Communication with Contextual Uncertainty

We introduce a simple model illustrating the utility of context in compressing 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∈ { 0 , 1 } ⁿ and Bob gets Y∈ { 0 , 1 } ⁿ, where (X, Y) is drawn from a known distribution, and Bob wishes to compute some function g(X, Y) or some close approximation to it (i.e., the output is 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 a very close approximation to g. Thus, the function being computed forms the context for the communication. It is an enormous implicit input, potentially described by a truth table of size 2ⁿ. Imprecise knowledge of this function models the (mild) uncertainty in this context. We show that uncertainty can lead to a huge cost in communication. Specifically, we construct a distribution μ over (X, Y) ∈ { 0 , 1 } ⁿ× { 0 , 1 } ⁿ and a class of function pairs (f, g) which are very close (i.e., disagree with o(1) probability when (X, Y) are sampled according to μ), for which the communication complexity of f or g in the standard setting is one bit, whereas the (two-way) communication complexity in the uncertain setting is at least Ω(n) bits even when allowing a constant probability of error. It turns out that this blow-up in communication complexity can be attributed in part to the mutual information between X and Y. In particular, we give an efficient protocol for communication under contextual uncertainty that incurs only a small blow-up in communication if this mutual information is small. Namely, we show that if g has a communication protocol with complexity k in the standard setting and the mutual information between X and Y is I, then g has a one-way communication protocol with complexity O((1 + I) · 2 ^k) in the uncertain setting. This result is an immediate corollary of an even stronger result which shows that if g has one-way communication complexity k, then it has one-way uncertain-communication complexity at most O((1 + I) · k). In the particular case where the input distribution is a product distribution (and so I = 0), the protocol in the uncertain setting only incurs a constant factor blow-up in one-way communication and error.