  Federated learning has attracted significant recent attention due to its
applicability across a wide range of settings where data is collected and
analyzed across disparate locations. In this paper, we study federated
nonparametric goodness-of-fit testing in the white-noise-with-drift model under
distributed differential privacy (DP) constraints.
  We first establish matching lower and upper bounds, up to a logarithmic
factor, on the minimax separation rate. This optimal rate serves as a benchmark
for the difficulty of the testing problem, factoring in model characteristics
such as the number of observations, noise level, and regularity of the signal
class, along with the strictness of the $(\epsilon,\delta)$-DP requirement. The
results demonstrate interesting and novel phase transition phenomena.
Furthermore, the results reveal an interesting phenomenon that distributed
one-shot protocols with access to shared randomness outperform those without
access to shared randomness. We also construct a data-driven testing procedure
that possesses the ability to adapt to an unknown regularity parameter over a
large collection of function classes with minimal additional cost, all while
maintaining adherence to the same set of DP constraints.
