# Distributional robustness of K-class estimators and the PULSE

arXiv: https://arxiv.org/abs/2005.03353 · HTML: https://mlsanity.com/paper/2005.03353/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 May 2020.

## Built on by (most cited first, 11 of 11)

- [Identifying Representations for Intervention Extrapolation](https://mlsanity.com/paper/2310.04295/) (2023)
- [Regularizing towards Causal Invariance: Linear Models with Proxies](https://mlsanity.com/paper/2103.02477/) (2021)
- [Exploiting Independent Instruments: Identification and Distribution Generalization](https://mlsanity.com/paper/2202.01864/) (2022)
- [Invariant Policy Learning: A Causal Perspective](https://mlsanity.com/paper/2106.00808/) (2021)
- [Identifying Causal Effects using Instrumental Time Series: Nuisance IV and Correcting for the Past](https://mlsanity.com/paper/2203.06056/) (2022)
- [Identifiability of Sparse Causal Effects using Instrumental Variables](https://mlsanity.com/paper/2203.09380/) (2022)
- [AI for Science: An Emerging Agenda](https://mlsanity.com/paper/2303.04217/) (2023)
- [Achievable distributional robustness when the robust risk is only partially identified](https://mlsanity.com/paper/2502.02710/) (2025)
- [Boosted Control Functions: Distribution generalization and invariance in confounded models](https://mlsanity.com/paper/2310.05805/) (2023)
- [Invariant Probabilistic Prediction](https://mlsanity.com/paper/2309.10083/) (2023)
- [How Useful is Causal Invariance for Domain Adaptation in Finite-Sample Settings?](https://mlsanity.com/paper/2606.12680/) (2026)

## Abstract (arXiv)

While causal models are robust in that they are prediction optimal under arbitrarily strong interventions, they may not be optimal when the interventions are bounded. We prove that the classical K-class estimator satisfies such optimality by establishing a connection between K-class estimators and anchor regression. This connection further motivates a novel estimator in instrumental variable settings that minimizes the mean squared prediction error subject to the constraint that the estimator lies in an asymptotically valid confidence region of the causal coefficient. We call this estimator PULSE (p-uncorrelated least squares estimator), relate it to work on invariance, show that it can be computed efficiently as a data-driven K-class estimator, even though the underlying optimization problem is non-convex, and prove consistency. We evaluate the estimators on real data and perform simulation experiments illustrating that PULSE suffers from less variability. There are several settings including weak instrument settings, where it outperforms other estimators.