Information Lower Bounds via Self-Reducibility

Abstract

We use self-reduction methods to prove strong information lower bounds on two of the most studied functions in the communication complexity literature: Gap Hamming Distance (GHD) and Inner Product (IP). In our first result we affirm the conjecture that the information cost of GHD is linear even under the uniform distribution, which strengthens the (n) bound recently shown by Kerenidis et al. (2012), and answers an open problem from Chakrabarti et al. (2012). In our second result we prove that the information cost of I P n is arbitrarily close to the trivial upper bound n as the permitted error tends to zero, again strengthening the (n) lower bound recently proved by Braverman and Weinstein (Electronic Colloquium on Computational Complexity (ECCC) 18, 164 2011). Our proofs demonstrate that self-reducibility makes the connection between information complexity and communication complexity lower bounds a two-way connection. Whereas numerous results in the past (Chakrabarti et al. 2001; Bar-Yossef et al. J. Comput. Syst. Sci. 68(4), 702-732 2004;Barak et al. 2010) used information complexity techniques to derive new communication complexity lower bounds, we explore a generic way in which communication complexity lower bounds imply information complexity lower bounds in a black-box manner.

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“…Call a distribution ν on X × Y optimal if IC( f , 0) = IC ν ( f , 0). Braverman et al [6] showed that IC ν ( f , 0) is continuous in ν, and this implies that optimal distributions exist, and moreover the set of optimal distributions is closed. It is also convex, due to the concavity of IC ν ( f , 0) (see [5]).…”

“…Intuitively, an upper bound like Lemma 7.1 is essentially a compression result. Besides, as DISJ n has a self-reducible structure (see [6]), one can make use of this fact together with the Braverman-Rao [7] compression. A difficulty is that what we want to solve is [DISJ n , ε], that is, the error allowed is non-distributional, while the error unavoidably introduced in the compression phase is distributional.…”

“…As in the work of Braverman et al, we obtain our result by analyzing the information complexity of the 2-bit AND function (in which each player gets one bit). Roughly speaking, the information complexity IC µ ( f , ε) of a function f with respect to a distribution µ on the inputs is the minimal amount of information that the players need to leak in any protocol that computes f correctly with probability at THEORY OF COMPUTING, Volume 14 (6), 2018, pp. 1-73 least 1 − ε on every input.…”

“…THEORY OF COMPUTING, Volume 14(6), 2018, pp. For the first summand, we have that for every (x, y), E 1 z = f (x,y) µ (x, y) = ∑ Pr[XY = xy, π reaches ]1 z = f (x,y) (4.70)= ∑ Pr[π reaches | XY = xy]µ(xy)1 z = f (x,y) (4.71) = µ(xy) ∑ Pr[π x,y reaches ]1 z = f (x,y) = µ(xy) Pr[π(x, y) = f (x, y)] (4.72) ≤ µ(xy)ε ≤ ε, (4.73)where we used that by definition µ (xy) = Pr[XY = xy | π reaches ], and the fact that the protocol π performs the task [ f , ε].…”