Covariate Balancing for Estimating Continuous Treatment Effects Using Supremum of Integral Probability Metrics

Estimating causal effects from observational data requires controlling for confounding. Weighting approaches seek to remove dependence between treatment and confounders by reweighting the units. We propose Covariate Balancing using Supremum of Integral Probability Metrics (CBSupIPM), a weighting method for continuous treatments. CBSupIPM learns weights by minimizing the largest discrepancy, over treatment levels, between the weighted conditional covariate distribution and the population covariate distribution. We establish uniform convergence of the weighted Nadaraya-Watson estimator of the average dose-response function (ADRF) based on CBSupIPM weights. When an ideal weight function is sufficiently smooth and the true response surface belongs to a suitable class, the estimator matches the nonparametric convergence rate of the corresponding regression estimator in an unconfounded setting, up to logarithmic factors. Also, uniform consistency is guaranteed even when the true response surface belongs to a broader class, such as uniformly Lipschitz continuous functions in the covariates. For the numerical experiments, we construct a realistic semi-synthetic dataset using the National Medical Expenditure Survey (NMES). We also use repeated simulations with synthetic data to assess method performance more reliably and examine how it changes as different data-generating factors vary. In these numerical experiments, CBSupIPM shows the most favorable empirical performance in terms of ADRF estimation error among the comparators in most settings.