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Recently, invariant risk minimization (IRM) (Arjovsky et al.) was proposed asa promising solution to address out-of-distribution (OOD) generalization. InAhuja et al., it was shown that solving for the Nash equilibria of a new classof "ensemble-games" is equivalent to solving IRM. In this work, we extend theframework in Ahuja et al. for linear regressions by projecting theensemble-game on an $\ell_{\infty}$ ball. We show that such projections helpachieve non-trivial OOD guarantees despite not achieving perfect invariance.For linear models with confounders, we prove that Nash equilibria of thesegames are closer to the ideal OOD solutions than the standard empirical riskminimization (ERM) and we also provide learning algorithms that provablyconverge to these Nash Equilibria. Empirical comparisons of the proposedapproach with the state-of-the-art show consistent gains in achieving OODsolutions in several settings involving anti-causal variables and confounders.

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