# Augmented balancing weights as linear regression

arXiv: https://arxiv.org/abs/2304.14545 · HTML: https://mlsanity.com/paper/2304.14545/

## Key facts

- Cited by 21 papers in the map (data through 14 Oct 2026).
- Builds on 34 of its 91 references that are in the map.
- First posted on arXiv 27 Apr 2023.

## Builds on (most cited first, 30 of 34)

- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Benign Overfitting in Linear Regression](https://mlsanity.com/paper/1906.11300/) (2019)
- [Statistics of Robust Optimization: A Generalized Empirical Likelihood Approach](https://mlsanity.com/paper/1610.03425/) (2016)
- [Sobolev Norm Learning Rates for Regularized Least-Squares Algorithm](https://mlsanity.com/paper/1702.07254/) (2017)
- [An Introduction to Proximal Causal Learning](https://mlsanity.com/paper/2009.10982/) (2020)
- [Approximate Residual Balancing: De-Biased Inference of Average Treatment Effects in High Dimensions](https://mlsanity.com/paper/1604.07125/) (2016)
- [Causal Inference Under Unmeasured Confounding With Negative Controls: A Minimax Learning Approach](https://mlsanity.com/paper/2103.14029/) (2021)
- [RieszNet and ForestRiesz: Automatic Debiased Machine Learning with Neural Nets and Random Forests](https://mlsanity.com/paper/2110.03031/) (2021)
- [Synthetic Difference in Differences](https://mlsanity.com/paper/1812.09970/) (2018)
- [Generalized Optimal Matching Methods for Causal Inference](https://mlsanity.com/paper/1612.08321/) (2016)
- [Robust Inference on Average Treatment Effects with Possibly More Covariates than Observations](https://mlsanity.com/paper/1309.4686/) (2013)
- [Kernel Methods for Causal Functions: Dose, Heterogeneous, and Incremental Response Curves](https://mlsanity.com/paper/2010.04855/) (2020)
- [Covariate Balancing Propensity Score by Tailored Loss Functions](https://mlsanity.com/paper/1601.05890/) (2016)
- [De-Biased Machine Learning of Global and Local Parameters Using Regularized Riesz Representers](https://mlsanity.com/paper/1802.08667/) (2018)
- [Augmented Minimax Linear Estimation](https://mlsanity.com/paper/1712.00038/) (2017)
- [On Reproducing Kernel Banach Spaces: Generic Definitions and Unified Framework of Constructions](https://mlsanity.com/paper/1901.01002/) (2019)
- [The Augmented Synthetic Control Method](https://mlsanity.com/paper/1811.04170/) (2018)
- [Minimax Kernel Machine Learning for a Class of Doubly Robust Functionals with Application to Proximal Causal Inference](https://mlsanity.com/paper/2104.02929/) (2021)
- [The Balancing Act in Causal Inference](https://mlsanity.com/paper/2110.14831/) (2021)
- [Hierarchical Shrinkage: improving the accuracy and interpretability of tree-based methods](https://mlsanity.com/paper/2202.00858/) (2022)
- [Minimal Dispersion Approximately Balancing Weights: Asymptotic Properties and Practical Considerations](https://mlsanity.com/paper/1705.00998/) (2017)
- [Causal Inference with Corrupted Data: Measurement Error, Missing Values, Discretization, and Differential Privacy](https://mlsanity.com/paper/2107.02780/) (2021)
- [Regularized calibrated estimation of propensity scores with model misspecification and high-dimensional data](https://mlsanity.com/paper/1710.08074/) (2017)
- [Entropy balancing is doubly robust](https://mlsanity.com/paper/1501.03571/) (2015)
- [Balancing Covariates in Randomized Experiments with the Gram-Schmidt Walk Design](https://mlsanity.com/paper/1911.03071/) (2019)
- [Kernel Ridge Riesz Representers: Generalization, Mis-specification, and the Counterfactual Effective Dimension](https://mlsanity.com/paper/2102.11076/) (2021)
- [Kernel-based off-policy estimation without overlap: Instance optimality beyond semiparametric efficiency](https://mlsanity.com/paper/2301.06240/) (2023)
- [Outcome Assumptions and Duality Theory for Balancing Weights](https://mlsanity.com/paper/2203.09557/) (2022)
- [Double Robustness for Complier Parameters and a Semiparametric Test for Complier Characteristics](https://mlsanity.com/paper/1909.05244/) (2019)
- [Automatic Double Machine Learning for Continuous Treatment Effects](https://mlsanity.com/paper/2104.10334/) (2021)

