Szabó Botond (Bocconi Milano University): Optimal distributed testing under communication constraints in high-dimensional and nonparametric Gaussian white noise model

We study the problem of signal detection in Gaussian noise in a distributed setting both for high-dimensional and nonparametric signals. We consider both the public and private coin protocols, i.e. when the machines have and don't have access to a shared source of randomness, respectively. We derive lower bounds on the size that the signal needs to have in order to be detectable. We also derive matching upper bounds based on constructive algorithms. We distinguish different regimes based on the dimension of the model (or the smoothness of the signal in the nonparametric setting), the number of machines and the number of transmitted bits between the machines. We show that in certain regimes under the more flexible public coin protocol one can achieve lower detection boundaries than using private coins, while in other regimes the two type of protocols results in the same testing limitations and guarantees. Finally in the nonparametric framework we derive both lower and upper bounds for adaptation.