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Rerandomization with diminishing covariate imbalance and diverging number of covariates

Completely randomized experiments have been the gold standard for drawing causal inference because they can balance all potential confounding on average. However, they may suffer from unbalanced covariates for realized treatment assignments. Rerandomization, a design that rerandomizes the treatment assignment until a prespeciﬁed covariate balance criterion is met, has recently got attention due to its easy implementation, improved covariate balance and more efﬁcient inference. Researchers have then suggested to use the treatment assignments that minimize the covariate imbalance, namely the optimally balanced design. This has caused again the long-time controversy between two philosophies for designing experiments: randomization versus optimal and thus almost deterministic designs. Existing literature argued that rerandomization with overly balanced observed covariates can lead to highly imbalanced unobserved covariates, making it vulnerable to model misspeciﬁcation. On the contrary, rerandomization with properly balanced covariates can provide robust inference for treatment effects while sacriﬁc-ing some efﬁciency compared to the ideally optimal design. In this paper, we show it is possible that, by making the covariate imbalance diminishing at a proper rate as the sample size increases, rerandomization can achieve its ideally optimal precision that one can expect with perfectly balanced covariates, while still maintaining its robustness. We further investigate conditions on the number of covariates for achieving the desired optimality. Our results rely on a more delicate asymptotic analysis for rerandomization, allowing both diminishing covariate imbalance threshold (or equivalently the acceptance probability) and diverging number of covariates. The derived theory for rerandomization provides a deeper understanding of its large-sample property and can better guide its practical implementation. Furthermore, it also helps reconcile the controversy between randomized and optimal designs in an asymptotic sense.