# Outcome Assumptions and Duality Theory for Balancing Weights

arXiv: https://arxiv.org/abs/2203.09557 · HTML: https://mlsanity.com/paper/2203.09557/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 50 references that are in the map.
- First posted on arXiv 17 Mar 2022.

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

- [Domain-Adversarial Training of Neural Networks](https://mlsanity.com/paper/1505.07818/) (2015)
- [Generalization Bounds and Representation Learning for Estimation of Potential Outcomes and Causal Effects](https://mlsanity.com/paper/2001.07426/) (2020)
- [Approximate Residual Balancing: De-Biased Inference of Average Treatment Effects in High Dimensions](https://mlsanity.com/paper/1604.07125/) (2016)
- [Counterfactual Representation Learning with Balancing Weights](https://mlsanity.com/paper/2010.12618/) (2020)
- [DeepMatch: Balancing Deep Covariate Representations for Causal Inference Using Adversarial Training](https://mlsanity.com/paper/1802.05664/) (2018)
- [Generalized Optimal Matching Methods for Causal Inference](https://mlsanity.com/paper/1612.08321/) (2016)
- [Generalized Resilience and Robust Statistics](https://mlsanity.com/paper/1909.08755/) (2019)
- [KALE Flow: A Relaxed KL Gradient Flow for Probabilities with Disjoint Support](https://mlsanity.com/paper/2106.08929/) (2021)
- [Covariate Balancing Propensity Score by Tailored Loss Functions](https://mlsanity.com/paper/1601.05890/) (2016)
- [Optimal Bounds between f-Divergences and Integral Probability Metrics](https://mlsanity.com/paper/2006.05973/) (2020)
- [Bridging the Gap Between f-GANs and Wasserstein GANs](https://mlsanity.com/paper/1910.09779/) (2019)
- [The Balancing Act in Causal Inference](https://mlsanity.com/paper/2110.14831/) (2021)
- [Minimal Dispersion Approximately Balancing Weights: Asymptotic Properties and Practical Considerations](https://mlsanity.com/paper/1705.00998/) (2017)
- [Regularized calibrated estimation of propensity scores with model misspecification and high-dimensional data](https://mlsanity.com/paper/1710.08074/) (2017)
- [Optimizing Variational Representations of Divergences and Accelerating their Statistical Estimation](https://mlsanity.com/paper/2006.08781/) (2020)
- [Permutation Weighting](https://mlsanity.com/paper/1901.01230/) (2019)
- [Entropy balancing is doubly robust](https://mlsanity.com/paper/1501.03571/) (2015)
- [Minimax Linear Estimation of the Retargeted Mean](https://mlsanity.com/paper/1901.10296/) (2019)
- [Formulation and properties of a divergence used to compare probability measures without absolute continuity](https://mlsanity.com/paper/1911.07422/) (2019)
- [Adversarial Balancing for Causal Inference](https://mlsanity.com/paper/1810.07406/) (2018)

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

- [Augmented balancing weights as linear regression](https://mlsanity.com/paper/2304.14545/) (2023)
- [Towards Representation Learning for Weighting Problems in Design-Based Causal Inference](https://mlsanity.com/paper/2409.16407/) (2024)
- [Generalized Balancing Weights via Deep Neural Networks](https://mlsanity.com/paper/2211.07533/) (2022)
- [Tailoring Strictly Proper Scoring Rules for Downstream Tasks: An Application to Causal Inference](https://mlsanity.com/paper/2606.03332/) (2026)
- [Learning density ratios in causal inference using Bregman-Riesz regression](https://mlsanity.com/paper/2510.16127/) (2025)

## Abstract (arXiv)

We study balancing weight estimators, which reweight outcomes from a source population to estimate missing outcomes in a target population. These estimators minimize the worst-case error by making an assumption about the outcome model. In this paper, we show that this outcome assumption has two immediate implications. First, we can replace the minimax optimization problem for balancing weights with a simple convex loss over the assumed outcome function class. Second, we can replace the commonly-made overlap assumption with a more appropriate quantitative measure, the minimum worst-case bias. Finally, we show conditions under which the weights remain robust when our assumptions on the outcomes are wrong.