This paper provides the first sample complexity lower bounds for the estimation of simple diffusion models, including the Bass model (used in modeling consumer adoption) and the SIR model (used in modeling epidemics). We show that one cannot hope to learn such models until quite late in the diffusion. Specifically, we show that the time required to collect a number of observations that exceeds our sample complexity lower bounds is large. For Bass models with low innovation rates, our results imply that one cannot hope to predict the eventual number of adopting customers until one is at least two-thirds of the way to the time at which the rate of new adopters is at its peak. In a similar vein, our results imply that in the case of an SIR model, one cannot hope to predict the eventual number of infections until one is approximately two-thirds of the way to the time at which the infection rate has peaked. These limits are borne out in both product adoption data (Amazon), as well as epidemic data (COVID-19).

翻译：本文为估算简单的传播模型,包括Bass模型(用于模拟消费者采用)和SIR模型(用于模拟流行病)提供了第一个样本复杂性较低的下限,以估计简单的传播模型,包括Bass模型(用于模拟消费者采用)和SIR模型(用于模拟流行病)。我们表明,人们不可能期望在传播时间过后再学习这种模型。具体地说,我们表明,收集超过我们抽样复杂程度的多点观测数据所需的时间要长得多。 对于创新率低的Bass模型,我们的结果意味着,人们不可能指望预测最终接受客户的人数,直到一个是至少达到新采用者比率高峰三分之二的路程。 类似地,我们的结果意味着,在SIR模型中,人们不可能期望预测最终感染人数,直到感染率达到高峰的时间的大约三分之二。 对于产品采用数据(亚马逊)和流行病数据(COVID-19)都证明了这些限制。