Manifold Markov chain Monte Carlo algorithms have been introduced to sample more effectively from challenging target densities exhibiting multiple modes or strong correlations. Such algorithms exploit the local geometry of the parameter space, thus enabling chains to achieve a faster convergence rate when measured in number of steps. However, acquiring local geometric information can often increase computational complexity per step to the extent that sampling from high-dimensional targets becomes inefficient in terms of total computational time. This paper analyzes the computational complexity of manifold Langevin Monte Carlo and proposes a geometric adaptive Monte Carlo sampler aimed at balancing the benefits of exploiting local geometry with computational cost to achieve a high effective sample size for a given computational cost. The suggested sampler is a discrete-time stochastic process in random environment. The random environment allows to switch between local geometric and adaptive proposal kernels with the help of a schedule. An exponential schedule is put forward that enables more frequent use of geometric information in early transient phases of the chain, while saving computational time in late stationary phases. The average complexity can be manually set depending on the need for geometric exploitation posed by the underlying model.
翻译:引入了 Manifold Markov 链 Monte Carlo 算法,以便更有效地从具有多种模式或密切关联的具有挑战性的目标密度中取样。这种算法利用了参数空间的本地几何测量法,从而使链子能够在按步数测量时实现更快的趋同率。然而,获取本地几何信息往往会提高每步的计算复杂性,以至于从高维目标取样在总计算时间方面变得效率低下。本文分析了多重Langevin Monte Carlo 的计算复杂性,并提出了一个几何适应性蒙特卡洛 采样器,旨在平衡利用本地几何测量法的好处和计算成本之间的平衡,以达到特定计算成本的高有效采样大小。建议的采样器是一种随机环境中的离开时间分解过程。随机环境允许在本地几何和适应性建议内核内核在总计算时间上变得效率低下。 一份指数表将提前,以便能够在链的早期中转阶段更频繁地使用几何信息,同时将计算时间保存在较晚的固定阶段。平均复杂程度可以根据基本模型的几何开发需要而手工设定。