Optimal Scaling and Diffusion Limits for the Langevin Algorithm in High Dimensions
Optimal Scaling and Diffusion Limits for the Langevin Algorithm in High Dimensions
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DOI：
10.1214/11-aap828
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发表时间：
2011-03
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影响因子：
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通讯作者：
N. Pillai;A. Stuart;Alexandre H. Thi'ery
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文献类型：
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作者：
N. Pillai;A. Stuart;Alexandre H. Thi'ery
Metropolis-adjusted Langevin（MALA）算法是一种采样算法，其通过结合关于目标密度的对数的梯度的信息来进行局部移动。本文研究了无穷维Hilbert空间上一类自然目标测度上MALA的有效性。这些自然的措施有密度相对于一个高斯随机场的措施，并出现在许多应用，如贝叶斯非参数统计和理论的条件扩散。我们证明，开始在平稳性，一个适当的插值和缩放版本的马尔可夫链对应的MALA收敛到一个无限维的扩散过程。我们的研究结果意味着，在平稳性，MALA算法应用到一个N维近似的目标将采取$\mathcal{O}（N^{1/3}）$的步骤来探索不变的措施，比较有利的随机行走大都会这是最近被证明需要$\mathcal{O}（N）$步骤时，适用于同一类问题。
The Metropolis-adjusted Langevin (MALA) algorithm is a sampling algorithm which makes local moves by incorporating information about the gradient of the logarithm of the target density. In this paper we study the efficiency of MALA on a natural class of target measures supported on an infinite dimensional Hilbert space. These natural measures have density with respect to a Gaussian random field measure and arise in many applications such as Bayesian nonparametric statistics and the theory of conditioned diffusions. We prove that, started in stationarity, a suitably interpolated and scaled version of the Markov chain corresponding to MALA converges to an infinite dimensional diffusion process. Our results imply that, in stationarity, the MALA algorithm applied to an N-dimensional approximation of the target will take $\mathcal{O}(N^{1/3})$ steps to explore the invariant measure, comparing favorably with the Random Walk Metropolis which was recently shown to require $\mathcal{O}(N)$ steps when applied to the same class of problems.