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Differential privacy (DP) is by far the most widely accepted framework formitigating privacy risks in machine learning. However, exactly how small theprivacy parameter $\epsilon$ needs to be to protect against certain privacyrisks in practice is still not well-understood. In this work, we study datareconstruction attacks for discrete data and analyze it under the framework ofmultiple hypothesis testing. We utilize different variants of the celebratedFano's inequality to derive upper bounds on the inferential power of a datareconstruction adversary when the model is trained differentially privately.Importantly, we show that if the underlying private data takes values from aset of size $M$, then the target privacy parameter $\epsilon$ can be $O(\logM)$ before the adversary gains significant inferential power. Our analysisoffers theoretical evidence for the empirical effectiveness of DP against datareconstruction attacks even at relatively large values of $\epsilon$.

Analyzing Privacy Leakage in Machine Learning via Multiple Hypothesis Testing: A Lesson From Fano