Communication with Imperfectly Shared Randomness

by Clement Canonne, Venkatesan Guruswami, Raghu Meka, and Madhu Sudan

Oded's comments

This work that falls under the general theme of communication among parties that are not in full a priori agreement: In A Theory of Goal-Oriented Communication this theme was understood as lack of any a priori common language, which lead to rather wasteful alas optimal solutions. This indicated that better solutions may be possible only in setting that take a less pessimistic view and assume some common understanding, albeit a non-perfect one. In the current work the imperfect agreement takes two forms. The first, which follows a prior work on compression, refers to partial (i.e., imperfect or approximate) agreement about the distribution to which compression is employed. The second (and novel) aspect is of having correlated randomness rather than access to the same source of randomness; indeed, this model is inbetween private randomness and shared randomness, but as indicated in the introduction the small gap between the models is too big in the current context.

The original abstract

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.

See ECCC TR14-153.

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