Quantitative contraction rates for Markov chains on general state spaces
Quantitative contraction rates for Markov chains on general state spaces
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DOI：
10.1214/19-ejp287
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发表时间：
2018-08
影响因子：
1.4
通讯作者：
A. Eberle;Mateusz B. Majka
中科院分区：
文献类型：
--
作者：
A. Eberle;Mateusz B. Majka
本文研究了Kantorovich（$L^1$ Wasserstein）距离中Markov转移核的收缩系数的量化问题。对于扩散过程，收缩率的相对精确的定量界限最近已通过结合适当的耦合精心设计的康托洛维奇距离。在本文中，我们部分结转这种方法从扩散到马尔可夫链。我们推导出马尔可夫链的收缩率的一般状态空间，是强大的，如果动态是由小的局部移动为主的定量下界。对于具有各向同性转移核的$\mathbb{R^d}$上的马尔可夫链，可以有效地使用一般界以及结合最大耦合和反射耦合的耦合。结果适用于欧拉离散的随机微分方程的非全局压缩漂移，和大都会调整Langevin算法的抽样从一类概率措施的高维状态空间，不是全球性的日志凹。
We investigate the problem of quantifying contraction coefficients of Markov transition kernels in Kantorovich ($L^1$ Wasserstein) distances. For diffusion processes, relatively precise quantitative bounds on contraction rates have recently been derived by combining appropriate couplings with carefully designed Kantorovich distances. In this paper, we partially carry over this approach from diffusions to Markov chains. We derive quantitative lower bounds on contraction rates for Markov chains on general state spaces that are powerful if the dynamics is dominated by small local moves. For Markov chains on $\mathbb{R^d}$ with isotropic transition kernels, the general bounds can be used efficiently together with a coupling that combines maximal and reflection coupling. The results are applied to Euler discretizations of stochastic differential equations with non-globally contractive drifts, and to the Metropolis adjusted Langevin algorithm for sampling from a class of probability measures on high dimensional state spaces that are not globally log-concave.