Does Privacy Always Harm Fairness? Data-Dependent Trade-offs via Chernoff Information Neural Estimation

Fairness and privacy are two central pillars of trustworthy machine learning, yet their interaction remains poorly understood. To better characterize this relationship, we introduce Chernoff Difference (CD), an information-theoretic notion of data fairness based on Chernoff Information, and its noisy variant for analyzing fairness, privacy, and accuracy jointly. To make this framework usable beyond closed-form settings, we develop CINE (Chernoff Information Neural Estimator), to our knowledge the first neural estimator of Chernoff Information from samples of unknown distributions. Using CD and CINE together, we identify three qualitatively distinct regimes of the privacy--fairness interaction: privacy can hurt fairness, have little effect, or improve it ("free fairness"). We prove the existence of all three regimes analytically in the Gaussian case and observe them empirically on mixtures of Gaussians and real datasets. Overall, our results provide the first principled, data-dependent characterization of when privacy and fairness may align or conflict.