Balancing covariates in randomized experiments with the Gram-Schmidt Walk design

The design of experiments involves an inescapable compromise between covariate balance and robustness. This paper provides a formalization of this trade-off and introduces an experimental design that allows experimenters to navigate it. The design is specified by a robustness parameter that bounds the worst-case mean squared 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 potentially many covariates. The achieved level of balance is better than previously known possible and considerably better than what a fully random assignment would produce. We show that the mean squared error of the estimator is bounded by the minimum of the loss function of an implicit ridge regression of the potential outcomes on the covariates. The estimator does not itself conduct covariate adjustment, so one can interpret the approach as regression adjustment by design. Finally, we provide both a central limit theorem and non-asymptotic tail bounds for the estimator, which facilitate the construction of confidence intervals.