# A New Central Limit Theorem for the Augmented IPW Estimator: Variance Inflation, Cross-Fit Covariance and Beyond

arXiv: https://arxiv.org/abs/2205.10198 · HTML: https://mlsanity.com/paper/2205.10198/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 33 of its 122 references that are in the map.
- First posted on arXiv 20 May 2022.

## Builds on (most cited first, 30 of 33)

- [The generalization error of random features regression: Precise asymptotics and double descent curve](https://mlsanity.com/paper/1908.05355/) (2019)
- [Semiparametric doubly robust targeted double machine learning: a review](https://mlsanity.com/paper/2203.06469/) (2022)
- [Statistical physics of inference: Thresholds and algorithms](https://mlsanity.com/paper/1511.02476/) (2015)
- [Spectral Methods for Data Science: A Statistical Perspective](https://mlsanity.com/paper/2012.08496/) (2020)
- [A Model of Double Descent for High-dimensional Binary Linear Classification](https://mlsanity.com/paper/1911.05822/) (2019)
- [Universality of empirical risk minimization](https://mlsanity.com/paper/2202.08832/) (2022)
- [Approximate Residual Balancing: De-Biased Inference of Average Treatment Effects in High Dimensions](https://mlsanity.com/paper/1604.07125/) (2016)
- [Precise Statistical Analysis of Classification Accuracies for Adversarial Training](https://mlsanity.com/paper/2010.11213/) (2020)
- [A unifying tutorial on Approximate Message Passing](https://mlsanity.com/paper/2105.02180/) (2021)
- [The phase transition for the existence of the maximum likelihood estimate in high-dimensional logistic regression](https://mlsanity.com/paper/1804.09753/) (2018)
- [A Precise High-Dimensional Asymptotic Theory for Boosting and Minimum-ℓ1-Norm Interpolated Classifiers](https://mlsanity.com/paper/2002.01586/) (2020)
- [A framework to characterize performance of LASSO algorithms](https://mlsanity.com/paper/1303.7291/) (2013)
- [The Lasso with general Gaussian designs with applications to hypothesis testing](https://mlsanity.com/paper/2007.13716/) (2020)
- [Robust Inference on Average Treatment Effects with Possibly More Covariates than Observations](https://mlsanity.com/paper/1309.4686/) (2013)
- [On the role of surrogates in the efficient estimation of treatment effects with limited outcome data](https://mlsanity.com/paper/2003.12408/) (2020)
- [The generalization error of max-margin linear classifiers: Benign overfitting and high dimensional asymptotics in the overparametrized regime](https://mlsanity.com/paper/1911.01544/) (2019)
- [The Likelihood Ratio Test in High-Dimensional Logistic Regression Is Asymptotically a Rescaled Chi-Square](https://mlsanity.com/paper/1706.01191/) (2017)
- [Regularized calibrated estimation of propensity scores with model misspecification and high-dimensional data](https://mlsanity.com/paper/1710.08074/) (2017)
- [A unifying approach for doubly-robust ℓ1 regularized estimation of causal contrasts](https://mlsanity.com/paper/1904.03737/) (2019)
- [Consistent Risk Estimation in Moderately High-Dimensional Linear Regression](https://mlsanity.com/paper/1902.01753/) (2019)
- [Algorithmic Analysis and Statistical Estimation of SLOPE via Approximate Message Passing](https://mlsanity.com/paper/1907.07502/) (2019)
- [Out-of-sample error estimate for robust M-estimators with convex penalty](https://mlsanity.com/paper/2008.11840/) (2020)
- [A scalable estimate of the extra-sample prediction error via approximate leave-one-out](https://mlsanity.com/paper/1801.10243/) (2018)
- [Debiased Inverse Propensity Score Weighting for Estimation of Average Treatment Effects with High-Dimensional Confounders](https://mlsanity.com/paper/2011.08661/) (2020)
- [Sharp global convergence guarantees for iterative nonconvex optimization: A Gaussian process perspective](https://mlsanity.com/paper/2109.09859/) (2021)
- [Sparsity Double Robust Inference of Average Treatment Effects](https://mlsanity.com/paper/1905.00744/) (2019)
- [Minimum ℓ1-norm interpolators: Precise asymptotics and multiple descent](https://mlsanity.com/paper/2110.09502/) (2021)
- [Approximate Leave-One-Out for High-Dimensional Non-Differentiable Learning Problems](https://mlsanity.com/paper/1810.02716/) (2018)
- [Asymptotic normality of robust M-estimators with convex penalty](https://mlsanity.com/paper/2107.03826/) (2021)
- [Observable adjustments in single-index models for regularized M-estimators](https://mlsanity.com/paper/2204.06990/) (2022)

## Built on by (most cited first, 5 of 5)

- [Preventing Model Collapse Under Overparametrization: Optimal Mixing Ratios for Interpolation Learning and Ridge Regression](https://mlsanity.com/paper/2509.22341/) (2025)
- [Kernel-based off-policy estimation without overlap: Instance optimality beyond semiparametric efficiency](https://mlsanity.com/paper/2301.06240/) (2023)
- [Doubly Robust Proximal Causal Learning for Continuous Treatments](https://mlsanity.com/paper/2309.12819/) (2023)
- [Assumption-lean falsification tests of rate double-robustness of double-machine-learning estimators](https://mlsanity.com/paper/2306.10590/) (2023)
- [Optimal and Provable Calibration in High-Dimensional Binary Classification: Angular Calibration and Platt Scaling](https://mlsanity.com/paper/2502.15131/) (2025)

## Abstract (arXiv)

Estimation of the average treatment effect (ATE) is a central problem in causal inference. In recent times, inference for the ATE in the presence of high-dimensional covariates has been extensively studied. Among the diverse approaches that have been proposed, augmented inverse probability weighting (AIPW) with cross-fitting has emerged a popular choice in practice. In this work, we study this cross-fit AIPW estimator under well-specified outcome regression and propensity score models in a high-dimensional regime where the number of features and samples are both large and comparable. Under assumptions on the covariate distribution, we establish a new central limit theorem for the suitably scaled cross-fit AIPW that applies without any sparsity assumptions on the underlying high-dimensional parameters. Our CLT uncovers two crucial phenomena among others: (i) the AIPW exhibits a substantial variance inflation that can be precisely quantified in terms of the signal-to-noise ratio and other problem parameters, (ii) the asymptotic covariance between the pre-cross-fit estimators is non-negligible even on the root-n scale. These findings are strikingly different from their classical counterparts. On the technical front, our work utilizes a novel interplay between three distinct tools--approximate message passing theory, the theory of deterministic equivalents, and the leave-one-out approach. We believe our proof techniques should be useful for analyzing other two-stage estimators in this high-dimensional regime. Finally, we complement our theoretical results with simulations that demonstrate both the finite sample efficacy of our CLT and its robustness to our assumptions.