# Inconsistency of cross-validation for structure learning in Gaussian graphical models

arXiv: https://arxiv.org/abs/2312.17047 · HTML: https://mlsanity.com/paper/2312.17047/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 13 of its 82 references that are in the map.
- First posted on arXiv 28 Dec 2023.

## Builds on (most cited first, 13 of 13)

- [Inconsistency of Bayesian Inference for Misspecified Linear Models, and a Proposal for Repairing It](https://mlsanity.com/paper/1412.3730/) (2014)
- [Cross-validation: what does it estimate and how well does it do it?](https://mlsanity.com/paper/2104.00673/) (2021)
- [On Causal Discovery with Equal Variance Assumption](https://mlsanity.com/paper/1807.03419/) (2018)
- [Learning linear structural equation models in polynomial time and sample complexity](https://mlsanity.com/paper/1707.04673/) (2017)
- [False Discoveries Occur Early on the Lasso Path](https://mlsanity.com/paper/1511.01957/) (2015)
- [Concave Penalized Estimation of Sparse Gaussian Bayesian Networks](https://mlsanity.com/paper/1401.0852/) (2014)
- [Learning Identifiable Gaussian Bayesian Networks in Polynomial Time and Sample Complexity](https://mlsanity.com/paper/1703.01196/) (2017)
- [Cross-Validation with Confidence](https://mlsanity.com/paper/1703.07904/) (2017)
- [Approximate Cross-validation: Guarantees for Model Assessment and Selection](https://mlsanity.com/paper/2003.00617/) (2020)
- [On the Sample Complexity of Learning Bayesian Networks](https://mlsanity.com/paper/1302.3579/) (2013)
- [Estimating the error variance in a high-dimensional linear model](https://mlsanity.com/paper/1712.02412/) (2017)
- [Learning High-dimensional Gaussian Graphical Models under Total Positivity without Adjustment of Tuning Parameters](https://mlsanity.com/paper/1906.05159/) (2019)
- [Don't Fall for Tuning Parameters: Tuning-Free Variable Selection in High Dimensions With the TREX](https://mlsanity.com/paper/1404.0541/) (2014)

## Abstract (arXiv)

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.