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Trading Information Complexity for Error

Authors Yuval Dagan, Yuval Filmus, Hamed Hatami, Yaqiao Li

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Yuval Dagan
Yuval Filmus
Hamed Hatami
Yaqiao Li

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Yuval Dagan, Yuval Filmus, Hamed Hatami, and Yaqiao Li. Trading Information Complexity for Error. In 32nd Computational Complexity Conference (CCC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 79, pp. 16:1-16:59, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017) https://doi.org/10.4230/LIPIcs.CCC.2017.16

Abstract

We consider the standard two-party communication model. The central problem studied in this article is how much can one save in information complexity by allowing a certain error.

* For arbitrary functions, we obtain lower bounds and upper bounds indicating a gain that is of  order Omega(h(epsilon)) and O(h(sqrt{epsilon})). Here h denotes the binary entropy function.

* We analyze the case of the two-bit AND function in detail to show that for this function the gain is Theta(h(epsilon)). This answers a question of Braverman et al. [Braverman, STOC 2013].

* We obtain sharp bounds for the set disjointness function of order n. For the case of the distributional error, we introduce a new protocol that achieves a gain of Theta(sqrt{h(epsilon)}) provided that n is sufficiently large.  We apply these results to answer another of question of Braverman et al. regarding the randomized communication complexity of the set disjointness function.

* Answering a question of Braverman [Braverman, STOC 2012], we apply our analysis of the set disjointness function to establish a gap between the two different notions of the prior-free information cost. In light of [Braverman, STOC 2012], this implies that amortized randomized communication complexity is not necessarily equal to the amortized distributional communication complexity with respect to the hardest distribution.


As a consequence, we show that the epsilon-error randomized communication complexity of the set disjointness function of order n is n[C_{DISJ} - Theta(h(epsilon))] + o(n), where C_{DISJ} ~ 0.4827$ is the constant found by Braverman et al. [Braverman, STOC 2012].

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Keywords
  • communication complexity
  • information complexity

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