# Regularizing Extrapolation in Causal Inference

arXiv: https://arxiv.org/abs/2509.17180 · HTML: https://mlsanity.com/paper/2509.17180/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 55 references that are in the map.
- First posted on arXiv 21 Sep 2025.

## Builds on (most cited first, 20 of 20)

- [Generalized Random Forests](https://mlsanity.com/paper/1610.01271/) (2016)
- [A Brief Review of Domain Adaptation](https://mlsanity.com/paper/2010.03978/) (2020)
- [Generalization Bounds and Representation Learning for Estimation of Potential Outcomes and Causal Effects](https://mlsanity.com/paper/2001.07426/) (2020)
- [Balancing Covariates via Propensity Score Weighting](https://mlsanity.com/paper/1609.07494/) (2016)
- [Engression: Extrapolation through the Lens of Distributional Regression](https://mlsanity.com/paper/2307.00835/) (2023)
- [A Review of Generalizability and Transportability](https://mlsanity.com/paper/2102.11904/) (2021)
- [De-Biased Machine Learning of Global and Local Parameters Using Regularized Riesz Representers](https://mlsanity.com/paper/1802.08667/) (2018)
- [The Augmented Synthetic Control Method](https://mlsanity.com/paper/1811.04170/) (2018)
- [First Steps Toward Understanding the Extrapolation of Nonlinear Models to Unseen Domains](https://mlsanity.com/paper/2211.11719/) (2022)
- [Augmented balancing weights as linear regression](https://mlsanity.com/paper/2304.14545/) (2023)
- [The Balancing Act in Causal Inference](https://mlsanity.com/paper/2110.14831/) (2021)
- [Learning to Extrapolate: A Transductive Approach](https://mlsanity.com/paper/2304.14329/) (2023)
- [Why do Random Forests Work? Understanding Tree Ensembles as Self-Regularizing Adaptive Smoothers](https://mlsanity.com/paper/2402.01502/) (2024)
- [Minimax Linear Estimation of the Retargeted Mean](https://mlsanity.com/paper/1901.10296/) (2019)
- [Who Are We Missing? A Principled Approach to Characterizing the Underrepresented Population](https://mlsanity.com/paper/2401.14512/) (2024)
- [Importance Sampling with Unequal Support](https://mlsanity.com/paper/1611.03451/) (2016)
- [Interpretable Sensitivity Analysis for Balancing Weights](https://mlsanity.com/paper/2102.13218/) (2021)
- [Extrapolation-Aware Nonparametric Statistical Inference](https://mlsanity.com/paper/2402.09758/) (2024)
- [Towards Understanding Extrapolation: a Causal Lens](https://mlsanity.com/paper/2501.09163/) (2025)
- [On the implied weights of linear regression for causal inference](https://mlsanity.com/paper/2104.06581/) (2021)

## Built on by (most cited first, 1 of 1)

- [Mind the Sim-to-Real Gap & Think Like a Scientist](https://mlsanity.com/paper/2605.21458/) (2026)

## Abstract (arXiv)

Many common estimators in machine learning and causal inference are linear smoothers, where the prediction is a weighted average of the training outcomes. Some estimators, such as ordinary least squares and kernel ridge regression, allow for arbitrarily negative weights, which improve feature imbalance but often at the cost of increased dependence on parametric modeling assumptions and higher variance. By contrast, estimators like importance weighting and random forests (sometimes implicitly) restrict weights to be non-negative, reducing dependence on parametric modeling and variance at the cost of worse imbalance. In this paper, we propose a unified framework that directly penalizes the level of extrapolation, replacing the current practice of a hard non-negativity constraint with a soft constraint and corresponding hyperparameter. We derive a worst-case extrapolation error bound and introduce a novel "bias-bias-variance" tradeoff, encompassing biases due to feature imbalance, model misspecification, and estimator variance; this tradeoff is especially pronounced in high dimensions, particularly when positivity is poor. We then develop an optimization procedure that regularizes this bound while minimizing imbalance and outline how to use this approach as a sensitivity analysis for dependence on parametric modeling assumptions. We demonstrate the effectiveness of our approach through synthetic experiments and a real-world application, involving the generalization of randomized controlled trial estimates to a target population of interest.