# Achievable distributional robustness when the robust risk is only partially identified

arXiv: https://arxiv.org/abs/2502.02710 · HTML: https://mlsanity.com/paper/2502.02710/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 37 of its 63 references that are in the map.
- First posted on arXiv 4 Feb 2025.

## Builds on (most cited first, 30 of 37)

- [Explaining and Harnessing Adversarial Examples](https://mlsanity.com/paper/1412.6572/) (2014)
- [Towards Deep Learning Models Resistant to Adversarial Attacks](https://mlsanity.com/paper/1706.06083/) (2017)
- [Invariant Risk Minimization](https://mlsanity.com/paper/1907.02893/) (2019)
- [In Search of Lost Domain Generalization](https://mlsanity.com/paper/2007.01434/) (2020)
- [Out-of-Distribution Generalization via Risk Extrapolation (REx)](https://mlsanity.com/paper/2003.00688/) (2020)
- [Causal inference using invariant prediction: identification and confidence intervals](https://mlsanity.com/paper/1501.01332/) (2015)
- [Learning Models with Uniform Performance via Distributionally Robust Optimization](https://mlsanity.com/paper/1810.08750/) (2018)
- [Data-driven Distributionally Robust Optimization Using the Wasserstein Metric: Performance Guarantees and Tractable Reformulations](https://mlsanity.com/paper/1505.05116/) (2015)
- [The Risks of Invariant Risk Minimization](https://mlsanity.com/paper/2010.05761/) (2020)
- [Gradient Matching for Domain Generalization](https://mlsanity.com/paper/2104.09937/) (2021)
- [Invariance Principle Meets Information Bottleneck for Out-of-Distribution Generalization](https://mlsanity.com/paper/2106.06607/) (2021)
- [Invariant Risk Minimization Games](https://mlsanity.com/paper/2002.04692/) (2020)
- [Adversarial Training and Robustness for Multiple Perturbations](https://mlsanity.com/paper/1904.13000/) (2019)
- [Wasserstein Distributionally Robust Optimization: Theory and Applications in Machine Learning](https://mlsanity.com/paper/1908.08729/) (2019)
- [Variance-based regularization with convex objectives](https://mlsanity.com/paper/1610.02581/) (2016)
- [Domain Adaptation by Using Causal Inference to Predict Invariant Conditional Distributions](https://mlsanity.com/paper/1707.06422/) (2017)
- [Invariance, Causality and Robustness](https://mlsanity.com/paper/1812.08233/) (2018)
- [Anchor regression: heterogeneous data meets causality](https://mlsanity.com/paper/1801.06229/) (2018)
- [Kernel Instrumental Variable Regression](https://mlsanity.com/paper/1906.00232/) (2019)
- [Does Invariant Risk Minimization Capture Invariance?](https://mlsanity.com/paper/2101.01134/) (2021)
- [Learning Linear Causal Representations from Interventions under General Nonlinear Mixing](https://mlsanity.com/paper/2306.02235/) (2023)
- [Deep Generalized Method of Moments for Instrumental Variable Analysis](https://mlsanity.com/paper/1905.12495/) (2019)
- [A causal framework for distribution generalization](https://mlsanity.com/paper/2006.07433/) (2020)
- [Empirical or Invariant Risk Minimization? A Sample Complexity Perspective](https://mlsanity.com/paper/2010.16412/) (2020)
- [Active Invariant Causal Prediction: Experiment Selection through Stability](https://mlsanity.com/paper/2006.05690/) (2020)
- [Active Learning for Optimal Intervention Design in Causal Models](https://mlsanity.com/paper/2209.04744/) (2022)
- [An Online Learning Approach to Interpolation and Extrapolation in Domain Generalization](https://mlsanity.com/paper/2102.13128/) (2021)
- [CausalBench: A Large-scale Benchmark for Network Inference from Single-cell Perturbation Data](https://mlsanity.com/paper/2210.17283/) (2022)
- [Domain adaptation under structural causal models](https://mlsanity.com/paper/2010.15764/) (2020)
- [Transfer of Adversarial Robustness Between Perturbation Types](https://mlsanity.com/paper/1905.01034/) (2019)

## Built on by (most cited first, 3 of 3)

- [Partial Transportability for Domain Generalization](https://mlsanity.com/paper/2503.23605/) (2025)
- [Domain Generalization and Adaptation in Intensive Care with Anchor Regression](https://mlsanity.com/paper/2507.21783/) (2025)
- [How Useful is Causal Invariance for Domain Adaptation in Finite-Sample Settings?](https://mlsanity.com/paper/2606.12680/) (2026)

## Abstract (arXiv)

In safety-critical applications, machine learning models should generalize well under worst-case distribution shifts, that is, have a small robust risk. Invariance-based algorithms can provably take advantage of structural assumptions on the shifts when the training distributions are heterogeneous enough to identify the robust risk. However, in practice, such identifiability conditions are rarely satisfied -- a scenario so far underexplored in the theoretical literature. In this paper, we aim to fill the gap and propose to study the more general setting when the robust risk is only partially identifiable. In particular, we introduce the worst-case robust risk as a new measure of robustness that is always well-defined regardless of identifiability. Its minimum corresponds to an algorithm-independent (population) minimax quantity that measures the best achievable robustness under partial identifiability. While these concepts can be defined more broadly, in this paper we introduce and derive them explicitly for a linear model for concreteness of the presentation. First, we show that existing robustness methods are provably suboptimal in the partially identifiable case. We then evaluate these methods and the minimizer of the (empirical) worst-case robust risk on real-world gene expression data and find a similar trend: the test error of existing robustness methods grows increasingly suboptimal as the fraction of data from unseen environments increases, whereas accounting for partial identifiability allows for better generalization.