The Metropolis-Adjusted Langevin Algorithm (MALA) is a Markov Chain Monte Carlo method which creates a Markov chain reversible with respect to a given target distribution, \(\pi ^N\), with Lebesgue density on \({\mathbb {R}}^N\); it can hence be used to approximately sample the target distribution. When the dimension N is large a key question is to determine the computational cost of the algorithm as a function of N. The measure of efficiency that we consider in this paper is the expected squared jumping distance (ESJD), introduced in Roberts et al. (Ann Appl Probab 7(1):110–120, 1997). To determine how the cost of the algorithm (in terms of ESJD) increases with dimension N, we adopt the widely used approach of deriving a diffusion limit for the Markov chain produced by the MALA algorithm. We study this problem for a class of target measures which is not in product form and we address the situation of practical relevance in which the algorithm is started out of stationarity. We thereby significantly extend previous works which consider either measures of product form, when the Markov chain is started out of stationarity, or non-product measures (defined via a density with respect to a Gaussian), when the Markov chain is started in stationarity. In order to work in this non-stationary and non-product setting, significant new analysis is required. In particular, our diffusion limit comprises a stochastic PDE coupled to a scalar ordinary differential equation which gives a measure of how far from stationarity the process is. The family of non-product target measures that we consider in this paper are found from discretization of a measure on an infinite dimensional Hilbert space; the discretised measure is defined by its density with respect to a Gaussian random field. The results of this paper demonstrate that, in the non-stationary regime, the cost of the algorithm is of \({{\mathcal {O}}}(N^{1/2})\) in contrast to the stationary regime, where it is of \({{\mathcal {O}}}(N^{1/3})\).

中文翻译：

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MALA算法的非平稳阶段

大都会调整的Langevin算法（MALA）是一种马尔可夫链蒙特卡罗方法，它针对给定的目标分布\（\ pi ^ N \）创建了可逆的马尔可夫链，其Lebesgue密度为\（{\ mathbb {R }} ^ N \） ; 因此，它可以用于近似采样目标分布。当维数N大时，关键问题是确定算法的计算成本作为N的函数。我们在本文中考虑的效率度量是在Roberts等人中引入的预期平方跳距（ESJD）。（Ann Appl Probab 7（1）：110-120，1997）。确定算法的成本（就ESJD而言）如何随维度N增加，我们采用了广泛使用的方法来推导MALA算法产生的马尔可夫链的扩散极限。我们对一类目标的措施研究这个问题，这是不以产品形式提供服务，我们解决了实用性强的问题，即算法出于平稳性而开始。因此，我们极大地扩展了先前的工作，当马尔可夫链不平稳时开始考虑产品形式的度量，或者当马尔可夫链平稳时开始考虑非产品度量（通过相对于高斯的密度定义）。为了在这种非平稳和非产品环境中工作，需要进行大量的新分析。尤其是，我们的扩散极限包括与标量常微分方程耦合的随机PDE，它可以度量过程与平稳性之间的距离。我们从无穷维希尔伯特空间上的一个度量离散化中发现了我们在本文中考虑的非产品目标度量族。离散量度由其相对于高斯随机场的密度定义。本文的结果表明，在非平稳状态下，该算法的成本为\（{{\ mathcal {O}}}（N ^ {1/2}）\）与固定状态相反，后者是\（{{\ mathcal {O}}}（N ^ {1 / 3}）\）。