Distributed Simulation and Distributed Inference

Independent samples from an unknown probability distribution $\mathbf{p}$ on a domain of size $k$ are distributed across $n$ players, with each player holding one sample. Each player can communicate $\ell$ bits to a central referee in a simultaneous message passing (SMP) model of communication to help the referee infer a property of the unknown $\mathbf{p}$. When $\ell\geq\log k$ bits, the problem reduces to the well-studied collocated case where all the samples are available in one place. In this work, we focus on the communication-starved setting of $\ell < \log k$, in which the landscape may change drastically. We propose a general formulation for inference problems in this distributed setting, and instantiate it to two prototypical inference questions: learning and uniformity testing.