A Bias-Variance-Privacy Trilemma for Statistical Estimation

Abstract

The canonical algorithm for differentially private mean estimation is to first clip the samples to a bounded range and then add noise to their empirical mean. Clipping controls the sensitivity and, hence, the variance of the noise that we add for privacy. But clipping also introduces statistical bias. We prove that this tradeoff is inherent: no algorithm can simultaneously have low bias, low variance, and low privacy loss for arbitrary distributions.

On the positive side, we show that unbiased mean estimation is possible under approx- imate differential privacy if we assume that the distribution is symmetric. Relaxing to approximate differential privacy is necessary. We show that, even when the data is sampled from a Gaussian, unbiased mean estimation is impossible under pure or concentrated differential privacy.