Differentially Private Multi-Sampling from Distributions

• URL: http://arxiv.org/abs/2412.10512v1
• Date: Fri, 13 Dec 2024 19:14:05 GMT
• Title: Differentially Private Multi-Sampling from Distributions
• Authors: Albert Cheu, Debanuj Nayak,
• Abstract summary: We study the sample complexity of DP emphsingle-sampling i.e., the minimum number of samples needed to perform this task. We define two variants of emphmulti-sampling, where the goal is to privately approximate $m>1$ samples.
• Score: 4.292685318253575
• License:
• Abstract: Many algorithms have been developed to estimate probability distributions subject to differential privacy (DP): such an algorithm takes as input independent samples from a distribution and estimates the density function in a way that is insensitive to any one sample. A recent line of work, initiated by Raskhodnikova et al. (Neurips '21), explores a weaker objective: a differentially private algorithm that approximates a single sample from the distribution. Raskhodnikova et al. studied the sample complexity of DP \emph{single-sampling} i.e., the minimum number of samples needed to perform this task. They showed that the sample complexity of DP single-sampling is less than the sample complexity of DP learning for certain distribution classes. We define two variants of \emph{multi-sampling}, where the goal is to privately approximate $m>1$ samples. This better models the realistic scenario where synthetic data is needed for exploratory data analysis. A baseline solution to \emph{multi-sampling} is to invoke a single-sampling algorithm $m$ times on independently drawn datasets of samples. When the data comes from a finite domain, we improve over the baseline by a factor of $m$ in the sample complexity. When the data comes from a Gaussian, Ghazi et al. (Neurips '23) show that \emph{single-sampling} can be performed under approximate differential privacy; we show it is possible to \emph{single- and multi-sample Gaussians with known covariance subject to pure DP}. Our solution uses a variant of the Laplace mechanism that is of independent interest. We also give sample complexity lower bounds, one for strong multi-sampling of finite distributions and another for weak multi-sampling of bounded-covariance Gaussians.

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