This paper studies the problem of nonparametric estimation of a smooth
function with data distributed across multiple machines. We assume an
independent sample from a white noise model is collected at each machine, and
an estimator of the underlying true function needs to be constructed at a
central machine. We place limits on the number of bits that each machine can
use to transmit information to the central machine. Our results give both
asymptotic lower bounds and matching upper bounds on the statistical risk under
various settings. We identify three regimes, depending on the relationship
among the number of machines, the size of the data available at each machine,
and the communication budget. When the communication budget is small, the
statistical risk depends solely on this communication bottleneck, regardless of
the sample size. In the regime where the communication budget is large, the
classic minimax risk in the non-distributed estimation setting is recovered. In
an intermediate regime, the statistical risk depends on both the sample size
and the communication budget.

本文研究了分布在多个机器上的数据的非参数平滑函数估计问题，其中每个机器上收集了来自白噪声模型的独立样本，并需要在中央机器上构建潜在真实函数的估计值以及在每台机器上限制了传输信息所需的比特数。我们给出了各种设置下统计风险的渐近下限和匹配上限，并且确定了三个不同的区别（通信预算 / 机器数 / 每台机器上可用数据大小），在这些区别中当通信预算很小时，统计风险仅取决于通信瓶颈，而与样本量无关；在通信预算很大的区域内，与非分布式估计的经典极小风险的情况相同；在一种中间机制中，统计风险取决于样本大小和通信预算。