论文标题
在低温限制中分析和优化某些平行的蒙特卡洛方法
Analysis and optimization of certain parallel Monte Carlo methods in the low temperature limit
论文作者
论文摘要
亚竞争性是对马尔可夫链蒙特卡洛方法的巨大挑战。在本文中，我们介绍了算法设计的方法来应对这一挑战。我们认为的设计问题是无限交换方案的温度选择，这是当交换速率倾向于无穷大时获得的广泛使用的平行回火方案的极限。我们使用最近开发的工具来分析小噪声扩散的经验度量，以将差异问题转化为明确的优化问题。我们对优化问题的首次分析是在双井模型的设置中，它表明温度比的最佳选择是几何序列，除了可能的最高温度。在同一情况下，我们确定了降低方差的两种不同来源，并显示他们的竞争如何决定最佳的最高温度。在一般的多孔设置中，我们证明温度比的纯几何序列始终是最佳的，其性能差距在温度的数量中几何衰减。
Metastability is a formidable challenge to Markov chain Monte Carlo methods. In this paper we present methods for algorithm design to meet this challenge. The design problem we consider is temperature selection for the infinite swapping scheme, which is the limit of the widely used parallel tempering scheme obtained when the swap rate tends to infinity. We use a recently developed tool for the analysis of the empirical measure of a small noise diffusion to transform the variance reduction problem into an explicit optimization problem. Our first analysis of the optimization problem is in the setting of a double well model, and it shows that the optimal selection of temperature ratios is a geometric sequence except possibly the highest temperature. In the same setting we identify two different sources of variance reduction, and show how their competition determines the optimal highest temperature. In the general multi-well setting we prove that a pure geometric sequence of temperature ratios is always nearly optimal, with a performance gap that decays geometrically in the number of temperatures.
