An interactive information odometer and applications

We introduce a novel technique which enables two players to maintain an estimate of the internal information cost of their conversation in an online fashion without revealing much extra information. We use this construction to obtain new results about communication complexity and information-theoretic privacy. As a first corollary, we prove a strong direct product theorem for communication complexity in terms of information complexity: If I bits of information are required for solving a single copy of f under μ with probability 2/3, then any protocol attempting to solve n independent copies of f under μⁿ using o(n · I) communication, will succeed with probability 2^Ω(n). This is tight, as Braverman and Rao [BR11] previously showed that O(n · I) communication suffice to succeed with probability ~ (2/3)ⁿ. We then show how the information odometer can be used to achieve the best possible information-theoretic privacy between two untrusted parties: If the players' goal is to compute a function f(x,y), and f admits a protocol with information cost is I and communication cost C, then our odometer can be used to produce a "robust" protocol which: (i) Assuming both players are honest, computes f with high probability, and (ii) Even if one party is malicious, then for any k ∈ N, the probability that the honest player reveals more than O(k· (I + log C)) bits of information to the other player is at most 2^-Ω(k). Finally, we outline an approach which uses the odometer as a proxy for breaking state of the art interactive compression results: We show that our odometer allows to reduce interactive compression to the regime where I = O(logC), thereby opening a potential avenue for improving the compression result of [BBCR10] and to new direct sum and product theorems in communication complexity.