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Interventional Causal Structure Discovery over Graphical Models with Convergence and Optimality Guarantees

Qiu Chengbo, Yang Kai

TL;DR

The paper tackles the identifiability gap in causal structure learning from observational data by integrating interventional data within a unified bilevel polynomial optimization framework. Bloom formulates the problem over two levels, using a least-squares upper objective and a convex lower problem, and leverages SDP relaxations with correlative- and term-sparsity to obtain monotone convergence toward a global optimum. It further extends Bloom to a distributed setting to reduce data sharing and privacy risks, while preserving convergence guarantees via consensus-style coordination. Empirical results on synthetic and real data show Bloom outperforms leading baselines in TPR and SHD, and remain robust under imperfect interventions and latent confounders. The approach provides a principled, theoretically grounded path toward scalable, privacy-preserving causal structure discovery with strong practical impact for domains such as healthcare and complex systems.

Abstract

Learning causal structure from sampled data is a fundamental problem with applications in various fields, including healthcare, machine learning and artificial intelligence. Traditional methods predominantly rely on observational data, but there exist limits regarding the identifiability of causal structures with only observational data. Interventional data, on the other hand, helps establish a cause-and-effect relationship by breaking the influence of confounding variables. It remains to date under-explored to develop a mathematical framework that seamlessly integrates both observational and interventional data in causal structure learning. Furthermore, existing studies often focus on centralized approaches, necessitating the transfer of entire datasets to a single server, which lead to considerable communication overhead and heightened risks to privacy. To tackle these challenges, we develop a bilevel polynomial optimization (Bloom) framework. Bloom not only provides a powerful mathematical modeling framework, underpinned by theoretical support, for causal structure discovery from both interventional and observational data, but also aspires to an efficient causal discovery algorithm with convergence and optimality guarantees. We further extend Bloom to a distributed setting to reduce the communication overhead and mitigate data privacy risks. It is seen through experiments on both synthetic and real-world datasets that Bloom markedly surpasses other leading learning algorithms.

Interventional Causal Structure Discovery over Graphical Models with Convergence and Optimality Guarantees

TL;DR

The paper tackles the identifiability gap in causal structure learning from observational data by integrating interventional data within a unified bilevel polynomial optimization framework. Bloom formulates the problem over two levels, using a least-squares upper objective and a convex lower problem, and leverages SDP relaxations with correlative- and term-sparsity to obtain monotone convergence toward a global optimum. It further extends Bloom to a distributed setting to reduce data sharing and privacy risks, while preserving convergence guarantees via consensus-style coordination. Empirical results on synthetic and real data show Bloom outperforms leading baselines in TPR and SHD, and remain robust under imperfect interventions and latent confounders. The approach provides a principled, theoretically grounded path toward scalable, privacy-preserving causal structure discovery with strong practical impact for domains such as healthcare and complex systems.

Abstract

Learning causal structure from sampled data is a fundamental problem with applications in various fields, including healthcare, machine learning and artificial intelligence. Traditional methods predominantly rely on observational data, but there exist limits regarding the identifiability of causal structures with only observational data. Interventional data, on the other hand, helps establish a cause-and-effect relationship by breaking the influence of confounding variables. It remains to date under-explored to develop a mathematical framework that seamlessly integrates both observational and interventional data in causal structure learning. Furthermore, existing studies often focus on centralized approaches, necessitating the transfer of entire datasets to a single server, which lead to considerable communication overhead and heightened risks to privacy. To tackle these challenges, we develop a bilevel polynomial optimization (Bloom) framework. Bloom not only provides a powerful mathematical modeling framework, underpinned by theoretical support, for causal structure discovery from both interventional and observational data, but also aspires to an efficient causal discovery algorithm with convergence and optimality guarantees. We further extend Bloom to a distributed setting to reduce the communication overhead and mitigate data privacy risks. It is seen through experiments on both synthetic and real-world datasets that Bloom markedly surpasses other leading learning algorithms.
Paper Structure (41 sections, 4 theorems, 25 equations, 9 figures, 8 tables, 2 algorithms)

This paper contains 41 sections, 4 theorems, 25 equations, 9 figures, 8 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Background and Related works
  3. Causal Structure Learning
  4. Causal Structure Learning with Continuous Optimization
  5. Causal Structure Learning with Interventional Data
  6. Causal Structure Learning with Distributed Learning
  7. Bilevel Optimization
  8. Polynomial Optimization
  9. Correlative sparsity
  10. Term sparsity
  11. Methodology
  12. Scope and Assumptions
  13. Bilevel Polynomial Optimization for Causal Structure Learning
  14. SDP Relaxations with Structured Sparsity
  15. Generate Cliques
  16. ...and 26 more sections

Key Result

Proposition 1

Consider Eq. (e11) where the lower-level optimization problem is convex and satisfies the slater condition. Then $\left(\boldsymbol{q}, \boldsymbol{p}\right)$ is a global optimal solution of the bilevel polynomial optimization problem Eq. (e11) when there exist Lagrange multipliers $\boldsymbol{\lam

Figures (9)

  • Figure 1: Framework of the proposed algorithm.
  • Figure 2: Types of graphs under different experimental settings. In imperfect interventions, the weighted coefficients are changed. In latent confounder, a common cause is not observed.
  • Figure 3: Distributed Architecture
  • Figure 4: Experiments with Different $\lambda_{obs}$.
  • Figure 5: Experiments on Large-scale Datasets.
  • ...and 4 more figures

Theorems & Definitions (7)

  • Definition 1: Slater’s Condition
  • Definition 2: Chord and Chordal Graph
  • Definition 3: Clique and Maximal Clique
  • Proposition 1: Equivalent single-level reformulation
  • Proposition 2: Monotonic convergence
  • Proposition 3: Optimality condition
  • Lemma 1: Putinar’s Positivstellensatz Putinar