Communication with Contextual Uncertainty

TL;DR

This paper explores how uncertainty about the context affects communication complexity, demonstrating that protocols can be adapted with only a small overhead when the function has a simple protocol, but can require significantly more communication otherwise.

Contribution

It introduces a model for communication under contextual uncertainty and establishes bounds on the increased complexity, highlighting the role of mutual information and distribution type.

Findings

Protocols with low complexity under known functions can be adapted with minimal overhead in uncertain settings.

For product distributions, the increase in communication complexity is only a constant factor.

Certain functions require significantly more communication in uncertain settings, proving the dependence on mutual information is necessary.

Abstract

We introduce a simple model illustrating the role of context in communication and the challenge posed by uncertainty of knowledge of context. We consider a variant of distributional communication complexity where Alice gets some information $x$ and Bob gets $y$, where $(x,y)$ is drawn from a known distribution, and Bob wishes to compute some function $g(x,y)$ (with high probability over $(x,y)$). In our variant, Alice does not know $g$, but only knows some function $f$ which is an approximation of $g$. Thus, the function being computed forms the context for the communication, and knowing it imperfectly models (mild) uncertainty in this context. A naive solution would be for Alice and Bob to first agree on some common function $h$ that is close to both $f$ and $g$ and then use a protocol for $h$ to compute $h(x,y)$. We show that any such agreement leads to a large overhead in communication ruling out such a universal solution. In contrast, we show that if $g$ has a one-way communication protocol with complexity $k$ in the standard setting, then it has a communication protocol with complexity $O(k \cdot (1+I))$ in the uncertain setting, where $I$ denotes the mutual information between $x$ and $y$. In the particular case where the input distribution is a product distribution, the protocol in the uncertain setting only incurs a constant factor blow-up in communication and error. Furthermore, we show that the dependence on the mutual information $I$ is required. Namely, we construct a class of functions along with a non-product distribution over $(x,y)$ for which the communication complexity is a single bit in the standard setting but at least $\Omega(\sqrt{n})$ bits in the uncertain setting.