Strong Coordination over Noisy Channels

We study the problem of strong coordination of the actions of two nodes $\mathsf X$ and $\mathsf Y$ that communicate over a discrete memoryless channel (DMC) such that the actions follow a prescribed joint probability distribution. We propose two novel random coding schemes and a polar coding scheme for this noisy strong coordination problem, and derive inner and outer bounds for the respective strong coordination capacity region. The first scheme is a joint coordination-channel encoding scheme that utilizes the randomness provided by the communication channel to reduce the amount of local randomness required to generate the sequence of actions at Node $\mathsf Y$. Based on this random coding scheme, we provide a characterization of the capacity region for a special case of the noisy strong coordination setup, namely, when the DMC is a deterministic channel. The second scheme exploits separate coordination and channel encoding where local randomness is extracted from the channel after decoding. Moreover, by leveraging the random coding results for this problem, we present an example in which the proposed joint encoding scheme is able to strictly outperform the separate encoding scheme in terms of achievable communication rate for the same amount of injected randomness into both systems. Thus, we establish the sub-optimality of the separation of strong coordination and channel encoding with respect to the communication rate over the DMC in this problem. Finally, the third scheme is a joint coordination-channel polar coding scheme for strong coordination. We show that polar codes are able to achieve the established inner bound to the strong noisy coordination capacity region and thus provide a constructive alternative to a random coding proof. Our polar coding scheme also offers a constructive solution to a channel simulation problem where a DMC and shared randomness are employed together to simulate another DMC.