PAC Learning Guarantees Under Covariate Shift

We consider the Domain Adaptation problem, also known as the covariate shift problem, where the distributions that generate the training and test data differ while retaining the same labeling function. This problem occurs across a large range of practical applications, and is related to the more general challenge of transfer learning. Most recent work on the topic focuses on optimization techniques that are specific to an algorithm or practical use case rather than a more general approach. The sparse literature attempting to provide general bounds seems to suggest that efficient learning even under strong assumptions is not possible for covariate shift. Our main contribution is to recontextualize these results by showing that any Probably Approximately Correct (PAC) learnable concept class is still PAC learnable under covariate shift conditions with only a polynomial increase in the number of training samples. This approach essentially demonstrates that the Domain Adaptation learning problem is as hard as the underlying PAC learning problem, provided some conditions over the training and test distributions. We also present bounds for the rejection sampling algorithm, justifying it as a solution to the Domain Adaptation problem in certain scenarios.