Research map / Learning theory and algorithms / Combinatorial and search optimization
CoLiDE: Concomitant Linear DAG Estimation
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Proposes CoLiDE, a convex concomitant-scale least-squares score with a log-determinant acyclicity penalty that jointly estimates linear DAG structure and (homo- or heteroscedastic) noise levels from observational data.
Problem
Learning DAG structure from observational data under a linear SEM when exogenous noise variances are unknown and may differ across nodes, which makes lasso-type penalties require retuning and ordinary least squares biased.
Approach
Adapt smoothed concomitant lasso and generalized concomitant multi-task lasso ideas to DAG learning: a jointly convex scaled least-squares score in (W, sigma) decouples the sparsity parameter from noise level, optimized via DAGMA-style log-det penalty continuation and block coordinate descent with closed-form noise updates.
Key results
- On 200-node ER4 homoscedastic Gaussian graphs with noise variance 5.0, CoLiDE-EV reaches SHD 105.6±51.5 versus 194.4±36.2 for DAGMA.
- On 200-node ER4 heteroscedastic Gaussian graphs, CoLiDE-NV achieves SHD 390.7±35.6 versus 470.2±50.6 for DAGMA and 397.8±27.8 for SortNRegress.
- On the Sachs dataset, CoLiDE-NV reports SHD 12, the lowest among the compared methods (next best 13 for SortNRegress, GES and CoLiDE-EV).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use CoLiDE-EV instead of DAGMA | Linear DAG estimation · ER4, 200 nodes, homoscedastic Gaussian noise variance 1.0 | SHD: 100.1$\pm$41.8 -> 87.3$\pm$33.7 | Helped |
| Use CoLiDE-EV instead of DAGMA | Linear DAG estimation · ER4, 200 nodes, homoscedastic Gaussian noise variance 5.0 | SHD: 194.4$\pm$36.2 -> 105.6$\pm$51.5 | Helped |
| Use CoLiDE-NV instead of SortNRegress | Linear DAG estimation · ER4, 200 nodes, heteroscedastic Gaussian noise | SHD: 397.8$\pm$27.8 -> 390.7$\pm$35.6 | Helped |
| Use CoLiDE-NV instead of SortNRegress | Linear DAG estimation · ER4, 200 nodes, heteroscedastic Gaussian noise | SID: 22560$\pm$1749 -> 22734$\pm$1767 | Hurt |
| Use CoLiDE-NV instead of SortNRegress | Linear DAG estimation · ER4, 200 nodes, heteroscedastic Gaussian noise | SHD-C: 402.1$\pm$26.3 -> 407.9$\pm$37.5 | Hurt |
| Use CoLiDE-NV instead of SortNRegress | Linear DAG estimation · ER4, 200 nodes, heteroscedastic Gaussian noise | FDR: 0.20$\pm$0.01 -> 0.25$\pm$0.03 | Hurt |
| Use CoLiDE-NV instead of CoLiDE-EV | Linear DAG estimation · Sachs | SHD: 13 -> 12 | Helped |
| Use CoLiDE-NV instead of CoLiDE-EV | Linear DAG estimation · Sachs | SID: 47 -> 46 | Helped |
| Use CoLiDE-NV instead of CoLiDE-EV | Linear DAG estimation · Sachs | TPR: 0.29 -> 0.35 | Helped |
| Use CoLiDE-NV instead of CoLiDE-EV | Linear DAG estimation · ER4, 200 nodes, heteroscedastic Gaussian noise | TPR: 0.68$\pm$0.02 -> 0.68$\pm$0.01 | No clear effect |
Surprising findings
- CoLiDE-NV, although overparametrized for homoscedastic problems, performs on par with CoLiDE-EV or as second best.
- GOLEM's own Gaussian profile likelihood is outperformed by the more general CoLiDE formulation even in GOLEM's specific Gaussian scenario.
Limitations
- Scope limited to observational data under linear SEMs; nonlinear and interventional settings left for future work.
- Heteroscedastic DAG structure is non-identifiable from observational data, making recovery harder as d grows.
- Post-processing thresholding at 0.3 is still required in practice to reduce false positives.
Experimental setup
Datasets: Sachs dataset
Benchmarks: Sachs dataset
Metrics: SHD SID SHD-C FDR TPR noise estimation relative error
Baselines: GES GOLEM DAGMA SortNRegress DAGuerreotype NoTears
Builds on: DAGMA NOTEARS smoothed concomitant lasso GOLEM
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