It has been a long standing and consistent finding that models that describe seemingly very different data and dynamics, can still be subject to very similar phenomena, such as the asymptotic minimax risk coinciding as the number of samples grows. This finds mathematical substantiation in Le Cam’s theory of asymptotic equivalence. In a so called distributed setting, however, samples are not readily available in one « location », meaning that there is no access to the complete data. This scenario commonly occurs when data is observed or stored at multiple locations, such as hospitals, sensors or servers, from which the data cannot be shared in full because of communication constraints.

In this talk, we shall focus on distributed settings where the full data cannot be shared due to either bandwidth or local differential privacy constraints. We study the extent to which models are still asymptotically equivalent given these constraints and present general sufficient conditions for asymptotic equivalence under bandwidth or privacy constraints.

As an application, we take on minimax testing in distributed setting, with bandwidth or local differential privacy constraints. Until now, the minimax performance for these settings is only known for Gaussian models. We shall use the derived theory to obtain distributed inference performance bounds in more complex models which are known to be asymptotically equivalent to the many-normal-means model and the infinite dimensional signal-in-white-noise model. Furthermore, we shall also demonstrate asymptotic nonequivalence results between certain models following from our theory, which hold even without the presence of communication constraints.