信息复杂度的一个偏差下界
A discrepancy lower bound for information complexity
摘要：
本文提供了第一个通用的技术证明信息下界的两方无界轮通信问题。我们表明，差异下界，这适用于随机通信复杂性，也适用于信息复杂性。更准确地说，如果两方函数$f$相对于分布$μ$的差异是$Disc_μf$，则计算$f$的任何两方随机化协议必须向参与者揭示至少$Ω(\log (1/Disc_μf))$比特的信息。作为推论，我们得到，任何两方协议计算$\{0,1\}^n\times\{0,1\}^n$上的随机函数必须揭示$Ω(n)$位的信息的参与者。此外，我们证明了Greater-Than函数的差异是$Ω(1/\sqrt{n})$，这为Viola \cite{Viola11}最近证明的关于该函数的通信复杂度的$Ω(\log n)$下界提供了一个替代证明，并结合我们的主要结果，证明了其信息复杂度的紧$Ω(\log n)$下界.我们的主要结果的证明开发了一个新的模拟程序，可能是一个独立的利益。在Kereneirs等人最近的一项突破性工作中，\cite{kerenidis2012lower}，这个模拟过程是证明几乎所有已知的通信复杂度下限技术（而不仅仅是差异）适用于信息复杂度的主要构建块。
英文摘要：
This paper provides the first general technique for proving information lower bounds on two-party unbounded-rounds communication problems. We show that the discrepancy lower bound, which applies to randomized communication complexity, also applies to information complexity. More precisely, if the discrepancy of a two-party function $f$ with respect to a distribution $μ$ is $Disc_μf$, then any two party randomized protocol computing $f$ must reveal at least $Ω(\log (1/Disc_μf))$ bits of information to the participants. As a corollary, we obtain that any two-party protocol for computing a random function on $\{0,1\}^n\times\{0,1\}^n$ must reveal $Ω(n)$ bits of information to the participants. In addition, we prove that the discrepancy of the Greater-Than function is $Ω(1/\sqrt{n})$, which provides an alternative proof to the recent proof of Viola \cite{Viola11} of the $Ω(\log n)$ lower bound on the communication complexity of this well-studied function and, combined with our main result, proves the tight $Ω(\log n)$ lower bound on its information complexity. The proof of our main result develops a new simulation procedure that may be of an independent interest. In a very recent breakthrough work of Kerenidis et al. \cite{kerenidis2012lower}, this simulation procedure was the main building block for proving that almost all known lower bound techniques for communication complexity (and not just discrepancy) apply to information complexity.
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