Recently, Eldan, Koehler, and Zeitouni (2020) showed that Glauber dynamics mixes rapidly for general Ising models so long as the difference between the largest and smallest eigenvalues of the coupling matrix is at most $1 - \epsilon$ for any fixed $\epsilon > 0$. We give evidence that Glauber dynamics is in fact optimal for this "general-purpose sampling" task. Namely, we give an average-case reduction from hypothesis testing in a Wishart negatively-spiked matrix model to approximately sampling from the Gibbs measure of a general Ising model for which the difference between the largest and smallest eigenvalues of the coupling matrix is at most $1 + \epsilon$ for any fixed $\epsilon > 0$. Combined with results of Bandeira, Kunisky, and Wein (2019) that analyze low-degree polynomial algorithms to give evidence for the hardness of the former spiked matrix problem, our results in turn give evidence for the hardness of general-purpose sampling improving on Glauber dynamics. We also give a similar reduction to approximating the free energy of general Ising models, and again infer evidence that simulated annealing algorithms based on Glauber dynamics are optimal in the general-purpose setting.

翻译：近期，Eldan、Koehler与Zeitouni（2020）证明：对于任意固定ε>0，若耦合矩阵的最大与最小特征值之差不超过1-ε，则Glauber动力学在通用Ising模型中可快速混合。我们给出证据表明，在此"通用抽样"任务中Glauber动力学实际上具有最优性。具体而言，我们构建了从Wishart负尖峰矩阵模型中的假设检验到通用Ising模型吉布斯测度近似抽样的平均情况归约，其中耦合矩阵的最大与最小特征值之差对于任意固定ε>0不超过1+ε。结合Bandeira、Kunisky与Wein（2019）对低阶多项式算法的分析（该分析为前述尖峰矩阵问题的难解性提供证据），我们的研究结果进而为改进Glauber动力学的通用抽样方法的难解性提供了证据。我们还给出了类似的归约方法以逼近通用Ising模型的自由能，并据此推断：在通用场景下，基于Glauber动力学的模拟退火算法具有最优性。