# Balancing Covariates in Randomized Experiments with the Gram-Schmidt Walk Design

arXiv: https://arxiv.org/abs/1911.03071 · HTML: https://mlsanity.com/paper/1911.03071/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Nov 2019.

## Built on by (most cited first, 13 of 13)

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- [Causal-EPIG: A Prediction-Oriented Active Learning Framework for CATE Estimation](https://mlsanity.com/paper/2509.21866/) (2025)
- [Synthetic Principal Component Design: Fast Covariate Balancing with Synthetic Controls](https://mlsanity.com/paper/2211.15241/) (2022)
- [Benchmarking Estimators for Natural Experiments: A Novel Dataset and a Doubly Robust Algorithm](https://mlsanity.com/paper/2409.04500/) (2024)
- [Observationally Informed Adaptive Causal Experimental Design](https://mlsanity.com/paper/2603.03785/) (2026)
- [Leveraging Causal Graphs for Blocking in Randomized Experiments](https://mlsanity.com/paper/2111.02306/) (2021)
- [Limits of Approximating the Median Treatment Effect](https://mlsanity.com/paper/2403.10618/) (2024)
- [WHOMP: Optimizing Randomized Controlled Trials via Wasserstein Homogeneity](https://mlsanity.com/paper/2409.18504/) (2024)
- [Fast Rerandomization for Balancing Covariates in Randomized Experiments: A Metropolis-Hastings Framework](https://mlsanity.com/paper/2602.07613/) (2026)
- [Self-Balancing Sequential Sampling: Fast Convergence with Controlled Predictability](https://mlsanity.com/paper/2607.20818/) (2026)

## Abstract (arXiv)

The design of experiments involves a compromise between covariate balance and robustness. This paper provides a formalization of this trade-off and describes 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. Less robustness allows for more balance. We show that the mean squared error of the estimator is bounded in finite samples by the minimum of the loss function of an implicit ridge regression of the potential outcomes on the covariates. Asymptotically, the design perfectly balances all linear functions of a growing number of covariates with a diminishing reduction in robustness, effectively allowing experimenters to escape the compromise between balance and robustness in large samples. Finally, we describe conditions that ensure asymptotic normality and provide a conservative variance estimator, which facilitate the construction of asymptotically valid confidence intervals.