Compression in a Distributed Setting.

Motivated by an attempt to understand the formation and development of (human) language, we introduce a compression problem. In our problem a sequence of pairs of players from a set of K players are chosen and tasked to communicate messages drawn from an unknown distribution Q. Arguably languages are created and evolve to compress frequently occurring messages, and we focus on this aspect.The only knowledge that players have about the distribution Q is from previously drawn samples, but these samples differ from player to player.The only common knowledge between the players is restricted to a common prior distribution P and some constant numberof bits of information (such as a learning algorithm). Letting T_epsilon denote the number of iterations it would take for a typical playerto obtain an epsilon-approximation to Q in total variation distance, we askwhether T_epsilon iterations suffice to compress the messages down roughly to theirentropy and give a partial positive answer.We show that a natural uniform algorithm can compress the communication down to an average cost permessage of O(H(Q) + log (D(P || Q)) in tilde{O}(T_epsilon) iterationswhile allowing for O(epsilon)-error,where D(. || .) denotes the KL-divergence between distributions.For large divergencesthis compares favorably with the static algorithm that ignores all samples andcompresses down to H(Q) + D(P || Q) bits, while not requiring T_epsilon * K iterations that it would take players to develop optimal but separate compressions for each pair of players.Along the way we introduce a data-structural view of the task ofcommunicating with a natural language and show that our natural algorithm can also beimplemented by an efficient data structure, whose storage is comparable to the storage requirements of Q and whose query complexity is comparable to the lengths of the message to becompressed.Our results give a plausible mathematical analogy to the mechanisms by whichhuman languages get created and evolve, and in particular highlights thepossibility of coordination towards a joint task (agreeing on a language)while engaging in distributed learning.