论文标题
随机步行大都市算法的可扩展耦合
Scalable couplings for the random walk Metropolis algorithm
论文作者
论文摘要
最近，人们对马尔可夫链蒙特卡洛算法的耦合方法引起了人们的兴趣：它们促进了收敛量化和公正的估计，同时利用了令人尴尬的平行计算能力。由这些动机，我们考虑了随机步行大都市算法的耦合的设计和分析，该算法随着目标度量的维度很好地扩展。从方法上讲，我们引入了对同步耦合的低级别修饰，该耦合在标准高维渐近性方案中被证明是最佳缩度的。我们揭示了反射耦合的缺点，在撰写本文时的最新技术状态，我们提出了一种减轻问题的修改。我们的分析将差距桥接到了最佳缩放文献中，并建立了可能具有独立关注的渐近优化框架。我们说明了我们提出的耦合的适用性，以及通过各种数值实验来扩展思想的潜力。
There has been a recent surge of interest in coupling methods for Markov chain Monte Carlo algorithms: they facilitate convergence quantification and unbiased estimation, while exploiting embarrassingly parallel computing capabilities. Motivated by these, we consider the design and analysis of couplings of the random walk Metropolis algorithm which scale well with the dimension of the target measure. Methodologically, we introduce a low-rank modification of the synchronous coupling that is provably optimally contractive in standard high-dimensional asymptotic regimes. We expose a shortcoming of the reflection coupling, the state of the art at the time of writing, and we propose a modification which mitigates the issue. Our analysis bridges the gap to the optimal scaling literature and builds a framework of asymptotic optimality which may be of independent interest. We illustrate the applicability of our proposed couplings, and the potential for extending our ideas, with various numerical experiments.
