高斯图模型中结构学习的交叉验证不一致性
Inconsistency of cross-validation for structure learning in Gaussian graphical models
AI总结：
本文指出交叉验证在高斯图模型结构学习中存在固有局限性，通过有限样本界限证明其不一致性，并经模拟实验对比其他信息准则，强调超参数精确校准的重要性。
摘要：
尽管多年来对各类模型选择标准的优劣与权衡进行了大量研究，但获得能阐明交叉验证行为的稳健结果仍是一项具有挑战性的工作。本文中，我们强调了交叉验证在用于识别高斯图模型结构时所固有的局限性。我们提供了有限样本界限，界定了在高斯图模型中，通过预测预言机优化的节点邻域Lasso估计器错误识别邻域的概率。我们的结果同时适用于无向图和有向无环图，涵盖了一般的稀疏协方差结构。为支持我们的理论发现，我们通过广泛的模拟研究，将我们的结果与其他常用信息准则进行对比，对该不一致性进行了实证调查。鉴于许多用于学习图模型结构的算法需要超参数选择，该超参数的精确校准对于准确估计固有结构至关重要。因此，我们的观察结果揭示了这一广为人知的实践挑战。
英文摘要：
Despite numerous years of research into the merits and trade-offs of various model selection criteria, obtaining robust results that elucidate the behavior of cross-validation remains a challenging endeavor. In this paper, we highlight the inherent limitations of cross-validation when employed to discern the structure of a Gaussian graphical model. We provide finite-sample bounds on the probability that the Lasso estimator for the neighborhood of a node within a Gaussian graphical model, optimized using a prediction oracle, misidentifies the neighborhood. Our results pertain to both undirected and directed acyclic graphs, encompassing general, sparse covariance structures. To support our theoretical findings, we conduct an empirical investigation of this inconsistency by contrasting our outcomes with other commonly used information criteria through an extensive simulation study. Given that many algorithms designed to learn the structure of graphical models require hyperparameter selection, the precise calibration of this hyperparameter is paramount for accurately estimating the inherent structure. Consequently, our observations shed light on this widely recognized practical challenge.
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