We examine the privacy-enhancing properties of importance sampling. In importance sampling, selection probabilities are heterogeneous and each selected data point is weighted by the reciprocal of its selection probability. Due to the heterogeneity of importance sampling, we express our results within the framework of personalized differential privacy. We first consider the general case where an arbitrary personalized differentially private mechanism is subsampled with an arbitrary importance sampling distribution and show that the resulting mechanism also satisfies personalized differential privacy. This constitutes an extension of the established privacy amplification by subsampling result to importance sampling. Then, for any fixed mechanism, we derive the sampling distribution that achieves the optimal sampling rate subject to a worst-case privacy constraint. Empirically, we evaluate the privacy, efficiency, and accuracy of importance sampling on the example of k-means clustering.

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