Distributed function estimation: adaptation using minimal communication

We investigate whether in a distributed setting, adaptive estimation of a smooth function at the optimal rate is possible under minimal communication. It turns out that the answer depends on the risk considered and on the number of servers over which the procedure is distributed. We show that for the $L_\infty$-risk, adaptively obtaining optimal rates under minimal communication is not possible. For the $L_2$-risk, it is possible over a range of regularities that depends on the relation between the number of local servers and the total sample size.