A piecewise Laplace mechanism attains pure differential privacy while scaling noise by per-dataset sensitivity, and it strictly dominates inverse sensitivity for sample-monotone functions.
Average-case averages: Private algorithms for smooth sensitivity and mean estimation
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Unifying Laplace Mechanism with Instance Optimality in Differential Privacy
A piecewise Laplace mechanism attains pure differential privacy while scaling noise by per-dataset sensitivity, and it strictly dominates inverse sensitivity for sample-monotone functions.