Greedy Equivalence Search (GES)

• Greedy Equivalence Search (GES) is a causal discovery algorithm that learns DAG structures by moving along the edges of the characteristic imset polytope.
• It reframes structure learning as a linear optimization problem, leveraging edge-addition, deletion, and reorientation moves for enhanced performance.
• Extensions like the greedy CIM algorithm integrate skeleton constraints with polytope moves to improve model recovery and computational efficiency.

The Greedy Equivalence Search (GES) algorithm is a foundational method for causal discovery, particularly for estimating the structure of directed acyclic graphs (DAGs) that encode observable conditional independencies. Recent geometric analysis reframes GES as a discrete edge-walk on the characteristic imset polytope ($\operatorname{CIM}_p$), thereby offering unified conceptual clarity and extending the algorithmic toolkit for structure learning. All technical definitions and results below follow the original account by Studený, Hemmecke, Lindner and subsequent researchers (Linusson et al., 2021).

1. The Characteristic Imset Polytope Framework

GES operates on the space of Markov equivalence classes of DAGs. Each DAG $G$ on $[p]$ nodes is associated with its characteristic imset $c_G: \{S \subseteq [p]: |S| \geq 2\} \to \{0,1\}$, defined by $c_G(S) = 1$ iff $\exists\,i \in S$ such that $S \setminus\{i\} \subseteq \text{pa}_G(i)$. The vector $c_G$ encodes all membership in Markov equivalence classes.

The polytope $$\operatorname{CIM}_p := \text{conv}\{c_G: G \text{ a DAG on } [p]\} \subset \mathbb{R}^{2^p-p-1}$$ has as vertices exactly the characteristic imsets, uniquely representing Markov equivalence classes. This geometric reformulation turns causal structure learning into a linear optimization problem.

2. Skeleton Constraints and Faces of $\operatorname{CIM}_p$

Hybrid approaches to structure learning often begin by estimating an undirected skeleton $H$ of the target DAG. Correspondingly, the subset $$\operatorname{CIM}_H := \text{conv}\{c_G: G \text{ a DAG with skeleton } H\}$$ forms a face of the full polytope $\operatorname{CIM}_p$. By selecting a linear functional $w_{\varnothing,H}$ supported only on 2-subsets, the maximizers are precisely DAGs whose skeleton is a subgraph of $H$. This structural correspondence clarifies the constraint-enforcement mechanism in hybrid algorithms.

3. Geometric Realization of GES Moves as Edge-Walks

GES comprises three classes of elementary moves, each corresponding precisely to an edge of the polytope $\operatorname{CIM}_p$:

A) Edge-addition (“forward phase”): For DAG $G$ and pair $(i,j)$ non-adjacent in the skeleton, one can sometimes add $j \to i$ without acyclicity violation. Proposition 3.10 and Theorem 3.12 show $$c_{G+(j\to i)} = c_G + \sum_{S \subseteq \text{pa}_G(i)} e_{S\cup\{i,j\}}$$ and $[c_G, c_{G+(j\to i)}]$ is an edge of $\operatorname{CIM}_p$. This move is called an edge pair.

B) Edge-deletion (“backward phase”): Removal of $i \to j$ from the essential graph is the reverse operation, and since $\operatorname{CIM}_p$ is a 0/1-polytope, if two vertices differ in exactly one coordinate they span an edge (Lemma 3.1). Every valid deletion is therefore an edge of $\operatorname{CIM}_p$.

C) Edge-reorientation (“turn phase”): GES also reverses existing arrows (“Chickering-turn” operations) within the same equivalence class; when this switch creates a new MEC, the corresponding imsets again span an edge—see Proposition 3.5 and Theorem 3.8 for the explicit combinatorics.

In summary, every GES add, delete, or turn move is a walk along an edge of $\operatorname{CIM}_p$.

4. Extension Beyond Classical GES: Larger Classes of Polytope Moves

The edge pairs and turn pairs in $\operatorname{CIM}_p$ strictly generalize the classical GES moves—there exist polytope edges corresponding to equivalence class transitions that do not arise from simple edge reorientations in any DAG of the current class (Example 3.7), but still realize valid greedy steps. This suggests potential improvements over standard GES by expanding the move set at each sweep.

5. The Greedy CIM Algorithm

Leveraging the geometric generalization, the “greedy CIM” algorithm performs a simplex-type walk on $\operatorname{CIM}_p$, considering all available edge-pairs (additions and deletions) and turn-pairs permitted by acyclicity. At each step, it selects the move yielding the maximal increase in a chosen score (e.g., BIC). Its algorithmic skeleton is:

• Initialize $G$ to the empty DAG.
• Edge-phase: evaluate all valid edge-pairs $\{G,H\}$ with $\text{BIC}(H) > \text{BIC}(G)$, and take the best.
• Turn-phase: evaluate all valid turn-pairs $\{G,H\}$ (consistent with acyclicity) with $\text{BIC}(H) > \text{BIC}(G)$, and take the best.
• Repeat until convergence.

All moves correspond to polytope edges, so the procedure is a greedy edge-walk in the sense of linear programming.

6. Empirical Performance and Hybrid Variants

On Gaussian linear-SEM data with $p=8$ and $n=10\,000$, the breadth-first, phased version of greedy CIM matches or outperforms GES and GIES in structure recovery. The “skeletal greedy CIM” hybrid variant—where a skeleton is first learned by CI tests, followed by greedy CIM turn-pairs—outperforms MMHC and PC in model and SHD metrics. Reported computational times are $1\,\text{s}$ (skeletal) to $10\,\text{s}$ (full) per replication in an R+Python implementation.

7. Synthesis: Unifying Geometric Perspective

GES is a restricted greedy edge-walk on the characteristic imset polytope $\operatorname{CIM}_p$. Casting GES, GIES, and MMHC in this geometric framework exposes the combinatorial structure of their operations, justifies new algorithmic extensions, and connects skeleton constraints with polytope faces. The greedy CIM algorithm demonstrates principled improvement by exploiting polytope edges not traversable by standard GES, offering both a theoretical unification and a practically robust methodology for greedy causal discovery (Linusson et al., 2021).