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Adaptive Online Experimental Design for Causal Discovery

Muhammad Qasim Elahi, Lai Wei, Murat Kocaoglu, Mahsa Ghasemi

TL;DR

A track-and-stop causal discovery algorithm that adaptively selects interventions from the graph separating system via allocation matching and learns the causal graph based on sampling history is proposed, which achieves higher accuracy and significantly fewer samples.

Abstract

Causal discovery aims to uncover cause-and-effect relationships encoded in causal graphs by leveraging observational, interventional data, or their combination. The majority of existing causal discovery methods are developed assuming infinite interventional data. We focus on data interventional efficiency and formalize causal discovery from the perspective of online learning, inspired by pure exploration in bandit problems. A graph separating system, consisting of interventions that cut every edge of the graph at least once, is sufficient for learning causal graphs when infinite interventional data is available, even in the worst case. We propose a track-and-stop causal discovery algorithm that adaptively selects interventions from the graph separating system via allocation matching and learns the causal graph based on sampling history. Given any desired confidence value, the algorithm determines a termination condition and runs until it is met. We analyze the algorithm to establish a problem-dependent upper bound on the expected number of required interventional samples. Our proposed algorithm outperforms existing methods in simulations across various randomly generated causal graphs. It achieves higher accuracy, measured by the structural hamming distance (SHD) between the learned causal graph and the ground truth, with significantly fewer samples.

Adaptive Online Experimental Design for Causal Discovery

TL;DR

A track-and-stop causal discovery algorithm that adaptively selects interventions from the graph separating system via allocation matching and learns the causal graph based on sampling history is proposed, which achieves higher accuracy and significantly fewer samples.

Abstract

Causal discovery aims to uncover cause-and-effect relationships encoded in causal graphs by leveraging observational, interventional data, or their combination. The majority of existing causal discovery methods are developed assuming infinite interventional data. We focus on data interventional efficiency and formalize causal discovery from the perspective of online learning, inspired by pure exploration in bandit problems. A graph separating system, consisting of interventions that cut every edge of the graph at least once, is sufficient for learning causal graphs when infinite interventional data is available, even in the worst case. We propose a track-and-stop causal discovery algorithm that adaptively selects interventions from the graph separating system via allocation matching and learns the causal graph based on sampling history. Given any desired confidence value, the algorithm determines a termination condition and runs until it is met. We analyze the algorithm to establish a problem-dependent upper bound on the expected number of required interventional samples. Our proposed algorithm outperforms existing methods in simulations across various randomly generated causal graphs. It achieves higher accuracy, measured by the structural hamming distance (SHD) between the learned causal graph and the ground truth, with significantly fewer samples.
Paper Structure (28 sections, 22 theorems, 100 equations, 5 figures, 1 table, 5 algorithms)

This paper contains 28 sections, 22 theorems, 100 equations, 5 figures, 1 table, 5 algorithms.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Preliminaries
  4. Algorithm Initialization
  5. Constructing Graph Separating System
  6. Enumerating Causal Effects
  7. Online Algorithm Design and Analysis
  8. The Exact Algorithm
  9. Practical Algorithm Implementation
  10. An Asymptotic Analysis of Algorithm
  11. Experiments
  12. Conclusion
  13. Acknowledgements
  14. Supplementary Material
  15. Procedure to construct $(n,k)$ seperating system
  16. ...and 13 more sections

Key Result

Lemma 1

Consider an MPDAG $\mathcal{M}(\mathbf{V},\mathbf{E})$ and two disjoint sets of variables $\mathbf{X},\mathbf{Y} \subseteq \mathbf{V}$. The interventional distribution $P_{\mathbf{x}}(\mathbf{y})$ is identifiable from any observational distribution consistent with $\mathcal{M}$ if there exists no po The assignment for $\mathsf{Pa}(\mathbf{b}_i,\mathcal{M})$ must be in consistence with $do(\mathbf{

Figures (5)

  • Figure 1: MPDAGs obtained by assigning orientations to edges $E[{V_1},\mathbf{V}\setminus{V_1}]$ in corresponding skeleton i.e. CPDAG
  • Figure 2: SHD vs interventional samples for complete Erdös-Rényi random chordal graphs with varying graph orders.
  • Figure 3: SHD vs interventional samples for Erdös-Rényi random chordal graphs with varying graph density.
  • Figure 4: SHD vs No. of samples for SACHS dataset.
  • Figure 5: SHD versus interventional samples for the discovery algorithms for Erdös-Rényi random chordal graphs starting with the incorrect CPDAG.

Theorems & Definitions (42)

  • Definition 1: Faithfulness zhang2012strong
  • Definition 2: Partial Causal Ordering
  • Definition 3: Graph Separating System
  • Lemma 1: Causal Identification Formula for MPDAG perkovic2020identifying
  • Lemma 2
  • Definition 4
  • Lemma 3: hauser2012characterization, Th. 10
  • proof : Proof of \ref{['our_lemma']}
  • Definition 5: Soundness of Algorithm
  • Theorem 1
  • ...and 32 more