# CVaR anchor regression protects against rare shifts

arXiv: https://arxiv.org/abs/2609.27034 · HTML: https://mlsanity.com/paper/2609.27034/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 13 of its 17 references that are in the map.
- First posted on arXiv 22 Sep 2026.

## Builds on (most cited first, 13 of 13)

- [WILDS: A Benchmark of in-the-Wild Distribution Shifts](https://mlsanity.com/paper/2012.07421/) (2020)
- [Distributionally Robust Neural Networks for Group Shifts: On the Importance of Regularization for Worst-Case Generalization](https://mlsanity.com/paper/1911.08731/) (2019)
- [Out-of-Distribution Generalization via Risk Extrapolation (REx)](https://mlsanity.com/paper/2003.00688/) (2020)
- [Learning Models with Uniform Performance via Distributionally Robust Optimization](https://mlsanity.com/paper/1810.08750/) (2018)
- [Distributionally Robust Language Modeling](https://mlsanity.com/paper/1909.02060/) (2019)
- [Invariance, Causality and Robustness](https://mlsanity.com/paper/1812.08233/) (2018)
- [Anchor regression: heterogeneous data meets causality](https://mlsanity.com/paper/1801.06229/) (2018)
- [Probable Domain Generalization via Quantile Risk Minimization](https://mlsanity.com/paper/2207.09944/) (2022)
- [Maximin effects in inhomogeneous large-scale data](https://mlsanity.com/paper/1406.0596/) (2014)
- [Minimax Regret Optimization for Robust Machine Learning under Distribution Shift](https://mlsanity.com/paper/2202.05436/) (2022)
- [Distributional Anchor Regression](https://mlsanity.com/paper/2101.08224/) (2021)
- [Minimax Regret Learning for Data with Heterogeneous Subgroups](https://mlsanity.com/paper/2405.01709/) (2024)
- [Domain Generalization and Adaptation in Intensive Care with Anchor Regression](https://mlsanity.com/paper/2507.21783/) (2025)

## Abstract (arXiv)

We study prediction in new environments when training data contain rare, large shifts. Anchor regression penalizes the average of the squared mean residual across environments. It protects against shifts in an ellipsoid determined by the second moment of the training shifts. Covering rare shifts may therefore require a large penalty, expanding the ellipsoid in every direction and reducing accuracy on common environments. We propose CVaR anchor regression, which replaces the average of the squared mean residuals with a tail average. Unlike CVaR or GroupDRO applied directly to prediction risks, it does not give environments more weight solely because their noise levels are high. We prove an exact worst-case risk guarantee under a linear structural model that allows for heteroscedastic noise. For discrete environments, decreasing the CVaR tail fraction expands the robustness set from an ellipsoid to a scaled convex hull of the training shifts and their negatives. A separate parameter controls its scale. Examples show how the method can improve protection against rare shifts while retaining accuracy on common environments. We illustrate the method on New York City taxi data.