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Regret-Based Federated Causal Discovery with Unknown Interventions

Federico Baldo, Charles K. Assaad

TL;DR

The paper tackles federated causal discovery when client data come with unknown, potentially heterogeneous interventions, preventing centralized pooling. It introduces I-PERI, a two-phase algorithm that first learns the CPDAG of the union of client graphs and then refines edge orientations by exploiting intervention-induced structural differences, resulting in the unique $\mathbf{\Phi}$-CPDAG within the $\mathbf{\Phi}$-MEC. The approach provides convergence guarantees and differential privacy by sharing regrets rather than raw graphs, and it demonstrates improved edge orientation and tighter equivalence classes on synthetic data. This work advances practical federated causal discovery by accounting for unknown interventions and privacy constraints, with implications for multi-center studies and decentralized datasets.

Abstract

Most causal discovery methods recover a completed partially directed acyclic graph representing a Markov equivalence class from observational data. Recent work has extended these methods to federated settings to address data decentralization and privacy constraints, but often under idealized assumptions that all clients share the same causal model. Such assumptions are unrealistic in practice, as client-specific policies or protocols, for example, across hospitals, naturally induce heterogeneous and unknown interventions. In this work, we address federated causal discovery under unknown client-level interventions. We propose I-PERI, a novel federated algorithm that first recovers the CPDAG of the union of client graphs and then orients additional edges by exploiting structural differences induced by interventions across clients. This yields a tighter equivalence class, which we call the $\mathbfΦ$-Markov Equivalence Class, represented by the $\mathbfΦ$-CPDAG. We provide theoretical guarantees on the convergence of I-PERI, as well as on its privacy-preserving properties, and present empirical evaluations on synthetic data demonstrating the effectiveness of the proposed algorithm.

Regret-Based Federated Causal Discovery with Unknown Interventions

TL;DR

The paper tackles federated causal discovery when client data come with unknown, potentially heterogeneous interventions, preventing centralized pooling. It introduces I-PERI, a two-phase algorithm that first learns the CPDAG of the union of client graphs and then refines edge orientations by exploiting intervention-induced structural differences, resulting in the unique -CPDAG within the -MEC. The approach provides convergence guarantees and differential privacy by sharing regrets rather than raw graphs, and it demonstrates improved edge orientation and tighter equivalence classes on synthetic data. This work advances practical federated causal discovery by accounting for unknown interventions and privacy constraints, with implications for multi-center studies and decentralized datasets.

Abstract

Most causal discovery methods recover a completed partially directed acyclic graph representing a Markov equivalence class from observational data. Recent work has extended these methods to federated settings to address data decentralization and privacy constraints, but often under idealized assumptions that all clients share the same causal model. Such assumptions are unrealistic in practice, as client-specific policies or protocols, for example, across hospitals, naturally induce heterogeneous and unknown interventions. In this work, we address federated causal discovery under unknown client-level interventions. We propose I-PERI, a novel federated algorithm that first recovers the CPDAG of the union of client graphs and then orients additional edges by exploiting structural differences induced by interventions across clients. This yields a tighter equivalence class, which we call the -Markov Equivalence Class, represented by the -CPDAG. We provide theoretical guarantees on the convergence of I-PERI, as well as on its privacy-preserving properties, and present empirical evaluations on synthetic data demonstrating the effectiveness of the proposed algorithm.
Paper Structure (16 sections, 12 theorems, 17 equations, 6 figures, 1 table)

This paper contains 16 sections, 12 theorems, 17 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries & Problem Setup
  3. Causal Graphs.
  4. Interventions.
  5. Greedy Equivalence Search.
  6. Regret-based Federated Causal Discovery
  7. Regret-Based Federated Causal Discovery with Unknown Interventions
  8. Federated CPDAG Aggregation
  9. Orientation Refinement
  10. $\mathbf{\Phi}$-Markov Equivalence Class.
  11. Privacy Guarantees
  12. Experiments
  13. Synthetic Data.
  14. Metrics.
  15. Results.
  16. ...and 1 more sections

Key Result

Theorem 3.2

Let $G$ denote the true server causal DAG, and let $\mathcal{C}(G)$ denote its corresponding CPDAG. Let $\mathbf{\Phi}$ denote the family of unknown intervention targets across the $K$ clients. For each client $k \in \{1,\ldots,K\}$, let $G_{\phi^k}$ denote the client-specific causal DAG. Let $\hat{

Figures (6)

  • Figure 1: (a) True causal DAG; (b) Client CPDAG with no interventions; (c) Client CPDAG with interventions.
  • Figure 2: (a) Server PDAG obtained at an intermediate iteration (prior to convergence); (b) Intersection Server CPDAG and CPDAG client 1; (c) Intersection Server CPDAG and CPDAG client 2; (d) Target CPDAG result of the second phase of I-PERI.
  • Figure 3: (a) Server PDAG obtained at an intermediate iteration (prior to convergence); (b) Intersection Server CPDAG and skeleton of client 1 CPDAG; (c) Intersection Server CPDAG and skeleton of client 2 CPDAG; (d) target $\Phi$-CPDAG result of the second phase of I-PERI.
  • Figure 4: Three sets of server DAGs with their associated intervention targets where each set contains server DAGs that are $\mathbf{\Phi}$-Markov equivalent.
  • Figure 5: Example showing that, for two server-level DAGs that are $\mathbf{\Phi}$-Markov equivalent, the corresponding skeleton of client-level mutilated graphs are not necessarily identical.
  • ...and 1 more figures

Theorems & Definitions (27)

  • Definition 2.1: General intervention
  • Definition 2.2: Mutilated graph
  • Definition 2.3: Intersection between Graphs
  • Definition 2.4: Inclusion between Graphs
  • Definition 2.5: Regret
  • Example 3.1
  • Theorem 3.2
  • Example 3.3
  • Definition 3.1: $\mathbf{\Phi}$-Markov Equivalence
  • Theorem 3.4: Characterization of $\mathbf{\interventions}$-Markov Equivalence Class
  • ...and 17 more