Linear Regression Games: Convergence Guarantees to Approximate Out-of-Distribution Solutions
This work addresses OOD generalization for machine learning models, particularly in linear regression with confounders, but it is incremental as it builds on prior ensemble-game frameworks.
The paper tackles out-of-distribution generalization in linear regression by extending ensemble-games with an ℓ∞ ball projection, proving that Nash equilibria provide better OOD solutions than empirical risk minimization and offering convergent algorithms, with empirical results showing consistent gains in settings with anti-causal variables and confounders.
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 $\ell_{\infty}$ 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.