Design-based finite-sample analysis for regression adjustment

2025-11-19

AI Summary

This paper addresses finite-sample inference for regression-adjusted average treatment effect (ATE) estimation in high-dimensional randomized experiments where $p > n$. We propose a design-based non-asymptotic analytical framework—distinct from existing approaches relying on correct model specification or large-sample approximations. For the first time, we establish a non-asymptotic theory for high-dimensional regression-adjusted estimators, leveraging Stein’s exchangeable pair technique, Doob martingales, and Freedman’s inequality to characterize how the geometric structure of covariates governs estimation precision. The resulting confidence intervals are instance-adaptive and data-driven, with explicitly computable widths that remain informative even when $p > n$. This work provides the first rigorous, practical, design-based theoretical foundation for robust causal inference in small-sample, high-dimensional settings.

Technology Category

• Reasoning under Uncertainty: Causality
• Machine Learning: Calibration & Uncertainty Quantification
• Intelligent Robots: State Estimation

Application Category

• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

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 also outline data-driven envelopes that are computable from observed data. Our approach hinges on a refined swap sensitivity analysis: stochastic fluctuation is controlled via a variance-adaptive Doob martingale and Freedman's inequality, while design bias is bounded using Stein's method of exchangeable pairs. The analysis suggests how covariate geometry governs concentration and bias through leverages and cross-leverages, shedding light on when and how regression adjustment improves on the difference-in-means baseline.

Problem

• Analyzing regression adjustment for ATE estimation in high-dimensional randomized experiments
• Developing finite-sample confidence intervals when covariates exceed observations
• Investigating covariate geometry's impact on concentration and bias properties

Innovation

• Non-asymptotic confidence intervals for high-dimensional regression
• Design-based analysis using martingale and exchangeable pairs
• Instance-adaptive inference leveraging covariate geometry properties