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 $${X \in \{0,1\}^n}$$X∈{0,1}n and Bob gets $${Y \in \{0,1\}^n}$$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 $${\mu}$$μ over $${(X,Y)\in \{0,1\}^n \times \{0,1\}^n}$$(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 $${\mu}$$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 $${\Omega(\sqrt{n})}$$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 $${O((1+I)\cdot 2^k)}$$O((1+I)·2k) 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 $${O((1+I)\cdot k)}$$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.

中文翻译：

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具有上下文不确定性的通信

我们介绍了一个简单的模型，说明上下文在压缩通信中的效用以及上下文知识的不确定性带来的挑战。我们考虑分布式通信复杂性的一种变体，其中 Alice 得到一些信息 $${X \in \{0,1\}^n}$$X∈{0,1}n 而 Bob 得到 $${Y \in \{ 0,1\}^n}$$Y∈{0,1}n，其中 (X, Y) 来自已知分布，并且 Bob 希望计算一些函数 g(X, Y) 或一些近似于它（即，输出是 g(X, Y)，其概率超过 (X, Y)）。在我们的变体中，Alice 不知道 g，但只知道一些非常接近 g 的函数 f。因此，正在计算的函数形成了通信的上下文。这是一个巨大的隐式输入，可能由大小为 2n 的真值表来描述。对该函数的不精确知识模拟了这种情况下的（轻度）不确定性。我们表明，不确定性会导致巨大的沟通成本。具体来说，我们在 $${(X,Y)\in \{0,1\}^n \times \{0,1\}^n}$$( X,Y)∈{0,1}n×{0,1}n 和一类非常接近的函数对 (f, g)（即，当 (X, Y) 为根据 $${\mu}$$μ) 采样，其中 f 或 g 在标准设置下的通信复杂度为一位，而不确定设置下的（双向）通信复杂度至少为 $${ \Omega(\sqrt{n})}$$Ω(n) 比特，即使在允许错误概率不变的情况下。事实证明，这种通信复杂性的激增部分归因于 X 和 Y 之间的互信息。 特别是，我们在上下文不确定性下提供了一种有效的通信协议，如果这种互信息很小，则只会导致通信中的小爆发。即，我们证明如果 g 在标准设置中有一个复杂度为 k 的通信协议，并且 X 和 Y 之间的互信息为 I，则 g 有一个复杂度为 $${O((1+I) \cdot 2^k)}$$O((1+I)·2k) 在不确定的环境中。这个结果是一个更强的结果的直接推论，该结果表明，如果 g 具有单向通信复杂度 k，那么它最多具有单向不确定通信复杂度 $${O((1+I)\cdot k) }$$O((1+I)·k)。在输入分布是乘积分布（因此 I = 0）的特定情况下，不确定设置中的协议只会在单向通信和错误中导致常数因子爆炸。