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Achieving optimal statistical performance while ensuring the privacy ofpersonal data is a challenging yet crucial objective in modern data analysis.However, characterizing the optimality, particularly the minimax lower bound,under privacy constraints is technically difficult.  To address this issue, we propose a novel approach called the score attack,which provides a lower bound on the differential-privacy-constrained minimaxrisk of parameter estimation. The score attack method is based on the tracingattack concept in differential privacy and can be applied to any statisticalmodel with a well-defined score statistic. It can optimally lower bound theminimax risk of estimating unknown model parameters, up to a logarithmicfactor, while ensuring differential privacy for a range of statisticalproblems. We demonstrate the effectiveness and optimality of this generalmethod in various examples, such as the generalized linear model in bothclassical and high-dimensional sparse settings, the Bradley-Terry-Luce modelfor pairwise comparisons, and nonparametric regression over the Sobolev class.

Score Attack: A Lower Bound Technique for Optimal Differentially Private Learning