The simplest and most widely applied method for guaranteeing differential is to add instance-independent noise to a statistic of interest that is to its global sensitivity. However, global sensitivity is a worst-case that is often too conservative for realized dataset instances. We methods for scaling noise in an instance-dependent way and demonstrate they provide greater accuracy under average-case distributional. Specifically, we consider the basic problem of privately estimating the mean a real distribution from i.i.d.~samples. The standard empirical mean can have arbitrarily-high global sensitivity. We propose the trimmed estimator, which interpolates between the mean and the median, as a way of much lower sensitivity on average while losing very little in terms statistical accuracy. To privately estimate the trimmed mean, we revisit the smooth sensitivity of Nissim, Raskhodnikova, and Smith (STOC 2007), which provides a for using instance-dependent sensitivity. We propose three new noise distributions which provide concentrated differential privacy scaled to smooth sensitivity. We provide theoretical and experimental showing that our noise distributions compare favorably to others in literature, in particular, when applied to the mean estimation problem.
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Bun et al. (2019) studied this question.