MDM-Privacy: A Derivative-Driven Framework for Differential Privacy

Ensuring rigorous privacy protection while maintaining data utility remains a central challenge in privacy-preserving data analysis. Traditional Differential Privacy mechanisms, such as the Laplace and Gaussian mechanisms, rely on additive noise calibrated to sensitivity, which often leads to degraded utility, particularly in high-dimensional or low ε regimes. Moreover, these mechanisms lack the flexibility to adapt to the local structure or the gradient behavior of the data. This paper introduces the MDM-Privacy framework, which leverages the ε-MDM-Privacy mechanism, to calibrate privacy noise following sample-level sensitivity. Unlike traditional additive noise mechanisms, ε-MDM-Privacy employs a non-additive, exponential noise model where the privatized output is computed as a function of the form <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i>(<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x</i>) = <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</i>∙<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">e</i><sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ε∙<i>L</i>_<i>p</i>(<i>x</i>)</sup>, enabling precise control over the privacy-utility trade-off through both the privacy budget ε and the norm order <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</i>. The constant <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</i> is estimated via Monte Carlo simulations to ensure probabilistic utility guarantees under confidence bounds. Extensive experiments on the UCI Adult dataset demonstrate that ε-MDM-Privacy achieves significantly lower Mean Absolute Error (MAE) and Mean Relative Error (MRE) compared to the classical Laplace mechanism, particularly under tight privacy regimes. The MDM-Privacy also exhibits robustness across different Minkowski norms, confirming its adaptability to various sensitivity geometries. Moreover, the ε-MDM-Privacy is consistent under both parallel and sequential compositions, enabling the execution of an unlimited number of queries without exhausting the privacy budget (ε). These results highlight the effectiveness of ε-MDM-Privacy as a tunable and utility-preserving alternative for differentially private data release.