We study a distributed sampling problem where a set of processors want to output (approximately) independent and identically distributed samples from a joint distribution with the help of a common message from a coordinator. Each processor has access to a subset of sources from a set of independent sources of "shared" randomness. We consider two cases -- in the "omniscient coordinator setting", the coordinator has access to all these sources of shared randomness, while in the "oblivious coordinator setting", it has access to none. All processors and the coordinator may privately randomize. In the omniscient coordinator setting, when the subsets at the processors are disjoint (individually shared randomness model), we characterize the rate of communication required from the coordinator to the processors over a multicast link. For the two-processor case, the optimal rate matches a special case of relaxed Wyner's common information proposed by Gastpar and Sula (2019), thereby providing an operational meaning to the latter. We also give an upper bound on the communication rate for the "randomness-on-the-forehead" model where each processor observes all but one source of randomness and we give an achievable strategy for the general case where the processors have access to arbitrary subsets of sources of randomness. Also, we consider a more general model where the processors observe components of correlated sources (with the coordinator observing all the components), where we characterize the communication rate when all the processors wish to output the same random sequence. In the oblivious coordinator setting, we completely characterize the trade-off region between the communication and shared randomness rates for the general case where the processors have access to arbitrary subsets of sources of randomness.
翻译:我们研究一个分布式抽样问题,即一组处理者希望在协调员的共同信息帮助下从联合分发的样本中输出(约)独立和同样分布的样本。每个处理者都可以从一组“共享”随机独立的来源获得一组来源的通信。我们考虑两个案例——在“无意识的协调者”的设置中,协调员可以接触所有这些共享随机性来源,而在“明显协调者”的设置中,协调员可以接触任何此类来源。所有处理者和协调员都可以私下随机地进行。在无意识的协调员设置中,当处理器的子集脱节(个人共享随机模式)时,我们确定协调员与处理者之间在多盘点链接上需要的通信速度。对于两个案例,即“无意识的协调者”和“明显协调员”提出的所有共享随机性来源,而协调员提出的一个特殊案例,因此后者具有操作意义。在“我们使用前头处理器的分解”模型的分解分解分解分解器中,我们描述每个处理者与处理者之间的通信速度的分解速度,每个分解的分解分解的分解式处理器的分解顺序,同时考虑总的分解的分解的分解过程,我们所有分解的分解的分解的分解的分解的分解过程。