## Built on by (most cited first, 21 of 21)

- [On the Curses of Future and History in Future-dependent Value Functions for Off-policy Evaluation](https://mlsanity.com/paper/2402.14703/) (2024)
- [Stabilized Inverse Probability Weighting via Isotonic Calibration](https://mlsanity.com/paper/2411.06342/) (2024)
- [Direct Bias-Correction Term Estimation for Average Treatment Effect Estimation](https://mlsanity.com/paper/2509.22122/) (2025)
- [Automatic Debiased Machine Learning for Smooth Functionals of Nonparametric M-Estimands](https://mlsanity.com/paper/2501.11868/) (2025)
- [Nearest Neighbor Matching as Least Squares Density Ratio Estimation and Riesz Regression](https://mlsanity.com/paper/2510.24433/) (2025)
- [Direct Debiased Machine Learning via Bregman Divergence Minimization](https://mlsanity.com/paper/2510.23534/) (2025)
- [Riesz Regression As Direct Density Ratio Estimation](https://mlsanity.com/paper/2511.04568/) (2025)
- [Optimally taming biases in black-box models for efficient semiparametric estimation](https://mlsanity.com/paper/2606.06368/) (2026)
- [Towards Representation Learning for Weighting Problems in Design-Based Causal Inference](https://mlsanity.com/paper/2409.16407/) (2024)
- [Doubly Robust Fusion of Many Treatments for Policy Learning](https://mlsanity.com/paper/2505.08092/) (2025)
- [A Unified Theory for Causal Inference: Direct Debiased Machine Learning via Bregman-Riesz Regression](https://mlsanity.com/paper/2510.26783/) (2025)
- [Regularizing Extrapolation in Causal Inference](https://mlsanity.com/paper/2509.17180/) (2025)
- [Semi-Supervised Treatment Effect Estimation with Unlabeled Covariates for Prediction-Powered Causal Inference](https://mlsanity.com/paper/2511.08303/) (2025)
- [Efficient Inference after Directionally Stable Adaptive Experiments](https://mlsanity.com/paper/2602.21478/) (2026)
- [Optimal use of a black-box learner in semiparametric estimation](https://mlsanity.com/paper/2607.21541/) (2026)
- [A Stable and Efficient Covariate-Balancing Estimator for Causal Survival Effects](https://mlsanity.com/paper/2310.02278/) (2023)
- [genriesz: A Python Package for Automatic Debiased Machine Learning with Generalized Riesz Regression](https://mlsanity.com/paper/2602.17543/) (2026)
- [Two Approaches to Direct Estimation of Riesz Representers](https://mlsanity.com/paper/2603.20936/) (2026)
- [Covariate Balancing and Riesz Regression Should Be Guided by the Neyman Orthogonal Score in Debiased Machine Learning](https://mlsanity.com/paper/2605.06386/) (2026)
- [Shrinkage priors for Bayesian Substitute Confounders](https://mlsanity.com/paper/2606.18535/) (2026)
- [Target-Aware Linear Regression Under Distribution Shift](https://mlsanity.com/paper/2606.22775/) (2026)

## Abstract (arXiv)

We provide a novel characterization of augmented balancing weights, also known as automatic debiased machine learning (AutoDML). These popular doubly robust or de-biased machine learning estimators combine outcome modeling with balancing weights - weights that achieve covariate balance directly in lieu of estimating and inverting the propensity score. When the outcome and weighting models are both linear in some (possibly infinite) basis, we show that the augmented estimator is equivalent to a single linear model with coefficients that combine the coefficients from the original outcome model and coefficients from an unpenalized ordinary least squares (OLS) fit on the same data. We see that, under certain choices of regularization parameters, the augmented estimator often collapses to the OLS estimator alone; this occurs for example in a re-analysis of the Lalonde 1986 dataset. We then extend these results to specific choices of outcome and weighting models. We first show that the augmented estimator that uses (kernel) ridge regression for both outcome and weighting models is equivalent to a single, undersmoothed (kernel) ridge regression. This holds numerically in finite samples and lays the groundwork for a novel analysis of undersmoothing and asymptotic rates of convergence. When the weighting model is instead lasso-penalized regression, we give closed-form expressions for special cases and demonstrate a ``double selection'' property. Our framework opens the black box on this increasingly popular class of estimators, bridges the gap between existing results on the semiparametric efficiency of undersmoothed and doubly robust estimators, and provides new insights into the performance of augmented balancing weights.