  Multi-source domain adaptation aims to reduce performance degradation when
applying machine learning models to unseen domains. A fundamental challenge is
devising the optimal strategy for feature selection. Existing literature is
somewhat paradoxical: some advocate for learning invariant features from source
domains, while others favor more diverse features. To address the challenge, we
propose a statistical framework that distinguishes the utilities of features
based on the variance of their correlation to label $y$ across domains. Under
our framework, we design and analyze a learning procedure consisting of
learning approximately shared feature representation from source tasks and
fine-tuning it on the target task. Our theoretical analysis necessitates the
importance of learning approximately shared features instead of only the
strictly invariant features and yields an improved population risk compared to
previous results on both source and target tasks, thus partly resolving the
paradox mentioned above. Inspired by our theory, we proposed a more practical
way to isolate the content (invariant+approximately shared) from environmental
features and further consolidate our theoretical findings.
