Learning performance in inverse Ising problems with sparse teacher couplings
Learning performance in inverse Ising problems with sparse teacher couplings
复制标题
具有稀疏教师耦合的伊辛逆问题的学习性能
DOI：
10.1088/1742-5468/ab8c3a
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发表时间：
2020
期刊：
影响因子：
--
通讯作者：
Xu Yingying
中科院分区：
文献类型：
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作者：
Abbara Alia;Kabashima Yoshiyuki;Obuchi Tomoyuki;Xu Yingying
我们研究了逆伊辛问题的伪李维最大化方法的学习性能。在教师-学生的情况下，教师的耦合是稀疏的，学生不知道的图形结构的假设下，学习曲线和顺序参数进行评估，在典型的情况下，使用复制品和空腔方法从统计力学。我们的公式也适用于具有局部性的某类成本函数;标准似然不属于该类。导出的解析公式表明，只要α= M/N> 2，在自旋数N为无穷大的热力学极限下，保持数据集大小M与N成正比，就可以完美地推断教师耦合的存在/不存在。同时，公式还表明，教师真实存在的耦合值所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。这些结果被认为是精确的热力学极限的局部树状网络，如规则的随机或Erdens-Rényi图。数值模拟结果完全支持理论预测。另外还讨论了在循环图的估计偏差。
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 α= 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–Rényi graphs. Numerical simulation results fully support the theoretical predictions. Additional biases in the estimators on loopy graphs are also discussed.