  Sensitive statistics are often collected across sets of users, with repeated
collection of reports done over time. For example, trends in users' private
preferences or software usage may be monitored via such reports. We study the
collection of such statistics in the local differential privacy (LDP) model,
and describe an algorithm whose privacy cost is polylogarithmic in the number
of changes to a user's value.
  More fundamentally---by building on anonymity of the users' reports---we also
demonstrate how the privacy cost of our LDP algorithm can actually be much
lower when viewed in the central model of differential privacy. We show, via a
new and general privacy amplification technique, that any permutation-invariant
algorithm satisfying $\varepsilon$-local differential privacy will satisfy
$(O(\varepsilon \sqrt{\log(1/\delta)/n}), \delta)$-central differential
privacy. By this, we explain how the high noise and $\sqrt{n}$ overhead of LDP
protocols is a consequence of them being significantly more private in the
central model. As a practical corollary, our results imply that several
LDP-based industrial deployments may have much lower privacy cost than their
advertised $\varepsilon$ would indicate---at least if reports are anonymized.
