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Less Greedy Equivalence Search

Adiba Ejaz, Elias Bareinboim

TL;DR

This work introduces Less Greedy Equivalence Search (LGES), a family of score-based causal discovery algorithms designed to overcome the scalability and finite-sample limitations of Greedy Equivalence Search (GES). By replacing the strict highest-scoring forward insertion with ConservativeInsert and SafeInsert strategies, LGES achieves significant speedups (up to $10$-fold) and reduces structural errors, while accommodating prior knowledge through guided search and remaining asymptotically correct under misspecification. The authors further extend the framework with I-Orient, a scalable, score-based procedure that refines observational MECs using interventional data, enabling identification of a smaller, more informative I-MEC. Empirical results across synthetic and real-world datasets demonstrate LGES’s robustness to misspecification, superior accuracy, and greater scalability compared with GES, PC, and NoTears, with interventional data providing additional gains. Collectively, the methods offer a practical, theoretically sound pathway for robust causal discovery in high-dimensional settings and with heterogeneous data sources.

Abstract

Greedy Equivalence Search (GES) is a classic score-based algorithm for causal discovery from observational data. In the sample limit, it recovers the Markov equivalence class of graphs that describe the data. Still, it faces two challenges in practice: computational cost and finite-sample accuracy. In this paper, we develop Less Greedy Equivalence Search (LGES), a variant of GES that retains its theoretical guarantees while partially addressing these limitations. LGES modifies the greedy step; rather than always applying the highest-scoring insertion, it avoids edge insertions between variables for which the score implies some conditional independence. This more targeted search yields up to a \(10\)-fold speed-up and a substantial reduction in structural error relative to GES. Moreover, LGES can guide the search using prior knowledge, and can correct this knowledge when contradicted by data. Finally, LGES can use interventional data to refine the learned observational equivalence class. We prove that LGES recovers the true equivalence class in the sample limit, even with misspecified knowledge. Experiments demonstrate that LGES outperforms GES and other baselines in speed, accuracy, and robustness to misspecified knowledge. Our code is available at https://github.com/CausalAILab/lges.

Less Greedy Equivalence Search

TL;DR

This work introduces Less Greedy Equivalence Search (LGES), a family of score-based causal discovery algorithms designed to overcome the scalability and finite-sample limitations of Greedy Equivalence Search (GES). By replacing the strict highest-scoring forward insertion with ConservativeInsert and SafeInsert strategies, LGES achieves significant speedups (up to -fold) and reduces structural errors, while accommodating prior knowledge through guided search and remaining asymptotically correct under misspecification. The authors further extend the framework with I-Orient, a scalable, score-based procedure that refines observational MECs using interventional data, enabling identification of a smaller, more informative I-MEC. Empirical results across synthetic and real-world datasets demonstrate LGES’s robustness to misspecification, superior accuracy, and greater scalability compared with GES, PC, and NoTears, with interventional data providing additional gains. Collectively, the methods offer a practical, theoretically sound pathway for robust causal discovery in high-dimensional settings and with heterogeneous data sources.

Abstract

Greedy Equivalence Search (GES) is a classic score-based algorithm for causal discovery from observational data. In the sample limit, it recovers the Markov equivalence class of graphs that describe the data. Still, it faces two challenges in practice: computational cost and finite-sample accuracy. In this paper, we develop Less Greedy Equivalence Search (LGES), a variant of GES that retains its theoretical guarantees while partially addressing these limitations. LGES modifies the greedy step; rather than always applying the highest-scoring insertion, it avoids edge insertions between variables for which the score implies some conditional independence. This more targeted search yields up to a -fold speed-up and a substantial reduction in structural error relative to GES. Moreover, LGES can guide the search using prior knowledge, and can correct this knowledge when contradicted by data. Finally, LGES can use interventional data to refine the learned observational equivalence class. We prove that LGES recovers the true equivalence class in the sample limit, even with misspecified knowledge. Experiments demonstrate that LGES outperforms GES and other baselines in speed, accuracy, and robustness to misspecified knowledge. Our code is available at https://github.com/CausalAILab/lges.

