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Tightening Causal Bounds via Covariate-Aware Optimal Transport
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Relaxes conditional optimal transport for causal partial identification into a standard OT problem with mirror covariates and a quadratic penalty, yielding narrower PI bounds that converge to the COT bound as the penalty grows.
Problem
Partial identification bounds for causal estimands depending on the joint distribution of potential outcomes can be tightened by covariates, but the exact conditional OT (COT) formulation is hard to estimate and compute.
Approach
Introduce mirror covariates Z(0), Z(1) with a penalty eta*||Z(0)-Z(1)||^2 so the problem becomes an unconditional OT problem that interpolates between the unconditional OT bound (eta=0) and the COT bound (eta to infinity), and can be solved with off-the-shelf OT solvers.
Key results
- On the STAR data, the correlation PI interval length at eta=50 is 1.479 versus 1.994 for unconditional OT (about 74%).
- Neymanian variance ratio with eta=50 is 0.951 relative to unconditional OT, roughly a 5% sample-size reduction, while DualBounds ratios exceed 1.
- Theory: V_ip(eta) is non-decreasing in eta, equals V_u at eta=0 and converges to V_c as eta goes to infinity; estimator error bound scales as lambda*eta*N^{-1/4} or N^{-1/d}.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Increase the mirror-covariate penalty from η = 0 to η = 50 (ablation) | V_ip(η) · STAR Neymanian confidence upper bound | relative sample size: 1.0 -> 0.951 | Helped |
| Increase the mirror-covariate penalty from η = 0 to η = 50 (ablation) | V_ip(η) · STAR PI set for correlation | PI interval length: 1.994 -> 1.479 | Helped |
| Use the covariate-aware method with η = 50 instead of unconditional OT (η = 0) | V_ip(η) · STAR PI set for correlation | upper bound: 0.997 -> 0.887 | Helped |
| Use the covariate-aware method with η = 50 instead of unconditional OT (η = 0) | V_ip(η) · STAR Neymanian confidence interval | relative sample size: 1.0 -> 0.951 | Helped |
| Use the covariate-aware method instead of DualBounds | STAR Neymanian confidence interval · STAR | relative sample size: 1.302 -> 0.951 | Helped |
Limitations
- Finite-sample estimator accuracy deteriorates with more covariates, particularly bias.
- No general method to bound the curvature constant with respect to eta.
- Convergence rate requires quadratic costs and smoothness assumptions on Brenier potentials.
Experimental setup
Datasets: STAR Demonstration Project
Benchmarks: STAR Demonstration Project
Metrics: L1 error to oracle V_c upper and lower bounds of PI sets relative sample size for Neymanian CI
Baselines: unconditional optimal transport DualBounds (ridge nuisance) DualBounds (knn nuisance)
Builds on: conditional optimal transport Neymanian confidence interval DualBounds
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