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.

翻译：本文通过修正能量景观，提出了有限状态空间上的新型Metropolis-Hastings算法和模拟退火算法。景观修正的核心思想在于引入参数$c$：当算法处于该阈值参数之上时，修正景观以促进探索；当算法处于阈值之下时，则利用原始景观进行开发。通过研究亚临界状态下含外磁场的Glauber动力学经典Curie-Weiss模型，我们展示了景观修正的能力与优势。由此推导出景观修正的平均场方程，通过合理选择参数$c$，自由能景观可从双势阱转变为单势阱，同时修正景观中全局最小值的位置保持不变。因此，在修正景观上运行算法可改善Curie-Weiss模型向基态的收敛性。在模拟退火框架下，我们证明通过适当选择$c$，景观修正能在低温区域内实现全局最小值之间平均隧穿时间的改善甚至次指数衰减，并通过降低临界高度的改进对数冷却计划给出收敛保证。本文还讨论了景观修正与其他加速技术（如Catoni能量变换算法、预处理、重要性采样和量子退火）之间的联系。本文发展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。