The insights from this work can be extended to settings where public data is not drawn from the same distribution as private data by considering the concept of domain adaptation. In such scenarios, the challenge lies in effectively leveraging the public data to improve the performance of differentially private (DP) models trained on private data. One approach could involve developing algorithms that incorporate techniques from transfer learning, where the model is first pre-trained on the public data and then fine-tuned on the private data. This would allow the model to learn generalizable features from the public dataset while adapting to the specific characteristics of the private dataset.
Additionally, the theoretical bounds established in this work can serve as a benchmark for understanding the limitations and potential improvements when dealing with out-of-distribution public data. By analyzing the error rates in the context of domain mismatch, researchers can identify conditions under which the use of public data still yields benefits, such as when the public data provides complementary information that is relevant to the private data. Furthermore, the development of robust algorithms that can handle distributional shifts, such as adversarial training or domain-invariant feature extraction, could enhance the effectiveness of DP training in these settings.