Mean estimation in the add-remove model of differential privacy

Differential privacy is often studied under two different models of neighboring datasets: the add-remove model and the swap model. While the swap model is used extensively in the academic literature, many practical libraries use the more conservative add-remove model. However, analysis under the add-remove model can be cumbersome, and obtaining results with tight constants requires some additional work. Here, we study the problem of one-dimensional mean estimation under the add-remove model of differential privacy. We propose a new algorithm and show that it is min-max optimal, that it has the correct constant in the leading term of the mean squared error, and that this constant is the same as the optimal algorithm in the swap model. Our results show that, for mean estimation, the add-remove and swap model give nearly identical error even though the add-remove model cannot treat the size of the dataset as public information. In addition, we demonstrate empirically that our proposed algorithm yields a factor of two improvement in mean squared error over algorithms often used in practice.