  The Invariant Risk Minimization (IRM) approach aims to address the challenge
of domain generalization by training a feature representation that remains
invariant across multiple environments. However, in noisy environments,
IRM-related techniques such as IRMv1 and VREx may be unable to achieve the
optimal IRM solution, primarily due to erroneous optimization directions. To
address this issue, we introduce ICorr (an abbreviation for \textbf{I}nvariant
\textbf{Corr}elation), a novel approach designed to surmount the above
challenge in noisy settings. Additionally, we dig into a case study to analyze
why previous methods may lose ground while ICorr can succeed. Through a
theoretical lens, particularly from a causality perspective, we illustrate that
the invariant correlation of representation with label is a necessary condition
for the optimal invariant predictor in noisy environments, whereas the
optimization motivations for other methods may not be. Furthermore, we
empirically demonstrate the effectiveness of ICorr by comparing it with other
domain generalization methods on various noisy datasets.
