Optimal Control Variates for Survey Sampling and Causal Inference

2026-08-15

AI Summary

This study addresses the excessive variance of inverse probability weighting estimators under small propensity scores in causal inference by proposing a family of optimal control variate estimators. We establish a unified theoretical framework that elucidates existing methods and employ stochastic optimization combined with alternating local search to construct optimal basis functions, effectively overcoming approximation challenges under network interference. Experimental results demonstrate that the proposed estimator significantly outperforms traditional approaches in finite-sample settings, achieving substantial variance reduction. Consequently, this work introduces a novel paradigm for robust causal effect estimation in high-dimensional complex data environments, offering a theoretically grounded and computationally feasible solution to long-standing stability issues in weighted estimation.

Abstract

We propose a family of control variate estimators for variance reduction in design-based survey sampling and causal inference, with and without interference. In these settings, inverse probability weighting (IPW) estimators are widely used, but may have large variance when sampling, treatment, or exposure probabilities are small. Building on the observation that several common estimators, including the Hajek, normalized, and augmented inverse probability weighting (AIPW) estimators, correct the Horvitz-Thompson estimator by canceling part of its randomness, we provide a unified interpretation of these estimators as special cases of a general control variate estimator. We then construct optimal control variates that can further reduce the finite sample variance compared to these common estimators. We parameterize the proposed control variates by their bases and characterize the optimal bases through a stochastic optimization formulation. In survey sampling and causal inference without interference, the optimal bases are characterized by leading eigenvectors of matrices that depend on both the design-based sampling structure and the model-based outcome uncertainty. In causal inference under network interference, the optimal bases solve a nonconvex quadratic optimization problem; we provide a $\frac{1}{2}$-approximate solution and an alternating local search heuristic. We apply the control variate estimators to the Swiss Environmental Panel survey data and the Chinese social network data, and conduct extensive simulations to show that the proposed control variate estimators can achieve substantial variance reduction.

Problem

• Variance Reduction
• Survey Sampling
• Causal Inference
• Control Variates
• Network Interference

Innovation

• Control Variates
• Variance Reduction
• Causal Inference
• Network Interference
• Stochastic Optimization