# High-dimensional consistency in score-based and hybrid structure learning

arXiv: https://arxiv.org/abs/1507.02608 · HTML: https://mlsanity.com/paper/1507.02608/

## Key facts

- Cited by 46 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Jul 2015.

## Built on by (most cited first, 30 of 46)

- [D'ya like DAGs? A Survey on Structure Learning and Causal Discovery](https://mlsanity.com/paper/2103.02582/) (2021)
- [A survey of Bayesian Network structure learning](https://mlsanity.com/paper/2109.11415/) (2021)
- [A Survey on Causal Discovery: Theory and Practice](https://mlsanity.com/paper/2305.10032/) (2023)
- [Causal Structure Learning: a Combinatorial Perspective](https://mlsanity.com/paper/2206.01152/) (2022)
- [Optimal estimation of Gaussian DAG models](https://mlsanity.com/paper/2201.10548/) (2022)
- [Optimizing NOTEARS Objectives via Topological Swaps](https://mlsanity.com/paper/2305.17277/) (2023)
- [MM-DAG: Multi-task DAG Learning for Multi-modal Data -- with Application for Traffic Congestion Analysis](https://mlsanity.com/paper/2306.02831/) (2023)
- [Causal Discovery with Fewer Conditional Independence Tests](https://mlsanity.com/paper/2406.01823/) (2024)
- [Characterization and Greedy Learning of Gaussian Structural Causal Models under Unknown Interventions](https://mlsanity.com/paper/2211.14897/) (2022)
- [DRCFS: Doubly Robust Causal Feature Selection](https://mlsanity.com/paper/2306.07024/) (2023)
- [Structure learning in polynomial time: Greedy algorithms, Bregman information, and exponential families](https://mlsanity.com/paper/2110.04719/) (2021)
- [Gene Regulatory Network Inference in the Presence of Dropouts: a Causal View](https://mlsanity.com/paper/2403.15500/) (2024)
- [Valid Inference After Causal Discovery](https://mlsanity.com/paper/2208.05949/) (2022)
- [Novel Ordering-based Approaches for Causal Structure Learning in the Presence of Unobserved Variables](https://mlsanity.com/paper/2208.06935/) (2022)
- [Structure Learning for Directed Trees](https://mlsanity.com/paper/2108.08871/) (2021)
- [Extremely Greedy Equivalence Search](https://mlsanity.com/paper/2502.19551/) (2025)
- [Learning linear non-Gaussian directed acyclic graph with diverging number of nodes](https://mlsanity.com/paper/2111.00740/) (2021)
- [A Fast Non-parametric Approach for Local Causal Structure Learning](https://mlsanity.com/paper/2111.14969/) (2021)
- [iSCAN: Identifying Causal Mechanism Shifts among Nonlinear Additive Noise Models](https://mlsanity.com/paper/2306.17361/) (2023)
- [Partitioned hybrid learning of Bayesian network structures](https://mlsanity.com/paper/2103.12188/) (2021)
- [Causal Discovery over High-Dimensional Structured Hypothesis Spaces with Causal Graph Partitioning](https://mlsanity.com/paper/2406.06348/) (2024)
- [Embracing Discrete Search: A Reasonable Approach to Causal Structure Learning](https://mlsanity.com/paper/2510.04970/) (2025)
- [Optimal estimation of Gaussian (poly)trees](https://mlsanity.com/paper/2402.06380/) (2024)
- [CIDGMed: Causal Inference-Driven Medication Recommendation with Enhanced Dual-Granularity Learning](https://mlsanity.com/paper/2403.00880/) (2024)
- [Score-Based Causal Discovery of Latent Variable Causal Models](https://mlsanity.com/paper/2605.20396/) (2026)
- [VAC2: Visual Analysis of Combined Causality in Event Sequences](https://mlsanity.com/paper/2206.05420/) (2022)
- [Reframed GES with a Neural Conditional Dependence Measure](https://mlsanity.com/paper/2206.08531/) (2022)
- [Greedy equivalence search for nonparametric graphical models](https://mlsanity.com/paper/2406.17228/) (2024)
- [Score-based Greedy Search for Structure Identification of Partially Observed Linear Causal Models](https://mlsanity.com/paper/2510.04378/) (2025)
- [Causal Discovery on Higher-Order Interactions](https://mlsanity.com/paper/2511.14206/) (2025)

## Abstract (arXiv)

Main approaches for learning Bayesian networks can be classified as constraint-based, score-based or hybrid methods. Although high-dimensional consistency results are available for constraint-based methods like the PC algorithm, such results have not been proved for score-based or hybrid methods, and most of the hybrid methods have not even shown to be consistent in the classical setting where the number of variables remains fixed and the sample size tends to infinity. In this paper, we show that consistency of hybrid methods based on greedy equivalence search (GES) can be achieved in the classical setting with adaptive restrictions on the search space that depend on the current state of the algorithm. Moreover, we prove consistency of GES and adaptively restricted GES (ARGES) in several sparse high-dimensional settings. ARGES scales well to sparse graphs with thousands of variables and our simulation study indicates that both GES and ARGES generally outperform the PC algorithm.