Hypocoercivity of piecewise deterministic Markov process-Monte Carlo
Hypocoercivity of piecewise deterministic Markov process-Monte Carlo
复制标题
DOI：
10.1214/20-aap1653
复制
发表时间：
2018-08
期刊：
影响因子：
--
通讯作者：
C. Andrieu;Alain Durmus;Nikolas Nusken;Julien Roussel
中科院分区：
文献类型：
--
作者：
C. Andrieu;Alain Durmus;Nikolas Nusken;Julien Roussel
在这项工作中，我们证明了最近在马尔可夫过程蒙特卡罗方法的背景下提出的一类广泛的分段确定性马尔可夫过程的{L}^2$指数收敛，特别地包括随机哈密顿蒙特卡罗、Zig-Zag过程和弹跳粒子采样器。这类过程的生成元对称部分的核是非平凡的，我们遵循(Dolbeault等人，2009,2015)最近提出的想法，在相当一般和统一的设置中开发了一个严格的亚矫顽力框架，同时根据动力学参数推导出所涉及的常数的易于处理的估计。作为副产品，我们刻画了这些算法关于问题类的维度的标度性质，从而提供了一些理论证据来支持它们的实际意义。
In this work, we establish $\mathrm{L}^2$-exponential convergence for a broad class of Piecewise Deterministic Markov Processes recently proposed in the context of Markov Process Monte Carlo methods and covering in particular the Randomized Hamiltonian Monte Carlo, the Zig-Zag process and the Bouncy Particle Sampler. The kernel of the symmetric part of the generator of such processes is non-trivial, and we follow the ideas recently introduced by (Dolbeault et al., 2009, 2015) to develop a rigorous framework for hypocoercivity in a fairly general and unifying set-up, while deriving tractable estimates of the constants involved in terms of the parameters of the dynamics. As a by-product we characterize the scaling properties of these algorithms with respect to the dimension of classes of problems, therefore providing some theoretical evidence to support their practical relevance.