Round-Preserving Asymptotic Compression of Prior-Free Interactive Protocols

There is a close relationship between the communication complexity and information complexity of communication problems, as demonstrated by results such as Shannon’s noiseless source coding theorem, and the Slepian–Wolf theorem. Here, we study this relationship in the prior-free and interactive setting, where we provide an alternate proof for the result of Braverman [SIAM Review, vol. 59, no. 4, 2017], that the amortized communication complexity of simulating a prior-free interactive communication protocol, is equal to its prior-free information cost. While this is a known result, our approach addresses the need for a more natural proof of it. We also improve on the result by achieving round preservation, and using a bounded quantity of shared randomness. We do this by showing that the communicating parties can produce a reliable estimate of the joint type, or empirical distribution, of their inputs. This estimate is then used in our protocol for the prior-free reverse Shannon theorem with side information at the receiver. These results are then generalized to the interactive setting to obtain our main result.