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Active Adaptive Experimental Design for Treatment Effect Estimation with Covariate Choice
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Proposes an active adaptive experiment for ATE estimation that jointly optimizes covariate density and propensity score, derives the semiparametric efficiency bound and its minimizers, and proposes an AIPW-with-importance-weighting estimator whose asymptotic variance attains the minimized bound.
Problem
Estimating average treatment effects with smaller asymptotic variance in adaptive experiments, where prior work only optimizes the treatment-assignment propensity score and not the covariate sampling density.
Approach
Generalize adaptive experiments by also choosing the covariate distribution from which units are sampled (covariate choice), derive efficient covariate density and propensity score (Neyman-type allocation) that minimize the semiparametric efficiency bound, and estimate them sequentially with an AIPW estimator with importance weighting (AAS-AIPWIW).
Key results
- Proposed AAS-AIPWIW experiment reduces empirical MSE relative to RCT, deterministic RCT, and AS-AIPW in simulations with T=2000 and 200 repetitions.
- Theory shows the efficient covariate density and propensity score give minimized bound tau* = (E_q[sigma(1)+sigma(0)])^2, which is at most the bound obtained when optimizing only the propensity score.
- AAS-AIPWIW asymptotic variance attains the minimized semiparametric efficiency bound (sqrt(T)(theta_hat - theta0) -> N(0, tau*)).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Optimize both covariate density and propensity score instead of optimizing only the propensity score | AAS-AIPWIS · ATE estimation with heterogeneous conditional variances | semiparametric efficiency bound | Helped |
| Use the efficient propensity score (Neyman allocation) instead of uniform treatment assignment | ATE experiment with fixed covariate density · ATE estimation | semiparametric efficiency bound | Helped |
| Use the AAS-AIPWIS experiment with estimated efficient covariate density and propensity score | ATE estimation · Asymptotic regime, T → ∞ | asymptotic variance | Helped |
| Use adaptive covariate-density and propensity-score optimization | ATE estimation; compared with DRCT, RCT, and AS-AIPW · Gaussian covariates, N(1, 25) | empirical MSE | Helped |
| Use adaptive covariate-density and propensity-score optimization | ATE estimation; compared with DRCT, RCT, and AS-AIPW · Uniform covariates, U(−10, 10) | empirical MSE | Helped |
| Use estimated probabilities rather than oracle efficient probabilities | AS-AIPW and AAS-AIPWIW · Simulation studies | empirical MSE | No clear effect |
Surprising findings
- AS-AIPW and AAS-AIPW perform almost identically to or better than their oracle versions that use the true optimal probabilities, which the authors attribute to a semiparametric paradox.
Limitations
- Assumes the covariate density q(x) is known to the experimenter.
- Asymptotic results rely on consistency of nuisance estimators and sample-splitting; high-dimensional case left to sparsity or benign-overfitting arguments.
- Empirical evaluation limited to one-dimensional covariates in simulations.
Experimental setup
Benchmarks: simulation study with Gaussian and uniform covariate distributions
Metrics: empirical MSE of ATE estimator
Baselines: randomized controlled trial deterministic RCT AS-AIPW AS-AIPW (Oracle)
Builds on: Neyman allocation A2IPW estimator double machine learning active learning under covariate shift
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