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Effective Causal Discovery under Identifiable Heteroscedastic Noise Model
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Proposes an identifiable heteroscedastic noise model (HNM) for multivariate DAG learning, with a two-phase iterative continuous-optimization algorithm (ICDH) that alternates between estimating noise variances and structural parameters.
Problem
Causal DAG structure learning from observational data when exogenous noise variances vary across variables and observations (heteroscedastic noise), which violates the homoscedastic assumptions of most continuous-optimization methods.
Approach
Relaxed sufficient conditions (nonlinear causal functions, piecewise noise-variance functions, Gaussian independent noise) prove identifiability of a multivariate heteroscedastic noise model; a shared-first-layer MLP parameterization models both mean and per-observation variance, and an alternating Phase-I (fit variances with fixed structure) / Phase-II (fit structure under acyclicity with augment
Key results
- On Sachs, ICDH achieves SHD 13 and auPRC 0.4673, matching GraN-DAG, GraN-DAG++ and HOST on SHD and beating NOTEARS-MLP (SHD 15) and GOLEM variants.
- On the 99 cause-effect pairs, ICDH correctly infers 52/99 pairs versus 39 for NOTEARS-MLP and 36 for NOTEARS, though it has lower weighted accuracy than HEC (0.71) and CAREFL (0.73).
- On 50-variable ER1 synthetic graphs, ICDH reaches SHD 134.5±23.4 versus 144.1±38.0 for NOTEARS-MLP and 161.1±30.8 for GraN-DAG++.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use ICDH instead of NOTEARS-MLP | DAG learning on Sachs · Sachs | auSHDC: 21.95 -> 19.27 | Helped |
| Use ICDH instead of the strongest listed baseline | DAG learning on Sachs · Sachs | SHD: 13 -> 13 | No clear effect |
| Use ICDH instead of NOTEARS-MLP | DAG learning on Sachs · Sachs | auPRC: 0.3427 -> 0.4673 | Helped |
| Use ICDH instead of NOTEARS-MLP | Whole-graph causal discovery · Cause-effect pairs | Accuracy: 39/99 -> 52/99 | Helped |
| Use ICDH instead of CAREFL | Whole-graph causal discovery · Cause-effect pairs | Weighted Accuracy: 0.73 -> 0.58 | Hurt |
| Use ICDH instead of NOTEARS-MLP | DAG learning · Synthetic ER1, 50 variables | SHD: 144.1±38.0 -> 134.5±23.4 | Helped |
| Use ICDH instead of GraN-DAG++ | DAG learning · Synthetic ER1, 50 variables | SHD: 161.1±30.8 -> 134.5±23.4 | Helped |
| Use ICDH instead of HOST | DAG learning · Synthetic ER1, 50 variables | SHD: 152.5±24.8 -> 134.5±23.4 | Helped |
| Model heteroscedastic noise with ICDH rather than the compared baselines | Synthetic heteroscedastic data · Synthetic heteroscedastic data | DAG accuracy | Helped |
| Use ICDH rather than the compared continuous SCM methods | Synthetic homoscedastic data · Synthetic homoscedastic data | DAG accuracy | No clear effect |
Surprising findings
- The paper reports that jointly optimizing the structure and variance parameters yields degenerate solutions with large reconstruction loss and unreasonably large variances, motivating the alternating scheme.
Limitations
- Identifiability guarantee is for sufficient conditions; the method only guarantees convergence to a stationary point.
- Weighted accuracy on cause-effect pairs is lower than bivariate methods HEC and CAREFL.
- Sachs and cause-effect results rely on post-thresholding and heuristic threshold choices for SHD.
Experimental setup
Datasets: Sachs cause-effect pairs synthetic ER graphs (homo-EV, homo-NV, hetero)
Benchmarks: Sachs cause-effect pairs synthetic ER1/ER3 graphs
Metrics: SHD auSHDC auPRC accuracy weighted accuracy
Baselines: NOTEARS-MLP GOLEM-EV GOLEM-NV GraN-DAG GraN-DAG++ GFBS HOST HEC CAREFL CAM LiNGAM GES
Builds on: NOTEARS NOTEARS-MLP GraN-DAG GOLEM HOST
Closest papers by meaning
- Strong and Weak Identifiability of Optimization-based Causal Discovery in Non-linear Additive Noise Models (ICML 2025 · similarity 0.90)
- Causal Discovery via Bayesian Optimization (ICLR 2025 · similarity 0.89)
- Interventional Causal Discovery in a Mixture of DAGs (NeurIPS 2024 · similarity 0.89)
- CoLiDE: Concomitant Linear DAG Estimation (ICLR 2024 · similarity 0.88)
- Coarsening Linear Non-Gaussian Causal Models with Cycles (NeurIPS 2026 · similarity 0.88)
- Boosting Causal Structure Learning: An Asymmetric Exponential Modulation Gaussian-Based Adaptive Sample Reweighting Framework (AAAI 2025 · similarity 0.88)
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