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 status quo at 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.

翻译：近期，马尔可夫链蒙特卡洛算法中的耦合方法引起了广泛关注：它们能够促进收敛量化与无偏估计，同时利用令人尴尬的并行计算能力。受此启发，我们研究面向随机游走Metropolis算法的耦合设计与分析，要求其能够随目标测度维度扩展。方法论上，我们引入同步耦合的低秩修正，该修正方法在标准高维渐近框架下被证明具有最优压缩性能。我们揭示了当前主流反射耦合的缺陷，并提出缓解该问题的改进方案。本文分析填补了与最优缩放研究的理论鸿沟，构建了可能具有独立研究价值的渐近最优性分析框架。通过多种数值实验，我们展示了所提耦合方法的适用性及思想扩展潜力。