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We study the fundamental problem of the construction of optimal randomizationin Differential Privacy. Depending on the clipping strategy or additionalproperties of the processing function, the corresponding sensitivity settheoretically determines the necessary randomization to produce the requiredsecurity parameters. Towards the optimal utility-privacy tradeoff, finding theminimal perturbation for properly-selected sensitivity sets stands as a centralproblem in DP research. In practice, l_2/l_1-norm clippings withGaussian/Laplace noise mechanisms are among the most common setups. However,they also suffer from the curse of dimensionality. For more generic clippingstrategies, the understanding of the optimal noise for a high-dimensionalsensitivity set remains limited.  In this paper, we revisit the geometry of high-dimensional sensitivity setsand present a series of results to characterize the non-asymptotically optimalGaussian noise for R\'enyi DP (RDP). Our results are both negative andpositive: on one hand, we show the curse of dimensionality is tight for a broadclass of sensitivity sets satisfying certain symmetry properties; but if,fortunately, the representation of the sensitivity set is asymmetric on somegroup of orthogonal bases, we show the optimal noise bounds need not beexplicitly dependent on either dimension or rank. We also revisit sampling inthe high-dimensional scenario, which is the key for both privacy amplificationand computation efficiency in large-scale data processing. We propose a novelmethod, termed twice sampling, which implements both sample-wise andcoordinate-wise sampling, to enable Gaussian noises to fit the sensitivitygeometry more closely. With closed-form RDP analysis, we prove twice samplingproduces asymptotic improvement of the privacy amplification given anadditional infinity-norm restriction, especially for small sampling rate.

Geometry of Sensitivity: Twice Sampling and Hybrid Clipping in Differential Privacy with Optimal Gaussian Noise and Application to Deep Learning