## The unbounded-error communication complexity of symmetric functions (2008)

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Venue: | In Proc. of the 49th Symposium on Foundations of Computer Science (FOCS |

Citations: | 17 - 9 self |

### BibTeX

@INPROCEEDINGS{Sherstov08theunbounded-error,

author = {Alexander A. Sherstov},

title = {The unbounded-error communication complexity of symmetric functions},

booktitle = {In Proc. of the 49th Symposium on Foundations of Computer Science (FOCS},

year = {2008},

pages = {384--393}

}

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### Abstract

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 �(k / log5 n) and �(k log n), where k is the number of value changes of D in {0, 1,..., n}. The unbounded-error model is the most powerful of the basic models of communication (both classical and quantum), and proving lower bounds in it is a considerable challenge. The only previous nontrivial lower bounds for explicit functions in this model appear in the groundbreaking work of Forster (2001) and its extensions. Our proof is built around two novel ideas. First, we show that a given predicate D gives rise to a rapidly mixing random walk on Zn 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 PARITY with respect to a smooth distribution on the inputs.

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Citation Context ... ∗ 1/3 ( f ) + log log [|X| + |Y |]� , where Q∗ 1/3 refers to the quantum model with prior entanglement. An identical inequality is clearly valid for the quantum model without prior entanglement. See =-=[3, 19]-=- for rigorous definitions of these various models; our sole intention was to point out that the unbounded-error model is at least as powerful. Unlike other models of communication complexity, the unbo... |

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Citation Context ..., the learning problem remains unsolved for such natural concept classes as DNF formulas of polynomial size and intersections of two halfspaces, whereas hardness results and lower bounds are abundant =-=[15, 16, 20, 6, 21, 19]-=-. There is, however, an important case when efficient PAC learning is possible. Let C be a given concept class. For notational convenience, view the functions in C as mappings {0, 1} n → {−1, +1} rath... |

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Citation Context ...ions). Symmetric functions have been a vehicle of progress in the study of communication complexity. An illustrative example is the DISJOINTNESS function, whose study has led to considerable advances =-=[3,13,29,31]-=- in randomized communication complexity. Symmetric functions have also contributed to the progress in quantum 3scommunication complexity, starting with the breakthrough result of Razborov [32] and con... |

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Citation Context ...pact on the complexity of certain functions. For example, the well-known disjointness function on n-bit strings has complexity O(log n) in the 9sunbounded-error model and �(n) in the randomized model =-=[11, 29]-=-. Furthermore, explicit functions are known [2,31] with unbounded-error complexity O(log n) that require �( √ n) communication in the randomized model to even achieve advantage 2 −√ n/5 over random gu... |

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Citation Context ... symmetric with ˆφ(S) = 0 for |S| > r. Then there is a polynomial p ∈ Pr with φ(x) = p(x1 + · · · + xn) for all x ∈ {0, 1} n . Minsky and Papert’s observation has seen numerous uses in the literature =-=[1,25,26]-=-. 2.2 The Unbounded-Error Model of Communication We continue the review started in the Introduction. Readers with background in communication complexity will note that the unbounded-error model is exa... |

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Citation Context ...hreshold circuit of superpolynomial size. Communication complexity has been crucial to the progress on this problem. Using randomized communication complexity, many explicit functions have been found =-=[9, 24, 33, 34]-=- that require depth-2 majority circuits of exponential size. Via the reductions due to Goldman et al. [8], these lower bounds remain valid for the broader class of majority-of-threshold circuits. This... |

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Citation Context ..., Razborov and Sherstov [33] solved a long-standing open problem regarding the comparative power of alternation (the classes �cc 2 and �cc 2 ) and unbounded-error communication, posed by Babai et al. =-=[2]-=-. This paper focuses on symmetric functions, i.e., functions f : {0, 1} n × {0, 1} n → {0, 1} of the form f (x, y) = D(|x ∧ y|) for a given predicate D : {0, 1, . . . , n} → {0, 1}. Here |x ∧ y| stand... |

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Citation Context ... symmetric with ˆφ(S) = 0 for |S| > r. Then there is a polynomial p ∈ Pr with φ(x) = p(x1 + · · · + xn) for all x ∈ {0, 1} n . Minsky and Papert’s observation has seen numerous uses in the literature =-=[1,25,26]-=-. 2.2 The Unbounded-Error Model of Communication We continue the review started in the Introduction. Readers with background in communication complexity will note that the unbounded-error model is exa... |

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Citation Context ...pact on the complexity of certain functions. For example, the well-known disjointness function on n-bit strings has complexity O(log n) in the 9sunbounded-error model and �(n) in the randomized model =-=[11, 29]-=-. Furthermore, explicit functions are known [2,31] with unbounded-error complexity O(log n) that require �( √ n) communication in the randomized model to even achieve advantage 2 −√ n/5 over random gu... |

