Communication with Contextual Uncertainty

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 · (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 Ω( √ n) bits in the uncertain setting.

Universal Communication, Universal Graphs, and Graph Labeling

“…The universal SMP model is also related to a recent line of work studying communication between parties with imperfect knowledge of each other's "context". The most relevant incarnation of this idea is the recent work [GS17,GKKS18], who study the 2-way communication model where Alice and Bob receive functions f and g respectively, with inputs x and y, and must compute f (x, y) under the guarantee that f and g are close in some metric. In other words, one party does not have full knowledge of the function to be computed.…”

The Power of Shared Randomness in Uncertain Communication

“…While early works were very abstract and general, later works (starting with Juba, Kalai, Khanna and Sudan [JKKS11]) tried to explore the ramifications of uncertainty in Yao's standard communication complexity model [Yao79]. In particular, the more recent works relax the different pieces of context that were assumed to be perfectly shared in Yao's model, such as shared randomness [CGMS15], and in a recent work of the authors with Komargodski and Kothari the function being computed [GKKS16].…”

“…Specifically, [GKKS16] study the following functional notion of uncertainty in communication. Their setup builds on -and generalizes -Yao's classical model of (distributional) communication complexity, where Alice has an input x and Bob has an input y, with (x, y) being sampled from a distribution µ.…”

“…The question studied by [GKKS16] is: How much of this gain in communication is preserved when the communicating parties do not exactly agree on the function being computed? (We further discuss the importance of this question in Section 1.2.)…”

“…The quantities owPubCCU µ ǫ (F) and owPrivCCU µ ǫ (F) are similarly defined by restricting to one-way protocols. 2 The previous work ( [GKKS16]) gave an upper bound on owPubCCU µ ǫ (F) whenever F consists of functions g whose one-way distributional complexity is small. More precisely, denote by owCC µ ǫ (g) the one-way communication complexity of g in the standard distributional model.…”