With increasingly deployed deep neural networks in sensitive application domains, such as healthcare and security, it's essential to understand what kind of sensitive information can be inferred from these models. Most known model-targeted attacks assume attackers have learned the application domain or training data distribution to ensure successful attacks. Can removing the domain information from model APIs protect models from these attacks? This paper studies this critical problem. Unfortunately, even with minimal knowledge, i.e., accessing the model as an unnamed function without leaking the meaning of input and output, the proposed adaptive domain inference attack (ADI) can still successfully estimate relevant subsets of training data. We show that the extracted relevant data can significantly improve, for instance, the performance of model-inversion attacks. Specifically, the ADI method utilizes a concept hierarchy extracted from a collection of available public and private datasets and a novel algorithm to adaptively tune the likelihood of leaf concepts showing up in the unseen training data. We also designed a straightforward hypothesis-testing-based attack -- LDI. The ADI attack not only extracts partial training data at the concept level but also converges fastest and requires the fewest target-model accesses among all candidate methods. Our code is available at \url{https://anonymous.4open.science/r/KDD-362D}.
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