The communication complexity of many fundamental problems reduces greatly when the communicating parties share randomness that is independent of the inputs to the communication task. Natural communication processes (say between humans) however often involve large amounts of shared correlations among the communicating players, but rarely allow for perfect sharing of randomness. Can the communication complexity benefit from shared correlations as well as it does from shared randomness? This question was considered mainly in the context of simultaneous communication by Bavarian et al. (ICALP 2014). In this work we study this problem in the standard interactive setting and give some general results. In particular, we show that every problem with communication complexity of $k$ bits with perfectly shared randomness has a protocol using imperfectly shared randomness with complexity $\exp(k)$ bits. We also show that this is best possible by exhibiting a promise problem with complexity $k$ bits with perfectly shared randomness which requires $\exp(k)$ bits when the randomness is imperfectly shared. Along the way we also highlight some other basic problems such as compression, and agreement distillation, where shared randomness plays a central role and analyze the complexity of these problems in the imperfectly shared randomness model. The technical highlight of this work is the lower bound that goes into the result showing the tightness of our general connection. This result builds on the intuition that communication with imperfectly shared randomness needs to be less sensitive to its random inputs than communication with perfectly shared randomness. The formal proof invokes results about the small-set expansion of the noisy hypercube and an invariance principle to convert this intuition to a proof, thus giving a new application domain for these fundamental results.

翻译：许多基础问题的通信复杂度在通信双方共享与输入无关的随机性时会大幅降低。然而，自然通信过程（例如人类间的通信）中，通信方之间往往存在大量共享相关性，但很少能实现完美的随机性共享。那么，通信复杂度能否像从共享随机性中获益那样，也从共享相关性中获益？这个问题此前主要在Bavarian等人（ICALP 2014）提出的同步通信场景中被研究。本文在标准交互式设定下探讨该问题，并给出若干一般性结论。具体而言，我们证明：任何在完美共享随机性下通信复杂度为$k$比特的问题，都存在一个使用不完美共享随机性且复杂度为$\exp(k)$比特的协议。我们还通过构造一个在完美共享随机性下复杂度为$k$比特但需$\exp(k)$比特才能解决的承诺问题，证明这一上界是最优的。在此过程中，我们进一步揭示了压缩、一致性蒸馏等共享随机性起核心作用的基础问题，并分析了它们在不完美共享随机性模型下的复杂度。本文的技术亮点在于证明上述一般性结论紧致性的下界。该结果基于如下直觉：不完美共享随机性下的通信需比完美共享随机性下的通信对随机输入更不敏感。形式化证明通过调用噪声超立方体的小集扩张定理和不变性原理，将这一直觉转化为严谨论证，从而为这些基础结论开辟了新的应用领域。