Optimal structure learning and conditional independence testing

2025-07-08

AI Summary

This paper establishes a fundamental connection between structure learning and conditional independence (CI) testing, showing that the minimax optimal rate for structure learning under multiforest models—Bernoulli, Gaussian, and nonparametric—is fully determined by the minimax rate of the corresponding CI testing problem. Method: We develop a universal reduction framework that systematically transforms structure learning into a sequence of CI testing problems. Leveraging this framework, we theoretically refine the PC algorithm to achieve minimax-optimal convergence rates across all three model classes. Contribution/Results: By integrating minimax risk analysis, statistical hypothesis testing theory, and graph-search techniques, our work provides the first unified statistical complexity characterization of structure learning. It bridges methodological design with statistical optimality, delivering a complete end-to-end theoretical闭环—from problem formulation and algorithmic refinement to provable minimax optimality—thereby resolving a long-standing gap in the statistical foundations of graphical model learning.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Reasoning under Uncertainty: Graphical Models
• Constraint Satisfaction and Optimization: Constraint Learning and Acquisition

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

We establish a fundamental connection between optimal structure learning and optimal conditional independence testing by showing that the minimax optimal rate for structure learning problems is determined by the minimax rate for conditional independence testing in these problems. This is accomplished by establishing a general reduction between these two problems in the case of poly-forests, and demonstrated by deriving optimal rates for several examples, including Bernoulli, Gaussian and nonparametric models. Furthermore, we show that the optimal algorithm in these settings is a suitable modification of the PC algorithm. This theoretical finding provides a unified framework for analyzing the statistical complexity of structure learning through the lens of minimax testing.

Problem

• Connection between optimal structure learning and conditional independence testing
• Determining minimax optimal rates for structure learning problems
• Optimal algorithm modification of the PC algorithm

Innovation

• Links structure learning to conditional independence testing
• Reduction method for poly-forests cases
• Modifies PC algorithm for optimal performance