基于景观修改的改进Metropolis-Hastings算法及其在模拟退火和Curie-Weiss模型中的应用
Improved Metropolis-Hastings algorithms via landscape modifcation with applications to simulated annealing and the Curie-Weiss model
AI总结：
本文通过引入阈值参数修改能量景观，提出了改进的Metropolis-Hastings和模拟退火算法，在鼓励探索与利用间取得平衡，并在Curie-Weiss模型及低温模拟退火中显著提升了向基态的收敛速度与全局最小值间的隧道穿越时间。
摘要：
在本文中，我们通过修改能量景观，提出了有限状态空间上的新Metropolis-Hastings和模拟退火算法。景观修改的核心思想在于引入参数$c$，当算法高于该阈值参数时修改景观以鼓励探索，而当算法低于阈值时利用原始景观以进行开发。我们通过研究景观修改对具有Glauber动力学和外部磁场的经典Curie-Weiss模型在亚临界状态下的影响，阐明了其强大能力和优势。这导出了一个景观修改后的平均场方程，并且通过适当选择$c$，自由能景观可以从双阱转变为单阱，同时全局最小值的位置在修改后的景观上得以保留。因此，在修改后的景观上运行算法可以改善Curie-Weiss模型中向基态的收敛。在模拟退火设置中，我们证明了通过适当选择$c$，景观修改可以在低温状态下产生改善的甚至亚指数的全局最小值间的平均隧道穿越时间，并给出了使用具有降低临界高度的改进对数冷却方案的收敛保证。我们还讨论了景观修改与Catoni能量变换算法、预条件、重要性采样和量子退火等其他加速技术之间的联系。本文开发的技术不仅限于模拟退火，通过改变景观可广泛适用于任何基于差分的离散优化算法。
英文摘要：
In this paper, we propose new Metropolis-Hastings and simulated annealing algorithms on finite state space via modifying the energy landscape. The core idea of landscape modification rests on introducing a parameter $c$, in which the landscape is modified once the algorithm is above this threshold parameter to encourage exploration, while the original landscape is utilized when the algorithm is below the threshold for exploitation purpose. We illustrate the power and benefits of landscape modification by investigating its effect on the classical Curie-Weiss model with Glauber dynamics and external magnetic field in the subcritical regime. This leads to a landscape-modified mean-field equation, and with appropriate choice of $c$ the free energy landscape can be transformed from a double-well into a single-well, while the location of the global minimum is preserved on the modified landscape. Consequently, running algorithms on the modified landscape can improve the convergence to the ground-state in the Curie-Weiss model. In the setting of simulated annealing, we demonstrate that landscape modification can yield improved or even subexponential mean tunneling time between global minima in the low-temperature regime by appropriate choice of $c$, and give convergence guarantee using an improved logarithmic cooling schedule with reduced critical height. We also discuss connections between landscape modification and other acceleration techniques such as Catoni's energy transformation algorithm, preconditioning, importance sampling and quantum annealing. The technique developed in this paper is not only limited to simulated annealing and is broadly applicable to any difference-based discrete optimization algorithm by a change of landscape.
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