Interactive Information Complexity

The primary goal of this paper is to define and study the interactive information complexity of functions. Let $f(x,y)$ be a function, and suppose Alice is given $x$ and Bob is given $y$. Informally, the interactive information complexity $\mathsf{IC}(f)$ of $f$ is the least amount of information Alice and Bob need to reveal to each other to compute $f$. Previously, information complexity has been defined with respect to a prior distribution on the input pairs $(x,y)$. Our first goal is to give a definition that is independent of the prior distribution. We show that several possible definitions are essentially equivalent. We establish some basic properties of the interactive information complexity $\mathsf{IC}(f)$. In particular, we show that $\mathsf{IC}(f)$ is equal to the amortized (randomized) communication complexity of $f$. We also show a direct sum theorem for $\mathsf{IC}(f)$ and give the first general connection between information complexity and (nonamortized) communication complexity. This conn...