We consider the fundamental problem of learning the parameters of an undirected graphical model or Markov Random Field (MRF) in the setting where the edge weights are chosen at random. For Ising models, we show that a multiplicative-weight update algorithm due to Klivans and Meka learns the parameters in polynomial time for any inverse temperature $\beta \leq \sqrt{\log n}$. This immediately yields an algorithm for learning the Sherrington-Kirkpatrick (SK) model beyond the high-temperature regime of $\beta < 1$. Prior work breaks down at $\beta = 1$ and requires heavy machinery from statistical physics or functional inequalities. In contrast, our analysis is relatively simple and uses only subgaussian concentration. Our results extend to MRFs of higher order (such as pure $p$-spin models), where even results in the high-temperature regime were not known.

翻译：我们研究在边权重随机选取的设置下，学习无向图模型或马尔可夫随机场参数的基本问题。对于伊辛模型，我们证明由Klivans和Meka提出的乘性权重更新算法可在多项式时间内学习任意逆温度$\beta \leq \sqrt{\log n}$下的参数。这直接产生了一种超越$\beta < 1$高温区、学习Sherrington-Kirkpatrick模型的方法。先前工作在$\beta = 1$处失效，且需依赖统计物理学或函数不等式的复杂工具。相比之下，我们的分析相对简洁，仅需次高斯集中性理论。我们的结果可推广至高阶马尔可夫随机场（如纯$p$-自旋模型），此类模型即使在高温区亦缺乏已知结果。