Livingstone Samuel, Zanella Giacomo
Department of Statistical Science University College London UK.
Department of Decision Sciences BIDSA and IGIER, Bocconi University Milan Italy.
J R Stat Soc Series B Stat Methodol. 2022 Apr;84(2):496-523. doi: 10.1111/rssb.12482. Epub 2022 Jan 11.
There is a tension between robustness and efficiency when designing Markov chain Monte Carlo (MCMC) sampling algorithms. Here we focus on robustness with respect to tuning parameters, showing that more sophisticated algorithms tend to be more sensitive to the choice of step-size parameter and less robust to heterogeneity of the distribution of interest. We characterise this phenomenon by studying the behaviour of spectral gaps as an increasingly poor step-size is chosen for the algorithm. Motivated by these considerations, we propose a novel and simple gradient-based MCMC algorithm, inspired by the classical Barker accept-reject rule, with improved robustness properties. Extensive theoretical results, dealing with robustness to tuning, geometric ergodicity and scaling with dimension, suggest that the novel scheme combines the robustness of simple schemes with the efficiency of gradient-based ones. We show numerically that this type of robustness is particularly beneficial in the context of adaptive MCMC, giving examples where our proposed scheme significantly outperforms state-of-the-art alternatives.
在设计马尔可夫链蒙特卡罗（MCMC）采样算法时，稳健性和效率之间存在着一种张力。在这里，我们关注关于调优参数的稳健性，表明更复杂的算法往往对步长参数的选择更为敏感，并且对目标分布的异质性稳健性较差。我们通过研究当为算法选择越来越差的步长时谱隙的行为来刻画这一现象。基于这些考虑，我们受经典的巴克接受 - 拒绝规则启发，提出了一种新颖且简单的基于梯度的MCMC算法，其具有改进的稳健性。大量关于调优稳健性、几何遍历性和维度缩放的理论结果表明，新方案将简单方案的稳健性与基于梯度方案的效率结合起来。我们通过数值表明，这种类型的稳健性在自适应MCMC的背景下特别有益，并给出了我们提出的方案显著优于现有最佳替代方案的示例。
