# Kernel Discrepancy-Based Rerandomization for Controlled Experiments

## TL;DR

This paper proposes a kernel discrepancy-based rerandomization method to improve causal inference precision in controlled experiments, adaptable to multiple treatment groups and complex covariate relationships.

## Contribution

It introduces a novel kernel discrepancy framework for rerandomization, addressing limitations of existing methods and providing a computationally efficient strategy for factorial experiments.

## Key findings

- Significantly reduces estimator variance in numerical studies.
- Linear kernel is optimal for linear relationships.
- L2-discrepancy performs robustly under model uncertainty.

## Abstract

This paper introduces a kernel discrepancy-based framework for rerandomization to enhance the precision of causal inference in controlled experiments. We demonstrate that the kernel discrepancy is the key part of the variance upper bound for the difference-in-means estimator, thereby establishing a theoretical rationale for its use. It quantifies the difference between empirical covariate distributions of treatment groups. We can choose a suitable kernel function and the corresponding discrepancy to accommodate simple or complex relationships between the outcome and the covariates. The proposed framework efficiently applies to any number of treatment groups, overcoming a significant limitation of existing methods. Furthermore, we develop a computationally efficient composite strategy for factorial experiments by recursively applying two- or multi-group rerandomizations. Numerical studies demonstrate that our approach significantly reduces estimator variance, with the linear kernel being optimal for linear relationships and the $\mathcal{L}_2$-discrepancy offering robust performance under model uncertainty.