Rényi Information Complexity and an Information Theoretic Characterization of the Partition Bound

Abstract

In this work we introduce a new information-theoretic complexity measure for 2-party functions, called Renyi information complexity. It is a lower-bound on communication complexity, and has the two leading lower-bounds on communication complexity as its natural relaxations: (external) information complexity and logarithm of partition complexity. These two lower-bounds had so far appeared conceptually quite different from each other, but we show that they are both obtained from Renyi information complexity using two different, but natural relaxations: 1. The relaxation of Renyi information complexity that yields information complexity is to change the order of Renyi mutual information used in its definition from infinity to 1. 2. The relaxation that connects Renyi information complexity with partition complexity is to replace protocol transcripts used in the definition of Renyi information complexity with what we term pseudotranscripts, which omits the interactive nature of a protocol, but only requires that the probability of any transcript given inputs x and y to the two parties, factorizes into two terms which depend on x and y separately. While this relaxation yields an apparently different definition than (log of) partition function, we show that the two are in fact identical. This gives us a surprising characterization of the partition bound in terms of an information-theoretic quantity. We also show that if both the above relaxations are simultaneously applied to Renyi information complexity, we obtain a complexity measure that is lower-bounded by the (log of) relaxed partition complexity, a complexity measure introduced by Kerenidis et al. (FOCS 2012). We obtain a sharper connection between (external) information complexity and relaxed partition complexity than Kerenidis et al., using an arguably more direct proof. Further understanding Renyi information complexity (of various orders) might have consequences for important direct-sum problems in communication complexity, as it lies between communication complexity and information complexity.