# A causal framework for distribution generalization

arXiv: https://arxiv.org/abs/2006.07433 · HTML: https://mlsanity.com/paper/2006.07433/

## Key facts

- Cited by 68 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 Jun 2020.

## Built on by (most cited first, 30 of 68)

- [Probable Domain Generalization via Quantile Risk Minimization](https://mlsanity.com/paper/2207.09944/) (2022)
- [A Systematic Review of Robustness in Deep Learning for Computer Vision: Mind the gap?](https://mlsanity.com/paper/2112.00639/) (2021)
- [An Online Learning Approach to Interpolation and Extrapolation in Domain Generalization](https://mlsanity.com/paper/2102.13128/) (2021)
- [Causal Reasoning Meets Visual Representation Learning: A Prospective Study](https://mlsanity.com/paper/2204.12037/) (2022)
- [Mitigating Data Heterogeneity in Federated Learning with Data Augmentation](https://mlsanity.com/paper/2206.09979/) (2022)
- [Unbiased Scene Graph Generation via Two-stage Causal Modeling](https://mlsanity.com/paper/2307.05276/) (2023)
- [Identifying Representations for Intervention Extrapolation](https://mlsanity.com/paper/2310.04295/) (2023)
- [Causal Balancing for Domain Generalization](https://mlsanity.com/paper/2206.05263/) (2022)
- [Mix and Reason: Reasoning over Semantic Topology with Data Mixing for Domain Generalization](https://mlsanity.com/paper/2210.07571/) (2022)
- [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)
- [Optimization-based Causal Estimation from Heterogenous Environments](https://mlsanity.com/paper/2109.11990/) (2021)
- [Visual Knowledge in the Big Model Era: Retrospect and Prospect](https://mlsanity.com/paper/2404.04308/) (2024)
- [Core Risk Minimization using Salient ImageNet](https://mlsanity.com/paper/2203.15566/) (2022)
- [A Causal Adjustment Module for Debiasing Scene Graph Generation](https://mlsanity.com/paper/2503.17862/) (2025)
- [Latent Covariate Shift: Unlocking Partial Identifiability for Multi-Source Domain Adaptation](https://mlsanity.com/paper/2208.14161/) (2022)
- [Identifying Causal Effects using Instrumental Time Series: Nuisance IV and Correcting for the Past](https://mlsanity.com/paper/2203.06056/) (2022)
- [Activate and Reject: Towards Safe Domain Generalization under Category Shift](https://mlsanity.com/paper/2310.04724/) (2023)
- [Domain Generalization via Contrastive Causal Learning](https://mlsanity.com/paper/2210.02655/) (2022)
- [HCVP: Leveraging Hierarchical Contrastive Visual Prompt for Domain Generalization](https://mlsanity.com/paper/2401.09716/) (2024)
- [Instrumental Variable-Driven Domain Generalization with Unobserved Confounders](https://mlsanity.com/paper/2110.01438/) (2021)
- [General Greedy De-bias Learning](https://mlsanity.com/paper/2112.10572/) (2021)
- [Adaptive Methods for Aggregated Domain Generalization](https://mlsanity.com/paper/2112.04766/) (2021)
- [Progression: an extrapolation principle for regression](https://mlsanity.com/paper/2410.23246/) (2024)
- [Invariant Ancestry Search](https://mlsanity.com/paper/2202.00913/) (2022)
- [Causality-oriented robustness: exploiting general noise interventions](https://mlsanity.com/paper/2307.10299/) (2023)
- [Towards Unified and Effective Domain Generalization](https://mlsanity.com/paper/2310.10008/) (2023)
- [Invariance & Causal Representation Learning: Prospects and Limitations](https://mlsanity.com/paper/2312.03580/) (2023)
- [Extrapolation-Aware Nonparametric Statistical Inference](https://mlsanity.com/paper/2402.09758/) (2024)
- [Unified Source-Free Domain Adaptation](https://mlsanity.com/paper/2403.07601/) (2024)

## Abstract (arXiv)

We consider the problem of predicting a response Y from a set of covariates X when test and training distributions differ. Since such differences may have causal explanations, we consider test distributions that emerge from interventions in a structural causal model, and focus on minimizing the worst-case risk. Causal regression models, which regress the response on its direct causes, remain unchanged under arbitrary interventions on the covariates, but they are not always optimal in the above sense. For example, for linear models and bounded interventions, alternative solutions have been shown to be minimax prediction optimal. We introduce the formal framework of distribution generalization that allows us to analyze the above problem in partially observed nonlinear models for both direct interventions on X and interventions that occur indirectly via exogenous variables A. It takes into account that, in practice, minimax solutions need to be identified from data. Our framework allows us to characterize under which class of interventions the causal function is minimax optimal. We prove sufficient conditions for distribution generalization and present corresponding impossibility results. We propose a practical method, NILE, that achieves distribution generalization in a nonlinear IV setting with linear extrapolation. We prove consistency and present empirical results.