  Local differential privacy (LDP) is a model where users send privatized data
to an untrusted central server whose goal it to solve some data analysis task.
In the non-interactive version of this model the protocol consists of a single
round in which a server sends requests to all users then receives their
responses. This version is deployed in industry due to its practical advantages
and has attracted significant research interest. Our main result is an
exponential lower bound on the number of samples necessary to solve the
standard task of learning a large-margin linear separator in the
non-interactive LDP model. Via a standard reduction this lower bound implies an
exponential lower bound for stochastic convex optimization and specifically,
for learning linear models with a convex, Lipschitz and smooth loss. These
results answer the questions posed in \citep{SmithTU17,DanielyF18}. Our lower
bound relies on a new technique for constructing pairs of distributions with
nearly matching moments but whose supports can be nearly separated by a large
margin hyperplane. These lower bounds also hold in the model where
communication from each user is limited and follow from a lower bound on
learning using non-adaptive \emph{statistical queries}.
