How to Compress Interactive Communication

Abstract

We describe new ways to simulate two-party communication protocols to get protocols with potentially less communication. We show that every communication protocol that communicates C bits and reveals I bits of information about the inputs to the participating parties can be simulated by a new protocol involving at mostÕ( √ CI) bits of communication. If the protocol reveals I bits of information about the inputs to an observer that watches the communication in the protocol, we show how to carry out the simulation withÕ(I) bits of communication. These results lead to a direct sum theorem for randomized communication complexity. Ignoring polylogarithmic factors, we show that for worst-case computation, computing n copies of a function requires √ n times the communication required for computing one copy of the function. For average case complexity, given any distribution μ on inputs, computing n copies of the function on n inputs sampled independently according to μ requires √ n times the communication for computing one copy. If μ is a product distribution, computing n copies on n independent inputs sampled according to μ requires n times the communication required for computing the function. We also study the complexity of computing the sum (or parity) of n evaluations of f , and obtain results analogous to those above. Our results give the first compression schemes for general randomized protocols and the first direct sum results in the general setting of randomized and distributional communication complexity, without requiring bound on the number of rounds in the protocol or that the distribution of inputs is independent.

Interactive Information Complexity

“…In [3,4], the information cost IC μ (π) of executing a protocol π over an a priori distribution μ of inputs is defined. This immediately yields a definition of the information cost of a function f with respect to a distribution μ: we simply look for a protocol π for f that minimizes IC μ (π).…”

“…The most basic definition is that of the information cost of a protocol. This notion is implicit in [3] and was explicitly defined in [4] (see also [12]). Note that the information of a protocol π depends on the prior distribution μ, as the mutual information between the transcript Π and the inputs depends on the prior distribution on the inputs.…”

“…Direct sum theorems (and the related direct product theorems) are theorems giving lower bounds for the complexity of n copies of a certain problem in terms of the complexity of one copy. In the context of communication complexity, direct sum theorems in various contexts have been the focus of much work [26,22,66,39,31,4,50,33]. More recently, information-theoretic quantities have been studied in the context of differential privacy as well [57].…”

“…Informationtheoretic reasoning has also been successfully applied in the simpler deterministic setting [25]. The internal information cost for protocols over distributions of inputs was defined implicitly in [3] and explicitly in [4]. As in our earlier works, we will refer to internal information cost as simply information cost in this paper.…”

“…It is the tool that allows one to obtain the only known direct sum results for the general randomized communication complexity. If f is a function and R(f ) is its randomized communication complexity, it has been shown in [4] that the randomized communication complexity of n copies of f satisfies R(f n ) =Ω( √ n · R(f )). In the follow-up work [12] a tight relationship between the amortized distributional communication complexity of a function and its internal information cost has been established.…”