Adaptive Divide and Conquer with Two Rounds of Communication

We introduce a two-round adaptive communication strategy that enables rate-optimal estimation in the white noise model without requiring prior knowledge of the underlying smoothness. In the first round, local machines send summary statistics using $(\log_2(n))^2$ bits to enable the central machine to select the tuning parameters of the procedure. In the second round, another set of statistics are transmitted using optimal number of bits, enabling the central machine to aggregate and produce a final estimator that adapts to the true smoothness level. This approach achieves optimal convergence rates across a wider range of regularities, offering a potential improvement in the adaptability and efficiency of distributed estimation compared to existing one-round methods.