Paper Structure

This paper contains 56 sections, 15 theorems, 4 equations, 13 figures, 1 table, 9 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Notation.
  4. Causal graphs bareinboim:etal20pearl:09.
  5. Markov equivalence classes pearl:88spirtes:etal00.
  6. Greedy Equivalence Search chickering:2002meek:97.
  7. Less Greedy Equivalence Search
  8. Generalizing GES
  9. An improved insertion strategy
  10. Learning with prior knowledge
  11. Initialization from prior assumptions.
  12. Guided search from prior assumptions.
  13. The Less Greedy Equivalence Search algorithm
  14. Score-based learning from interventional data
  15. Background on interventions.
  16. ...and 41 more sections

Key Result

Theorem 1

Let $\mathcal{E}$ denote the Markov equivalence class that results from GGES (Alg. alg:gges) initialised from an arbitrary MEC $\mathcal{E}_0$ and let $P(\mathbf{v})$ denote the distribution from which the data $\mathbf{D}$ was generated. Then, as the number of samples goes to infinity, $\mathcal{E}

Figures (13)

  • Figure 1: A CPDAG $\mathcal{E}$ and the three DAGs in the MEC it represents, encoding $X \perp_d Y \mid Z$.
  • Figure 2: Possible trajectories, $\tau_1$ and $\tau_2$, that GES may take in the forward phase to obtain an MEC with respect to which a given distribution $P(\mathbf{v})$ is Markov. The true MEC is $\mathcal{E}^*$ (top right). In each trajectory, $\mathcal{E}^{(t+1)}$ results from applying some Insert operator to $\mathcal{E}^{(t)}$.
  • Figure 3: Illustration of some Insert operators that may be applied to the MEC $\mathcal{E}^{(1)}$ at $t=1$ in Fig. \ref{['fig:ges-traj']}. These operators correspond to various edge additions to the DAGs $\mathcal{G}_1, \mathcal{G}_2 \in \mathcal{E}^{(1)}$, where $\mathcal{G}_1$ orients $Z-Y$ as $Z \to Y$ and $\mathcal{G}_2$ orients $Z-Y$ as $Y \to Z$.
  • Figure 4: Performance of algorithms on $50$ simulated datasets from Erdős–Rényi graphs with $p$ variables. Lower is better (more accurate / faster) across all plots. (a) LGES outperforms baselines in accuracy and runtime on graphs with $2p$ edges in expectation, given $n=10^4$ observational samples and no prior knowledge. (b) Less greedy insertion improves several GES variants on graphs with $p=150$ variables and $2p$ and $3p$ edges in expectation, given $n=10^4$ observational samples (and $n=10^3$ samples per intervention for GIES) and no prior knowledge. LGES-0, LGES, LGES+, and LGIES are the less greedy variants of GES, XGES-0, XGES, and GIES respectively. (c) Given prior knowledge in the form of $3m/4$ required edges when the true graph contains $m$ edges, LGES' prioritization strategy is more robust to misspecification in the knowledge than initialization with the same knowledge, given $n=10^3$ observational samples. See Sec. \ref{['adx:sec:experiments']} for additional results.
  • Figure 5: Performance of algorithms on 50 simulated datasets from Erdős–Rényi graphs with $p$ variables and (left)$2p$ edges (right)$3p$ edges in expectation, given $n=10^4$ observational samples and $n=10^3$ samples per intervention (Sec. \ref{['adx:sec:exp:int']}). Lower is better (more accurate / faster) across all plots. LGIES significantly outperforms GIES.
  • ...and 8 more figures

Theorems & Definitions (42)

  • Definition 1: Locally consistent scoring criterion chickering:2002
  • Example 1
  • Definition 2: Insert operator, chickering:2002
  • Theorem 1: Correctness of GGES
  • Proposition 1
  • Example 2
  • Proposition 2: Correctness of SafeInsert
  • Example 3
  • Corollary 1: Correctness of LGES
  • Remark 1
  • ...and 32 more