Does Label Differential Privacy Prevent Label Inference Attacks?

Label differential privacy (LDP) is a popular framework for training private ML models on datasets with public features and sensitive private labels. Despite its rigorous privacy guarantee, it has been observed that in practice LDP does not preclude label inference attacks (LIAs): Models trained with LDP can be evaluated on the public training features to recover, with high accuracy, the very private labels that it was designed to protect. In this work, we argue that this phenomenon is not paradoxical and that LDP merely limits the advantage of an LIA adversary compared to predicting training labels using the Bayes classifier. At LDP $\epsilon=0$ this advantage is zero, hence the optimal attack is to predict according to the Bayes classifier and is independent of the training labels. Finally, we empirically demonstrate that our result closely captures the behavior of simulated attacks on both synthetic and real world datasets.