PAC Generalization via Invariant Representations

Invariant representations are transformations of the covariates such that the best model on top of the representation is invariant across training environments. In the context of linear Structural Equation Models (SEMs), invariant representations might allow us to learn models with out-of-distribution guarantees, i.e., models that are robust to interventions in the SEM. To address the invariant representation problem in a *finite sample* setting, we consider the notion of $\epsilon$-approximate invariance. We study the following question: If a representation is approximately invariant with respect to a given number of training interventions, will it continue to be approximately invariant on a larger collection of unseen intervened SEMs? Inspired by PAC learning, we obtain finite-sample out-of-distribution generalization guarantees for approximate invariance that holds *probabilistically* over a family of linear SEMs without faithfulness assumptions.