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Balancing covariates in randomized experiments using the Gram-Schmidt walk

The design of experiments involves a compromise between covariate balance and robustness. This paper introduces an experimental design that admits precise control over this trade-off. The design is specified by a parameter that bounds the worst-case mean square error of an estimator of the average treatment effect. Subject to the experimenter's desired level of robustness, the design aims to simultaneously balance all linear functions of the targeted covariates. The achieved level of balance is considerably better than what a fully random assignment would produce, and it is close to optimal given the desired level of robustness. We show that the mean square error of the estimator is bounded by the minimum of the loss function of a ridge regression of the potential outcomes on the covariates. One may thus interpret the approach as regression adjustment by design. Finally, we provide non-asymptotic tail bounds for the estimator, which facilitate the construction of conservative confidence intervals.