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Domain adaptation (DA) is a statistical learning problem that arises when the distribution of the source data used to train a model differs from that of the target data used to test the model. While many DA algorithms have demonstrated considerable empirical success, the unavailability of target labels in DA makes it challenging to determine their effectiveness in new datasets without a theoretical basis. Therefore, it is essential to clarify the assumptions required for successful DA algorithms and quantify the corresponding theoretical guarantees under these assumptions. In this work, we focus on the assumption that conditionally invariant components (CICs) useful for prediction exist across the source and target data. Under this assumption, we demonstrate that CICs, which can be estimated through conditional invariant penalty (CIP), play three prominent roles in providing theoretical guarantees for DA algorithms. First, we introduce a new CIC-based algorithm called importance-weighted conditional invariant penalty (IW-CIP), which has target risk guarantees beyond simple settings such as covariate shift and label shift. Second, we show that CICs can be used to identify large discrepancies between source and target risks of other DA algorithms. Finally, we demonstrate that incorporating CICs into the domain invariant projection (DIP) algorithm helps to address its well-known failure scenario caused by label-flipping features. We support our new algorithms and theoretical findings via numerical experiments on synthetic data, MNIST, CelebA, and Camelyon17 datasets. |