[Paper Review] Balancing covariates in randomized experiments using the Gram-Schmidt walk.

Christopher Harshaw, Fredrik Sävje+2 more|arXiv (Cornell University)|Nov 8, 2019

Advanced Causal Inference Techniques22 references11 citations

TL;DR

This paper proposes a novel experimental design using the Gram-Schmidt walk to balance covariates in randomized experiments while controlling robustness via a tunable parameter. The method minimizes worst-case mean square error of average treatment effect estimators by achieving superior covariate balance compared to full randomization, effectively enabling regression adjustment through design.

ABSTRACT

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.

Motivation & Objective

• Address the trade-off between covariate balance and robustness in experimental design.
• Develop a design that ensures strong balance across all linear functions of covariates under a specified robustness constraint.
• Improve upon fully random assignment by achieving significantly better covariate balance without sacrificing robustness.
• Provide non-asymptotic tail bounds for the estimator to support conservative inference.
• Enable regression adjustment by design through explicit control over estimation error via ridge regression loss minimization.

Proposed method

• Use the Gram-Schmidt walk to generate assignment vectors that balance covariates while respecting a robustness parameter.
• Formulate the design as a constrained optimization problem that bounds the worst-case mean square error of the average treatment effect estimator.
• Link the estimator's mean square error to the loss function of a ridge regression of potential outcomes on covariates.
• Introduce a parameter that controls the trade-off between balance and robustness, allowing experimenter-defined precision levels.
• Derive non-asymptotic tail bounds for the estimator to support valid confidence intervals.
• Ensure the design is computationally feasible and achieves near-optimal balance given the robustness constraint.

Experimental results

Research questions

• RQ1 How can experimental design be structured to simultaneously balance all linear functions of covariates while controlling robustness?
• RQ2 What is the achievable level of covariate balance under a given robustness constraint, and how does it compare to full randomization?
• RQ3 Can the mean square error of the average treatment effect estimator be bounded using a ridge regression loss framework?
• RQ4 How do non-asymptotic tail bounds for the estimator support the construction of conservative confidence intervals?
• RQ5 To what extent does the proposed design enable regression adjustment by design rather than by post-hoc adjustment?

Key findings

• The proposed design achieves significantly better covariate balance than full randomization, approaching the theoretical optimum for the given robustness level.
• The worst-case mean square error of the average treatment effect estimator is bounded by the minimum of the ridge regression loss of potential outcomes on covariates.
• The method enables regression adjustment by design, as the estimator’s error is directly controlled through the ridge regression loss framework.
• Non-asymptotic tail bounds are derived, allowing for the construction of conservative confidence intervals under finite-sample conditions.
• The design is tunable via a single parameter that controls the trade-off between robustness and balance, offering practical flexibility for experimenters.