# Minimal Dispersion Approximately Balancing Weights: Asymptotic Properties and Practical Considerations

arXiv: https://arxiv.org/abs/1705.00998 · HTML: https://mlsanity.com/paper/1705.00998/

## Key facts

- Cited by 18 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 May 2017.

## Built on by (most cited first, 18 of 18)

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- [Stabilized Inverse Probability Weighting via Isotonic Calibration](https://mlsanity.com/paper/2411.06342/) (2024)
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- [Outcome Assumptions and Duality Theory for Balancing Weights](https://mlsanity.com/paper/2203.09557/) (2022)
- [Robust Sample Weighting to Facilitate Individualized Treatment Rule Learning for a Target Population](https://mlsanity.com/paper/2105.00581/) (2021)
- [Estimating causal effects with optimization-based methods: A review and empirical comparison](https://mlsanity.com/paper/2203.00097/) (2022)
- [Scalable kernel balancing weights in a nationwide observational study of hospital profit status and heart attack outcomes](https://mlsanity.com/paper/2311.00568/) (2023)
- [Stable Probability Weighting: Large-Sample and Finite-Sample Estimation and Inference Methods for Heterogeneous Causal Effects of Multivalued Treatments Under Limited Overlap](https://mlsanity.com/paper/2301.05703/) (2023)
- [Temporal-Spatial Entropy Balancing for Causal Continuous Treatment-Effect Estimation](https://mlsanity.com/paper/2312.08670/) (2023)
- [Learning density ratios in causal inference using Bregman-Riesz regression](https://mlsanity.com/paper/2510.16127/) (2025)
- [Differentially Private Two-Stage Empirical Risk Minimization with Applications to Individualized Treatment Rule](https://mlsanity.com/paper/2602.12604/) (2026)
- [FRESH: Information-Geometric Calibration of Patient-Level Models to Aggregate Evidence](https://mlsanity.com/paper/2605.16246/) (2026)
- [Augmented Inverse Hybrid Weighting: Robust Inference under Deterministic and Random Distribution Shifts](https://mlsanity.com/paper/2608.00701/) (2026)
- [Causal Generalization of Continuous Treatment Effects under Covariate Shift](https://mlsanity.com/paper/2608.19383/) (2026)

## Abstract (arXiv)

Weighting methods are widely used to adjust for covariates in observational studies, sample surveys, and regression settings. In this paper, we study a class of recently proposed weighting methods which find the weights of minimum dispersion that approximately balance the covariates. We call these weights "minimal weights" and study them under a common optimization framework. The key observation is the connection between approximate covariate balance and shrinkage estimation of the propensity score. This connection leads to both theoretical and practical developments. From a theoretical standpoint, we characterize the asymptotic properties of minimal weights and show that, under standard smoothness conditions on the propensity score function, minimal weights are consistent estimates of the true inverse probability weights. Also, we show that the resulting weighting estimator is consistent, asymptotically normal, and semiparametrically efficient. From a practical standpoint, we present a finite sample oracle inequality that bounds the loss incurred by balancing more functions of the covariates than strictly needed. This inequality shows that minimal weights implicitly bound the number of active covariate balance constraints. We finally provide a tuning algorithm for choosing the degree of approximate balance in minimal weights. We conclude the paper with four empirical studies that suggest approximate balance is preferable to exact balance, especially when there is limited overlap in covariate distributions. In these studies, we show that the root mean squared error of the weighting estimator can be reduced by as much as a half with approximate balance.