利用斯特劳森扰动加速趋近平衡并减少渐近方差的 Langevin 动力学
Accelerated convergence to equilibrium and reduced asymptotic variance for Langevin dynamics using Stratonovich perturbations
AI总结：
本文提出了一种新的采样方法，通过在标准过阻尼 Langevin 动力学中引入适当的斯特劳森扰动，保持原系统的不变测度，从而实现加速趋近平衡和减少渐近方差，得到一种计算上有优势的采样算法。新的扰动 Langevin 动力学相对于目标概率分布是可逆的，因此不具有非可逆 Langevin 采样器的缺点，同时保留了其在加速收敛和减少渐近方差方面的所有优点。特别是，动力学的可逆性确保了没有振荡的暂态行为。
摘要：
在本文中，我们提出了一种新的方法，用于从概率分布中采样，可能是在高维空间中。通过在标准过阻尼 Langevin 动力学中引入适当的斯特劳森扰动，以保持原系统的不变测度，我们证明可以实现加速趋近平衡和减少渐近方差，从而得到一种在计算上更有优势的采样算法。新的扰动 Langevin 动力学相对于目标概率分布是可逆的，因此不具有非可逆 Langevin 采样器的缺点，如[C.-R. Hwang, S.-Y. Hwang-Ma, and S.-J. Sheu, Ann. Appl. Probab. 1993]中引入的以及在例如[T. Lelievre, F. Nier, and G. A. Pavliotis J. Stat. Phys., 2013]和[A. B. Duncan, T. Lelièvre, and G. A. Pavliotis J. Stat. Phys., 2016]中研究的那些缺点，同时保留了在加速收敛和减少渐近方差方面的所有优点。特别是，动力学的可逆性确保了没有振荡的暂态行为。
英文摘要：
In this paper we propose a new approach for sampling from probability measures in, possibly, high dimensional spaces. By perturbing the standard overdamped Langevin dynamics by a suitable Stratonovich perturbation that preserves the invariant measure of the original system, we show that accelerated convergence to equilibrium and reduced asymptotic variance can be achieved, leading, thus, to a computationally advantageous sampling algorithm. The new perturbed Langevin dynamics is reversible with respect to the target probability measure and, consequently, does not suffer from the drawbacks of the nonreversible Langevin samplers that were introduced in~[C.-R. Hwang, S.-Y. Hwang-Ma, and S.-J. Sheu, Ann. Appl. Probab. 1993] and studied in, e.g. [T. Lelievre, F. Nier, and G. A. Pavliotis J. Stat. Phys., 2013] and [A. B. Duncan, T. Lelièvre, and G. A. Pavliotis J. Stat. Phys., 2016], while retaining all of their advantages in terms of accelerated convergence and reduced asymptotic variance. In particular, the reversibility of the dynamics ensures that there is no oscillatory transient behaviour. The improved performance of the proposed methodology, in comparison to the standard overdamped Langevin dynamics and its nonreversible perturbation, is illustrated on an example of sampling from a two-dimensional warped Gaussian target distribution.
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