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 2 n . 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.