  To avoid failures on out-of-distribution data, recent works have sought to
extract features that have an invariant or stable relationship with the label
across domains, discarding "spurious" or unstable features whose relationship
with the label changes across domains. However, unstable features often carry
complementary information that could boost performance if used correctly in the
test domain. In this work, we show how this can be done without test-domain
labels. In particular, we prove that pseudo-labels based on stable features
provide sufficient guidance for doing so, provided that stable and unstable
features are conditionally independent given the label. Based on this
theoretical insight, we propose Stable Feature Boosting (SFB), an algorithm
for: (i) learning a predictor that separates stable and
conditionally-independent unstable features; and (ii) using the stable-feature
predictions to adapt the unstable-feature predictions in the test domain.
Theoretically, we prove that SFB can learn an asymptotically-optimal predictor
without test-domain labels. Empirically, we demonstrate the effectiveness of
SFB on real and synthetic data.
