A fundamental problem associated with the task of network reconstruction from dynamical or behavioral data consists in determining the most appropriate model complexity in a manner that prevents overfitting, and produces an inferred network with a statistically justifiable number of edges. The status quo in this context is based on $L_{1}$ regularization combined with cross-validation. However, besides its high computational cost, this commonplace approach unnecessarily ties the promotion of sparsity with weight "shrinkage". This combination forces a trade-off between the bias introduced by shrinkage and the network sparsity, which often results in substantial overfitting even after cross-validation. In this work, we propose an alternative nonparametric regularization scheme based on hierarchical Bayesian inference and weight quantization, which does not rely on weight shrinkage to promote sparsity. Our approach follows the minimum description length (MDL) principle, and uncovers the weight distribution that allows for the most compression of the data, thus avoiding overfitting without requiring cross-validation. The latter property renders our approach substantially faster to employ, as it requires a single fit to the complete data. As a result, we have a principled and efficient inference scheme that can be used with a large variety of generative models, without requiring the number of edges to be known in advance. We also demonstrate that our scheme yields systematically increased accuracy in the reconstruction of both artificial and empirical networks. We highlight the use of our method with the reconstruction of interaction networks between microbial communities from large-scale abundance samples involving in the order of $10^{4}$ to $10^{5}$ species, and demonstrate how the inferred model can be used to predict the outcome of interventions in the system.

翻译：从动力学或行为数据重构网络的一项基本问题在于，以防止过拟合的方式确定最合适的模型复杂度，并生成具有统计上合理的边数量的推断网络。当前该领域的常规方法是基于$L_{1}$正则化结合交叉验证。然而，除计算成本高昂外，这种常见做法不必要地将稀疏性促进与权重“收缩”绑定在一起。这种组合迫使偏差（由收缩引入）与网络稀疏性之间形成权衡，即使经过交叉验证也常导致显著过拟合。在本工作中，我们提出一种基于分层贝叶斯推断与权重量化的替代性非参数正则化方案，该方案不依赖权重收缩来促进稀疏性。我们的方法遵循最小描述长度（MDL）原则，揭示能够最大化压缩数据的权重分布，从而无需交叉验证即可避免过拟合。后者特性使我们的方法在部署时显著更高效——仅需对完整数据进行单次拟合。因此，我们获得了一个可与多种生成模型配合使用的规范化高效推断方案，且无需预先知道边数量。我们进一步证明，该方案在人工网络与经验网络的重构中均系统性地提升了准确性。我们通过从涉及$10^{4}$至$10^{5}$物种数量级的大规模丰度样本中重构微生物群落交互网络来突出展示该方法的应用，并演示如何利用推断模型预测系统干预的结果。