  We study a basic private estimation problem: each of $n$ users draws a single
i.i.d. sample from an unknown Gaussian distribution, and the goal is to
estimate the mean of this Gaussian distribution while satisfying local
differential privacy for each user. Informally, local differential privacy
requires that each data point is individually and independently privatized
before it is passed to a learning algorithm. Locally private Gaussian
estimation is therefore difficult because the data domain is unbounded: users
may draw arbitrarily different inputs, but local differential privacy
nonetheless mandates that different users have (worst-case) similar privatized
output distributions. We provide both adaptive two-round solutions and
nonadaptive one-round solutions for locally private Gaussian estimation. We
then partially match these upper bounds with an information-theoretic lower
bound. This lower bound shows that our accuracy guarantees are tight up to
logarithmic factors for all sequentially interactive
$(\varepsilon,\delta)$-locally private protocols.
