Optimal structure learning and conditional independence testing

2026-06-16 · Source: stat.ML updates on arXiv.org · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Data Science & Analytics, Mathematics & Computational Sciences · Depth: Expert, extended

Summary

This research establishes a fundamental connection between optimal structure learning and optimal conditional independence (CI) testing, demonstrating that the minimax optimal rate for structure learning in poly-forests is determined by the minimax rate for CI testing. The study achieves this by reducing structure learning to CI testing for poly-forests, a tractable subclass of Directed Acyclic Graphs (DAGs). It derives optimal sample complexities for various models, including Bernoulli, Gaussian, and nonparametric continuous distributions. Furthermore, the paper shows that a suitable modification of the PC algorithm, utilizing an optimal CI test, achieves these optimal rates. This theoretical framework unifies the analysis of statistical complexity in structure learning through the lens of minimax testing.

Key takeaway

For research scientists developing or applying graphical models, understanding the statistical limits of structure learning is crucial for efficient model recovery. This work demonstrates that the sample complexity for learning poly-forests is directly tied to the difficulty of conditional independence testing, providing a unified framework. You should consider the intrinsic hardness of your data's distribution (e.g., parametric vs. nonparametric) when estimating required sample sizes, and leverage the PC-tree algorithm with optimal CI tests for robust and accurate graph inference.

Key insights

Optimal structure learning rates for poly-forests are directly determined by minimax optimal conditional independence testing rates.

Principles

• Minimax optimal structure learning rates for poly-forests are directly linked to minimax CI testing rates.
• The PC algorithm, when modified with an optimal CI test, achieves optimal sample complexity.
• Parametric distributions (Bernoulli, Gaussian) typically yield an exponent α=2 for sample complexity.

Method

The proposed method involves a modified PC-tree algorithm that takes a valid conditional independence (CI) test as input, efficiently determining edges by testing marginal and single-node conditional independencies.

In practice

• Apply the PC-tree algorithm with tailored CI tests for Bernoulli, Gaussian, and nonparametric data.
• Expect higher sample complexity for nonparametric models due to their intrinsic hardness (exponent α > 2).

Topics

• Structure Learning
• Conditional Independence Testing
• Graphical Models
• Poly-forests
• Minimax Optimality
• PC Algorithm
• Nonparametric Statistics

Code references

• ilmunk/UCI

Best for: AI Scientist, Research Scientist

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Editorial summary, takeaway, and curation by AIssential. Original article published by stat.ML updates on arXiv.org.