Design-Based Finite-Sample Analysis for Regression Adjustment

In randomized experiments, regression adjustment leverages covariates to improve the precision of average treatment effect (ATE) estimation without requiring a correctly specified outcome model. Although well understood in low-dimensional settings, its behavior in high-dimensional regimes---where the number of covariates $p$ may exceed the number of observations $n$---remains underexplored. Furthermore, existing theory is largely asymptotic, providing limited guidance for finite-sample inference. We develop a design‑based, non‑asymptotic analysis of the regression‑adjusted ATE estimator under complete randomization. Specifically, we derive finite-sample valid confidence intervals with explicit, instance-adaptive widths that remain informative even when $p > n$. These intervals rely on oracle (population-level) quantities, and we outline computable data-driven envelopes. Our approach hinges on a refined swap sensitivity analysis: stochastic fluctuation is controlled via a variance‑adaptive Doob martingale, while design bias is bounded using Stein’s method of exchangeable pairs. Covariate geometry---through leverages and cross-leverages---governs concentration and bias, clarifying when regression adjustment improves on the difference-in-means baseline.