Scalable Out-of-distribution Robustness in the Presence of Unobserved Confounders

We consider the task of out-of-distribution (OOD) generalization, where the distribution shift is due to an unobserved confounder ($Z$) affecting both the covariates ($X$) and the labels ($Y$). This confounding introduces heterogeneity in the predictor, i.e., $P(Y | X) = E_{P(Z | X)}[P(Y | X,Z)]$, making traditional covariate and label shift assumptions unsuitable. OOD generalization differs from traditional domain adaptation in that it does not assume access to the covariate distribution ($X^\text{te}$) of the test samples during training. These conditions create a challenging scenario for OOD robustness: (a) $Z^\text{tr}$ is an unobserved confounder during training, (b) $P^\text{te}(Z) \neq P^\text{tr}(Z)$, (c) $X^\text{te}$ is unavailable during training, and (d) the predictive distribution depends on $P^\text{te}(Z)$. While prior work has developed complex predictors requiring multiple additional variables for identifiability of the latent distribution, we explore a set of identifiability assumptions that yield a surprisingly simple predictor using only a single additional variable. Our approach demonstrates superior empirical performance on several benchmark tasks.