We investigate the learning performance of the pseudolikelihood maximization method for inverse Ising problems. In the teacher-student scenario under the assumption that the teacher's couplings are sparse and the student does not know the graphical structure, the learning curve and order parameters are assessed in the typical case using the replica and cavity methods from statistical mechanics. Our formulation is also applicable to a certain class of cost functions having locality; the standard likelihood does not belong to that class. The derived analytical formulas indicate that the perfect inference of the presence/absence of the teacher's couplings is possible in the thermodynamic limit taking the number of spins $N$ as infinity while keeping the dataset size $M$ proportional to $N$, as long as $\alpha=M/N > 2$. Meanwhile, the formulas also show that the estimated coupling values corresponding to the truly existing ones in the teacher tend to be overestimated in the absolute value, manifesting the presence of estimation bias. These results are considered to be exact in the thermodynamic limit on locally tree-like networks, such as the regular random or Erdős--Renyi graphs. Numerical simulation results fully support the theoretical predictions. Additional biases in the estimators on loopy graphs are also discussed.

中文翻译：

---

具有稀疏教师耦合的逆伊辛问题的学习性能

我们研究了用于逆伊辛问题的伪似然最大化方法的学习性能。在教师-学生场景中，假设教师耦合稀疏且学生不知道图形结构，则使用统计力学中的复制和空腔方法在典型情况下评估学习曲线和阶次参数。我们的公式也适用于某一类具有局部性的成本函数；标准似然不属于该类。推导出的分析公式表明，在热力学极限中可以完美推断存在/不存在教师耦合，其中自旋数 $N$ 为无穷大，同时保持数据集大小 $M$ 与 $N$ 成正比，只要因为 $\alpha=M/N > 2$。同时，公式还表明，与教师中真实存在的耦合值对应的估计耦合值在绝对值上往往被高估，表现出估计偏差的存在。这些结果被认为在局部树状网络的热力学极限中是准确的，例如常规随机图或 Erdős--Renyi 图。数值模拟结果完全支持理论预测。还讨论了循环图估计器中的其他偏差。数值模拟结果完全支持理论预测。还讨论了循环图估计器中的其他偏差。数值模拟结果完全支持理论预测。还讨论了循环图估计器中的其他偏差。