[论文解读] Diffusion Limit For The Random Walk Metropolis Algorithm Out Of stationarity

Juan Kuntz, Michela Ottobre+1 more|arXiv (Cornell University)|May 19, 2014

Markov Chains and Monte Carlo Methods参考文献 13被引用 6

一句话总结

该论文在高维情形下，针对非平稳初始化和非独立同分布目标测度的随机漫步梅特罗波利斯（RWM）算法，建立了扩散极限与最优缩放。证明了最优提议方差的缩放为 $O(N^{-1})$，即使在不假设平稳性或乘积形式目标测度的情况下，仍可实现扩散极限，并使探索不变测度的计算成本为 $O(N)$——将先前结果推广至贝叶斯非参数推断和条件扩散过程更现实的设定中。

ABSTRACT

The Random Walk Metropolis (RWM) algorithm is a Metropolis- Hastings MCMC algorithm designed to sample from a given target distribution πwith Lebesgue density on R^N. RWM constructs a Markov chain by randomly proposing a new position (the "proposal move"), which is then accepted or rejected according to a rule which makes the chain reversible with respect to π. When the dimension N is large a key question is to determine the optimal scaling with N of the proposal variance: if the proposal variance is too large, the algorithm will reject the proposed moves too often; if it is too small, the algorithm will explore the state space too slowly. Determining the optimal scaling of the proposal variance gives a measure of the cost of the algorithm as well. One approach to tackle this issue, which we adopt here, is to derive diffusion limits for the algorithm. Such an approach has been proposed in the seminal papers [RGG97, RR98]; in particular in [RGG97] the authors derive a diffusion limit for the RWM algorithm under the two following assumptions: i) the algorithm is started in stationarity; ii) the target measure $π$ is in product form. The present paper considers the situation of practical interest in which both assumptions i) and ii) are removed. That is a) we study the case (which occurs in practice) in which the algorithm is started out of stationarity and b) we consider target measures which are in non-product form. The target measures that we consider arise in Bayesian nonparametric statistics and in the study of conditioned diffusions. We prove that, out of stationarity, the optimal scaling for the proposal variance is O(N), as it is in stationarity. Notice that the optimal scaling in and out of stationatity need not be the same in general, and indeed they differ e.g. in the case of the MALA algorithm [KOS16].

研究动机与目标

• 分析在目标测度非乘积形式且链从非平稳状态开始的高维设定下RWM算法的行为。
• 在这些更现实的条件下，确定提议方差相对于维度 $N$ 的最优缩放。
• 在非平稳、非独立同分布情形下，为RWM算法建立扩散极限，扩展经典结果。
• 量化算法的计算成本，包括混合时间与探索效率。
• 为理解RWM在贝叶斯非参数统计与无限维模型中的性能，提供一个严格框架。

提出的方法

• 将目标测度形式化为无限维希尔伯特空间上高斯测度的扰动，其密度满足 $\frac{d\pi}{d\pi_0} \propto \exp(-\Psi)$。
• 将RWM算法建模为具有对称随机游走提议和梅特罗波利斯接受概率 $\alpha(x,y) = \min\left\{1, \frac{\pi(y)}{\pi(x)}\right\}$ 的马尔可夫链。
• 采用与维度相关的提议方差缩放 $\sigma_N^2 = \frac{\ell^2}{N}$，以在 $N \to \infty$ 时导出扩散极限。
• 应用泛函中心极限定理技术与鞅型中心极限定理，证明其收敛至扩散过程。
• 通过加权矩估计与指数矩控制，建立归一化过程的矩界与一致可积性。
• 利用对势函数 $\Psi$ 与协方差算子 $\mathcal{C}$ 的假设，确保极限动力学的正则性与可积性。

实验结果

研究问题

• RQ1 当链从非平稳状态开始时，RWM算法的提议方差最优缩放为何？
• RQ2 当目标测度非乘积形式时，RWM的扩散极限是否仍然成立？
• RQ3 在非平稳初始化下，RWM算法的计算成本如何随维度 $N$ 变化？
• RQ4 在非独立同分布目标分布下，经典 $O(N^{-1})$ 提议方差最优缩放是否仍可保持？
• RQ5 这些结果对贝叶斯非参数推断与无限维模型中的MCMC方法有何影响？

主要发现

• 即使链从非平稳状态开始且目标测度非乘积形式，RWM算法的最优提议方差缩放仍为 $O(N^{-1})$。
• 在非平稳、非独立同分布设定下，为RWM算法建立了扩散极限，确认在 $O(N^{-1})$ 缩放下收敛至极限SDE。
• 达到并探索不变测度的计算成本为 $O(N)$，与平稳情形下的成本一致。
• 该结果在目标测度为希尔伯特空间上具有迹类协方差算子的高斯测度的扰动时成立。
• 证明依赖于通过加权矩估计与指数矩控制实现的归一化过程的矩界与一致可积性。
• 关键技术贡献在于将经典扩散极限框架扩展至无平稳性或乘积结构的设定，而这类设定在实际贝叶斯推断中极为常见。