Probabilistic communication complexity

2011

Combinatorica

We prove an essentially tight lower bound on the unbounded-error communication complexity of every symmetric function, i.e., f (x, y) = D(|x∧y|), where D : {0, 1, . . . , n} → {0, 1} is a given predicate and x, y range over {0, 1} n . Specifically, we show that the communication complexity of f is between Θ(k/ log 5 n) and Θ(k log n), where k is the number of value changes of D in {0, 1, . . . , n}. Prior to this work, the problem was solved only for the parity predicate D (Forster 2001).Our proof is built around two new ideas. First, we show that a predicate D gives rise to a rapidly mixing random walk on Z n 2 , which allows us to reduce the problem to communication lower bounds for "typical" predicates. Second, we use Paturi's approximation lower bounds (1992), suitably generalized here to clusters of real nodes in [0, n] and interpreted in their dual form, to prove that a typical predicate behaves analogous to the parity predicate with respect to a smooth distribution on the inputs.

“…The unbounded-error model, due to Paturi and Simon [28], is a rich and elegant model of communication. Fix a function f : X ×Y → {0, 1}, where X and Y are some finite sets.…”

Section: Introductionmentioning

confidence: 99%

The unbounded-error communication complexity of symmetric functions

2011

Combinatorica

“…As mentioned in the proof, the function f in question satisfies dc(f ) ≥ 2 Θ(n) . This is equivalent to saying that f has communication complexity Θ(n) in the unbounded-error model of Paturi and Simon (1986).…”

Section: Moreover This Communication Cost Can Be Achieved With a Onementioning

confidence: 99%

“…Furthermore, the communication complexity of f remains Ω(n) even if one simply seeks a randomized/quantum protocol with advantage 2 −n/4 over random guessing, on every input. Finally, it is clear from our proof (see Remark 3.10) that f has complexity Ω(n) in the unbounded-error model due to Paturi and Simon (1986), which has an even weaker success criterion.…”

Section: Introductionmentioning

confidence: 99%

Communication Complexity Under Product and Nonproduct Distributions

2010

comput. complex.

We solve an open problem in communication complexity posed by Kushilevitz and Nisan (1997). Let R (f ) and D μ (f ) denote the randomized and μ-distributional communication complexities of f, respectively ( a small constant). Yao's well-known minimax principle states that R (f ) = max μ {D μ (f )} . Kushilevitz and Nisan (1997) ask whether this equality is approximately preserved if the maximum is taken over product distributions only, rather than all distributions μ. We give a strong negative answer to this question. Specifically, we prove the existence of a function f :We also obtain an exponential separation between the statistical query dimension and signrank, solving a problem previously posed by the author (2007).

“…For completeness, we note that a slightly worse bound can be obtained using well-known and more elementary ideas relating margin complexity and discrepancy (Forster et al 2001;Paturi & Simon 1986). Proposition 6.1 and Lemma 6.3 readily imply the main result of this section:…”

Section: Margin-dimension Gapmentioning

confidence: 99%

Halfspace Matrices

2008

comput. complex.

We introduce the notion of a halfspace matrix, which is a sign matrix A with rows indexed by linear threshold functions f, columns indexed by inputs x ∈ {−1, 1} n , and the entries given by A f,x = f (x). We use halfspace matrices to solve the following problems. In communication complexity, we exhibit a Boolean function f with discrepancy Ω(1/n 4 ) under every product distribution but O( √ n/2 n/4 ) under a certain non-product distribution. This partially solves an open problem of Kushilevitz and Nisan (1997).In learning theory, we give a short and simple proof that the statisticalquery (SQ) dimension of halfspaces in n dimensions is less than 2(n + 1) 2 under all distributions. This improves on the n O(1) estimate from the fundamental paper of Blum et al. (1998). We show that estimating the SQ dimension of natural classes of Boolean functions can resolve major open problems in complexity theory, such as separating PSPACE cc and PH cc . Finally, we construct a matrix A ∈ {−1, 1} N ×N log N with dimension complexity log N but margin complexity Ω(N 1/4 / √ log N ). This gap is an exponential improvement over previous work. We prove several other relationships among the complexity measures of sign matrices, complementing work by Linial et al. ( , 2007.