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，Elan、Koehler和Zeitouni（2020）指出，对于任意固定$\epsilon>0$，只要耦合矩阵的最大与最小特征值之差不超过$1-\epsilon$，格劳伯动力学在通用伊辛模型中即可快速混合。我们证明格劳伯动力学在此"通用采样"任务中实际上是最优的。具体而言，我们建立了一个从Wishart负尖峰矩阵模型中的假设检验到一般伊辛模型吉布斯测度近似采样的平均情况归约，其中对任意固定$\epsilon>0$，耦合矩阵的最大与最小特征值之差不超过$1+\epsilon$。结合Bandeira、Kunisky和Wein（2019）关于低次多项式算法分析所揭示的尖峰矩阵问题难解性证据，我们的结果进一步表明，在通用采样中改进格劳伯动力学是困难的。我们还将类似归约应用于一般伊辛模型的自由能近似，并推论出基于格劳伯动力学的模拟退火算法在通用场景下具有最优性。