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Differentiable Structure Learning with Ancestral Constraints
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Integrates ancestral (path-existence) constraints into differentiable NOTEARS-style DAG learning via a binary-masked continuous relaxation and an order-guided initialization.
Problem
Differentiable structure learning of causal DAGs lacks a smooth, equivalent way to encode ancestral (path existence) constraints, and gradient-based optimization suffers from path order violations among constrained paths.
Approach
A binary-masked continuous relaxation b(W)∘ReLU(ε−p(W)) that deactivates the path-existence penalty once a path already exists, making the characterization equivalent for a sufficiently large path threshold; plus an order-guided initialization obtained by first enforcing the transitive-closure partial orders implied by the path constraints.
Key results
- On Sachs with 100 samples, PE-NOTEARS-100 reaches SHD 15 and F1 0.47 versus NOTEARS (902 samples) baseline and NOTEARS at 100 samples with SHD 22 and F1 0.34.
- Ablation over initialization shows the partial-order-guided initialization gives the lowest SHD and highest F1 with the lowest variance across six random initial guesses.
- The binary-masked path loss yields lower FDR than the intuitive ReLU path loss across path thresholds, and remains stable as the path threshold increases.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Add path-existence constraints (100%) | NOTEARS · Sachs-100 | F1: 0.34 -> 0.47 | Helped |
| Add path-existence constraints (75%) | NOTEARS · Sachs-200 | F1: 0.41 -> 0.48 | Helped |
| Add path-existence constraints (100%) | NOTEARS · Sachs-500 | F1: 0.42 -> 0.54 | Helped |
| Add path-existence constraints (75%) | NOTEARS · Sachs-902 | F1: 0.41 -> 0.50 | Helped |
| Use binary-masked path loss instead of intuitive relaxed path loss (ablation) | PE-NOTEARS · Synthetic datasets | F1 | Helped |
| Initialize with zero instead of the order-guided solution (ablation) | PE-NOTEARS · Synthetic datasets | F1 | Hurt |
| Use random initialization instead of the order-guided solution (ablation) | PE-NOTEARS · Same dataset as Figure 2 | F1 | Hurt |
| Initialize with the NOTEARS solution instead of the order-guided solution (ablation) | PE-NOTEARS · Same dataset as Figure 2 | F1 | Hurt |
| Vary path threshold ϵ from 0.1 to 10 (ablation) | PE-NOTEARS · Same dataset as Figure 3 | F1 | Helped |
| Use intuitive relaxed loss ¯p instead of binary-masked loss ˆp (ablation) | PE-NOTEARS · Same dataset as Figure 3 | F1 | Mixed |
| Use intuitive relaxed loss ¯p with order-guided initialization (ablation) | PE-NOTEARS · Same dataset as Figure 3 | F1 | Hurt |
Limitations
- Path existence formulation lacks explicit mechanism to prevent path order violation
- Computationally slower than acyclicity losses
- Path existence constraints are soft losses, not strictly enforced
Experimental setup
Datasets: Sachs (Sachs et al., 2005) synthetic ER and SF DAGs
Benchmarks: Sachs
Metrics: SHD TPR FDR F1 path recovery rate
Baselines: NOTEARS DAGMA GOLEM PE-NOTEARS-zero PE-NOTEARS-intuitive
Builds on: NOTEARS DAGMA GOLEM Wang et al. 2024 path-existence constraints Ban et al. 2024 partial order constraints
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