# Linear Regression Games: Convergence Guarantees to Approximate Out-of-Distribution Solutions

arXiv: https://arxiv.org/abs/2010.15234 · HTML: https://mlsanity.com/paper/2010.15234/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Oct 2020.

## Built on by (most cited first, 7 of 7)

- [Invariance Principle Meets Information Bottleneck for Out-of-Distribution Generalization](https://mlsanity.com/paper/2106.06607/) (2021)
- [FL Games: A Federated Learning Framework for Distribution Shifts](https://mlsanity.com/paper/2211.00184/) (2022)
- [Locally Invariant Explanations: Towards Stable and Unidirectional Explanations through Local Invariant Learning](https://mlsanity.com/paper/2201.12143/) (2022)
- [Out-of-Distribution Optimality of Invariant Risk Minimization](https://mlsanity.com/paper/2307.11972/) (2023)
- [A call for better unit testing for invariant risk minimisation](https://mlsanity.com/paper/2106.03234/) (2021)
- [FL Games: A federated learning framework for distribution shifts](https://mlsanity.com/paper/2205.11101/) (2022)
- [Mutual Information Learned Regressor: an Information-theoretic Viewpoint of Training Regression Systems](https://mlsanity.com/paper/2211.12685/) (2022)

## Abstract (arXiv)

Recently, invariant risk minimization (IRM) (Arjovsky et al.) was proposed as a promising solution to address out-of-distribution (OOD) generalization. In Ahuja et al., it was shown that solving for the Nash equilibria of a new class of "ensemble-games" is equivalent to solving IRM. In this work, we extend the framework in Ahuja et al. for linear regressions by projecting the ensemble-game on an ℓ∞ ball. We show that such projections help achieve non-trivial OOD guarantees despite not achieving perfect invariance. For linear models with confounders, we prove that Nash equilibria of these games are closer to the ideal OOD solutions than the standard empirical risk minimization (ERM) and we also provide learning algorithms that provably converge to these Nash Equilibria. Empirical comparisons of the proposed approach with the state-of-the-art show consistent gains in achieving OOD solutions in several settings involving anti-causal variables and confounders.