  Recent empirical studies on domain generalization (DG) have shown that DG
algorithms that perform well on some distribution shifts fail on others, and no
state-of-the-art DG algorithm performs consistently well on all shifts.
Moreover, real-world data often has multiple distribution shifts over different
attributes; hence we introduce multi-attribute distribution shift datasets and
find that the accuracy of existing DG algorithms falls even further. To explain
these results, we provide a formal characterization of generalization under
multi-attribute shifts using a canonical causal graph. Based on the
relationship between spurious attributes and the classification label, we
obtain realizations of the canonical causal graph that characterize common
distribution shifts and show that each shift entails different independence
constraints over observed variables. As a result, we prove that any algorithm
based on a single, fixed constraint cannot work well across all shifts,
providing theoretical evidence for mixed empirical results on DG algorithms.
Based on this insight, we develop Causally Adaptive Constraint Minimization
(CACM), an algorithm that uses knowledge about the data-generating process to
adaptively identify and apply the correct independence constraints for
regularization. Results on fully synthetic, MNIST, small NORB, and Waterbirds
datasets, covering binary and multi-valued attributes and labels, show that
adaptive dataset-dependent constraints lead to the highest accuracy on unseen
domains whereas incorrect constraints fail to do so. Our results demonstrate
the importance of modeling the causal relationships inherent in the
data-generating process.
