Communication with Imperfectly Shared Randomness

TL;DR

This work investigates interactive communication when randomness is imperfectly shared between the communicating parties. It shows a general, tight connection: any problem solvable with $k$ bits under perfect shared randomness can be solved with at most $\exp(k)$ bits using imperfectly shared randomness, and proves a matching lower bound via a parameterized promise problem (SparseGapInnerProduct). The authors develop a robust framework based on an inner-product representation of strategies and an invariance principle that transfers low-influence Boolean behavior to Gaussian settings, enabling both upper-bounds and lower-bounds that sharply delineate the power of imperfect sharing. They also analyze compression with uncertain priors and the problem of agreement distillation, highlighting where shared randomness remains central and where robustness to randomness perturbations constrains performance. Overall, the paper illuminates how correlation in randomness affects communication complexity, yielding a principled, generalizable understanding with potential applications to natural communication processes and related information-theoretic tasks.

Abstract

The communication complexity of many fundamental problems reduces greatly when the communicating parties share randomness that is independent of the inputs to the communication task. Natural communication processes (say between humans) however often involve large amounts of shared correlations among the communicating players, but rarely allow for perfect sharing of randomness. Can the communication complexity benefit from shared correlations as well as it does from shared randomness? This question was considered mainly in the context of simultaneous communication by Bavarian et al. (ICALP 2014). In this work we study this problem in the standard interactive setting and give some general results. In particular, we show that every problem with communication complexity of $k$ bits with perfectly shared randomness has a protocol using imperfectly shared randomness with complexity $\exp(k)$ bits. We also show that this is best possible by exhibiting a promise problem with complexity $k$ bits with perfectly shared randomness which requires $\exp(k)$ bits when the randomness is imperfectly shared. Along the way we also highlight some other basic problems such as compression, and agreement distillation, where shared randomness plays a central role and analyze the complexity of these problems in the imperfectly shared randomness model. The technical highlight of this work is the lower bound that goes into the result showing the tightness of our general connection. This result builds on the intuition that communication with imperfectly shared randomness needs to be less sensitive to its random inputs than communication with perfectly shared randomness. The formal proof invokes results about the small-set expansion of the noisy hypercube and an invariance principle to convert this intuition to a proof, thus giving a new application domain for these fundamental results.

Figures (1)

• Figure 1: Illustration of the constraints on $\chi_{A}$ (\ref{['def:xa:xb']}).

Theorems & Definitions (61)

• Proposition 1
• Definition 1: Uncertain Compression
• Theorem 2.1
• Definition 2: Agreement distillation
• Proposition 2
• Theorem 2.2
• Definition 3: $\textsc{GapInnerProduct}\xspace^n_{c,s}$, $\textsc{SparseGapInnerProduct}\xspace^n_{q,c,s}$
• Theorem 2.3
• Theorem 2.4
• Theorem 3.1