The planted coloring problem is a prototypical inference problem for which thresholds for Bayes optimal algorithms, like Belief Propagation (BP), can be computed analytically. In this paper, we analyze the limits and performances of the Simulated Annealing (SA), a Monte Carlo-based algorithm that is more general and robust than BP, and thus of broader applicability. We show that SA is sub-optimal in the recovery of the planted solution because it gets attracted by glassy states that, instead, do not influence the BP algorithm. At variance with previous conjectures, we propose an analytic estimation for the SA algorithmic threshold by comparing the spinodal point of the paramagnetic phase and the dynamical critical temperature. This is a fundamental connection between thermodynamical phase transitions and out of equilibrium behavior of Glauber dynamics. We also study an improved version of SA, called replicated SA (RSA), where several weakly coupled replicas are cooled down together. We show numerical evidence that the algorithmic threshold for the RSA coincides with the Bayes optimal one. Finally, we develop an approximated analytical theory explaining the optimal performances of RSA and predicting the location of the transition towards the planted solution in the limit of a very large number of replicas. Our results for RSA support the idea that mismatching the parameters in the prior with respect to those of the generative model may produce an algorithm that is optimal and very robust.

翻译：植入着色问题是典型的推理问题，其贝叶斯最优算法（如置信传播）的阈值可通过解析计算。本文分析了模拟退火（SA）这种基于蒙特卡罗的算法的局限性与性能——该算法比置信传播更具通用性和鲁棒性，因而适用范围更广。研究表明，SA在恢复植入解方面表现次优，因其会受玻璃态吸引，而这些玻璃态并不影响置信传播算法。与以往假设不同，我们通过比较顺磁相的自旋节点点与动态临界温度，提出了SA算法阈值的解析估计，这揭示了热力学相变与格劳伯动力学非平衡行为之间的基本关联。我们还研究了SA的改进版本——复制模拟退火（RSA），其中多个弱耦合副本共同降温。数值证据表明RSA的算法阈值与贝叶斯最优阈值一致。最后，我们建立了近似解析理论，解释RSA的最优性能，并预测在副本数趋近无穷大时向植入解转变的位置。我们的RSA结果支持如下观点：将先验参数与生成模型参数失配，可能产生最优且高度鲁棒的算法。