a specific algorithm within the class, the user must select a balancing function $g:{\mathbb{R}}_+ \to {\mathbb{R}}_+$ satisfying $g(t) = tg(1/t)$ and a noise distribution for the proposal increment. Popular choices within the class are the Metropolis-adjusted Langevin algorithm and the recently introduced Barker proposal. We first establish a general limiting optimal acceptance rate of 57$\%$ and scaling of $n^{-1/3}$, as the dimension $n$ tends to infinity among all members of the class under mild smoothness assumptions on $g$ and when the target distribution for the algorithm is of product form. In particular, we obtain an explicit expression for the asymptotic efficiency of an arbitrary algorithm in the class, as measured by expected squared jumping distance. We then consider how to optimize this expression under various constraints. We derive an optimal choice of noise distribution for the Barker proposal, an optimal choice of balancing function under a Gaussian noise distribution, and an optimal choice of first-order locally balanced algorithm among the entire class, which turns out to depend on the specific target distribution. Numerical simulations confirm our theoretical findings, and in particular, show that a bimodal choice of noise distribution in the Barker proposal gives rise to a practical algorithm that is consistently more efficient than the original Gaussian version.

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笔记与分组

分类依据与纠错

Optimal design of the Barker proposal and other locally balanced Metropolis–Hastings algorithms

代码、数据与补充材料

Barker 提议与局部平衡 Metropolis–Hastings 算法的最优设计

原文摘要

同方向的基础文献

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