The problem of domain generalization concerns learning predictive models that are robust to distribution shifts when deployed in new, previously unseen environments. Existing methods typically require labeled data from multiple training environments, limiting their applicability when labeled data are scarce. In this work, we study domain generalization in an anti-causal setting, where the outcome causes the observed covariates. Under this structure, environment perturbations that affect the covariates do not propagate to the outcome, which motivates regularizing the model's sensitivity to these perturbations. Crucially, estimating these perturbation directions does not require labels, enabling us to leverage unlabeled data from multiple environments. We propose two methods that penalize the model's sensitivity to variations in the mean and covariance of the covariates across environments, respectively, and prove that these methods have worst-case optimality guarantees under certain classes of environments. Finally, we demonstrate the empirical performance of our approach on a controlled physical system and a physiological signal dataset.
Figures & tables
Figure 1: Illustration of MIR. Blue points represent the training environment perturbation means \EX[εei] , and the green star represents the test environment mean. The red ellipse represents the covariance structure of the regularization matrix Var(K) , with red arrows indicating its (scaled) eigenvectors. The annotated arrows show the directions of the OLS and MIR solutions with different regularization strengths γ , as well as the optimal solution βtest∗ for the test environment etst .
Figure 2: Diagram of the light tunnel and the subset of variables used in our experiment: the outcomes or intervention targets red , green and blue , and the light-intensity measurements ir_1 , vis_1 , …, vis_3 used as predictors. Figure adapted from Gamella et al. (2025) , licensed under CC BY 4.0.
Figure 3: Performance on the Light Tunnel dataset. We show average RMSE (with standard errors) for leave-one-environment-out cross-validation across all outcome-intervention combinations. The x-axis indicates the number of environments with labeled observations; MIR uses unlabeled data from all training environments regardless of this number.
Figure 4: Performance on VitalDB dataset when all 128 training subjects have labeled data. (a) CVaR (average nMSE for subjects whose errors are above a given quantile) as a function of quantile threshold. (b) Distribution of per-subject Spearman’s correlations between predicted and true stroke volume variations. VIR achieves improved robustness, especially on worse-performing subjects, while maintaining tracking performance comparable to baselines.
Figure 5: Performance on VitalDB dataset as a function of the number of labeled environments. VIR improves over baselines across all settings. The improvement is most pronounced when few labeled environments are available.
Appendix figures & tables2 assets
Supplementary material from the paper’s appendix.
Appendix
Figure 6: Stress-tests on violations of causal assumptions
Figure 7: Effect of unlabelled environments in VitalDB
School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, China · Key Laboratory of Big Data Mining and Knowledge Management, Chinese Academy of Sciences, Beijing, China