We study the generalization behavior of Markov Logic Networks (MLNs) across relational structures of different sizes. Multiple works have noticed that MLNs learned on a given domain generalize poorly across domains of different sizes. This behavior emerges from a lack of internal consistency within an MLN when used across different domain sizes. In this paper, we quantify this inconsistency and bound it in terms of the variance of the MLN parameters. The parameter variance also bounds the KL divergence between an MLN's marginal distributions taken from different domain sizes. We use these bounds to show that maximizing the data log-likelihood while simultaneously minimizing the parameter variance corresponds to two natural notions of generalization across domain sizes. Our theoretical results apply to Exponential Random Graphs and other Markov network based relational models. Finally, we observe that solutions known to decrease the variance of the MLN parameters, like regularization and Domain-Size Aware MLNs, increase the internal consistency of the MLNs. We empirically verify our results on four different datasets, with different methods to control parameter variance, showing that controlling parameter variance leads to better generalization.

翻译：我们研究了马尔可夫逻辑网络（MLNs）在不同规模关系结构上的泛化行为。多项研究注意到，在特定域上训练的MLNs难以在规模不同的域之间实现良好泛化。这种行为源于MLN在不同域规模下使用时缺乏内部一致性。本文量化了这种不一致性，并通过MLN参数方差对其进行界定。参数方差还能约束不同域规模下MLN边际分布之间的KL散度。我们利用这些界限证明，最大化数据对数似然同时最小化参数方差对应于两种自然的域规模泛化准则。本文的理论结果适用于指数随机图及其他基于马尔可夫网络的关系模型。最后，我们观察到，已知能降低MLN参数方差的解决方案（如正则化和域规模感知MLN）可增强MLN的内部一致性。我们通过四种不同数据集，采用不同控制参数方差的方法进行实证验证，结果表明控制参数方差能带来更好的泛化效果。