Analysis and Optimization of Certain Parallel Monte Carlo Methods in the Low Temperature Limit
Analysis and Optimization of Certain Parallel Monte Carlo Methods in the Low Temperature Limit
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DOI：
10.1137/21m1402029
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发表时间：
2020-11
期刊：
影响因子：
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通讯作者：
P. Dupuis;Guo-Jhen Wu
中科院分区：
文献类型：
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作者：
P. Dupuis;Guo-Jhen Wu
亚稳定性是马尔可夫链蒙特卡罗方法面临的一个严峻挑战。在本文中，我们提出了算法设计，以满足这一挑战的方法。我们考虑的设计问题是无限交换方案的温度选择，这是广泛使用的并行回火方案的限制时，交换率趋于无穷大。我们使用一个最近开发的工具的经验措施的小噪声扩散的分析转换成一个明确的优化问题的方差减少问题。我们的第一个分析的优化问题是在设置一个双井模型，它表明，最佳的选择的温度比是一个几何序列，除了可能的最高温度。在相同的设置中，我们确定了两个不同的来源的方差减少，并显示它们的竞争如何确定最佳的最高温度。在一般的多井设置，我们证明了一个纯粹的几何序列的温度比总是接近最佳的，与性能差距，几何衰减的温度的数量。
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.