Relative Discrepancy Does not Separate Information and Communication Complexity

Abstract

Does the information complexity of a function equal its communication complexity? We examine whether any currently known techniques might be used to show a separation between the two notions. Ganor et al. recently provided such a separation in the distributional case for a specific input distribution. We show that in the non-distributional setting, the relative discrepancy bound is smaller than the information complexity, hence it cannot separate information and communication complexity. In addition, in the distributional case, we provide a linear program formulation for relative discrepancy and relate it to variants of the partition bound, resolving also an open question regarding the relation of the partition bound and information complexity. Last, we prove the equivalence between the adaptive relative discrepancy and the public-coin partition, implying that the logarithm of the adaptive relative discrepancy bound is quadratically tight with respect to communication. 41 instance. This has been shown to be true in the simultaneous and one-way models [5,6], for bounded-round two-way protocols under product distributions [3,7] or non-product distributions [1], and also for specific functions like Disjointness [8]; non-trivial direct sum theorems have also been shown for general two-way randomized communication complexity [9]. Since the information complexity is equal to amortized communication complexity [1], the question of whether information and communication complexity are equal is equivalent to whether communication complexity has a direct sum property [1,10]. Note that in the case of deterministic, zero-error protocols, a separation between information and communication complexity is known for Equality [10].Since information complexity deals with the information Alice and Bob transmit about their inputs, it is necessary to define a distribution on these inputs. For each fixed distribution μ, we define the distributional information complexity of a function f (also known as the information cost) as the information Alice and Bob transmit about their inputs in any protocol that solves f with small error according to μ [1,5]. The (non-distributional) information complexity of the function f is defined as its distributional information complexity for the worst distribution μ [10]. In this paper we consider the internal information complexity.Similarly, for communication complexity, one may also consider a model with a distribution μ over the inputs, and the error probability of the protocol is taken over this distribution. This is called a distributional model, and Yao's minmax principle [11] states that the randomized communication complexity of f is equal to its distributional communication complexity for the worst distribution μ, where the randomized communication complexity of a function f is defined as the minimum number of bits exchanged, in the worst case over the inputs, for a randomized protocol to compute the function with small error [12].One can therefore ask whether the following stronger relation ho...

Exponential Separation of Information and Communication for Boolean Functions

“…Recently, [12] showed several interesting connections between the relative discrepancy method and other lower bounds techniques (see Section 3 for more details).…”

“…In a followup work, [12] showed that the relative discrepancy bound can be described as the optimum value of a maximization program, which is a relaxation of the partition bound defined by [17]. [12] noted that the variant in which ρ in Definition 1 is any function ρ : {0, 1} n × {0, 1} n → R (not necessarily non-negative), such that x,y ρ(x, y) = 1, is equivalent to the partition bound (up to multiplicative factors that depend on the error probability).…”

“…[12] noted that the variant in which ρ in Definition 1 is any function ρ : {0, 1} n × {0, 1} n → R (not necessarily non-negative), such that x,y ρ(x, y) = 1, is equivalent to the partition bound (up to multiplicative factors that depend on the error probability). [12] also showed that the adaptive relative discrepancy bound is equivalent to the public-coin partition bound defined by [18] (up to multiplicative factors that depend on the error probability). Therefore, by the result of [18], the logarithm of the adaptive relative discrepancy bound is quadratically tight with public-coin randomized communication complexity.…”