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Citation Context ...f D, denoted deg(D), is the total number of value changes of D. For example, the familiar predicate PARITY(t) = t mod 2 has degree n, whereas a constant predicate has degree 0. It is not hard to show =-=[2]-=- that deg(D) is the least degree of a real univariate polynomial p such that sgn(p(t)) = (−1) D(t) , t = 0, 1, . . . , n, hence the term degree. Finally, given two predicates D1, D2 : {0, 1, . . . , n... |

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Citation Context ...[3,13,29,31] in randomized communication complexity. Symmetric functions have also contributed to the progress in quantum 3scommunication complexity, starting with the breakthrough result of Razborov =-=[32]-=- and continuing with more recent work, e.g., [17, 38, 39]. Our main result settles the unbounded-error complexity of every symmetric function, to within logarithmic factors. Since the unbounded-error ... |

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Citation Context ...Using randomized communication complexity, many explicit functions have been found [9, 24, 33, 34] that require depth-2 majority circuits of exponential size. Via the reductions due to Goldman et al. =-=[8]-=-, these lower bounds remain valid for the broader class of majority-of-threshold circuits. This solves an important special case of the general problem. The unbounded-error model solves another import... |

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Citation Context ... the complexity of every function a constant, as one can easily verify. By contrast, introducing shared randomness into the randomized model has minimal impact on the complexity of any given function =-=[23]-=-. As one might expect, the weaker success criterion in the unbounded-error model has a drastic impact on the complexity of certain functions. For example, the well-known disjointness function on n-bit... |

82 | An Introduction to the Approximation of Functions - RIVLIN - 1969 |

75 |
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(Show Context)
Citation Context ... symmetric with ˆφ(S) = 0 for |S| > r. Then there is a polynomial p ∈ Pr with φ(x) = p(x1 + · · · + xn) for all x ∈ {0, 1} n . Minsky and Papert’s observation has seen numerous uses in the literature =-=[1,25,26]-=-. 2.2 The Unbounded-Error Model of Communication We continue the review started in the Introduction. Readers with background in communication complexity will note that the unbounded-error model is exa... |

74 |
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(Show Context)
Citation Context ..., the learning problem remains unsolved for such natural concept classes as DNF formulas of polynomial size and intersections of two halfspaces, whereas hardness results and lower bounds are abundant =-=[15, 16, 20, 6, 21, 19]-=-. There is, however, an important case when efficient PAC learning is possible. Let C be a given concept class. For notational convenience, view the functions in C as mappings {0, 1} n → {−1, +1} rath... |

67 |
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Citation Context ...Sn is symmetric. The symmetric functions on {0, 1} n are intimately related to univariate polynomials, as demonstrated by Minsky and Papert’s symmetrization argument: Proposition 2.3 (Minsky & Papert =-=[22]-=-). Let φ : {0, 1} n → R be symmetric with ˆφ(S) = 0 for |S| > r. Then there is a polynomial p ∈ Pr with φ(x) = p(x1 + · · · + xn) for all x ∈ {0, 1} n . Minsky and Papert’s observation has seen numero... |

49 | Spectral Methods for Matrix Rigidity with Applications to Size-Depth Tradeoffs and Communication Complexity
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Citation Context ...sign-preserving perturbations. The sensitivity of rank is an important and difficult subject in complexity theory. For example, much work has focused on the closely related concept of matrix rigidity =-=[12, 21]-=-. On the surface, unbounded-error complexity and sign-rank seem unrelated. In reality, they are equivalent notions! More specifically, let f : X × Y → {0, 1} be a given function. Consider its communic... |

48 | A linear lower bound on the unbounded error probabilistic communication complexity
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(Show Context)
Citation Context ...ix formulation offers more insight. The power of the unbounded-error model arguably makes it the most challenging model in which to prove communication lower bounds. In a breakthrough result, Forster =-=[5]-=- has recently proved the first nontrivial lower bound in the unbounded-error model for an explicit function. (By contrast, hard functions have long been known [3, 19] for all other communication model... |

45 | New results for learning noisy parities and halfspaces
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Citation Context ..., the learning problem remains unsolved for such natural concept classes as DNF formulas of polynomial size and intersections of two halfspaces, whereas hardness results and lower bounds are abundant =-=[15, 16, 20, 6, 21, 19]-=-. There is, however, an important case when efficient PAC learning is possible. Let C be a given concept class. For notational convenience, view the functions in C as mappings {0, 1} n → {−1, +1} rath... |

44 | Quantum and classical strong direct product theorems and optimal time-space tradeoffs, manuscript
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(Show Context)
Citation Context ...ty. Symmetric functions have also contributed to the progress in quantum 3scommunication complexity, starting with the breakthrough result of Razborov [32] and continuing with more recent work, e.g., =-=[17, 38, 39]-=-. Our main result settles the unbounded-error complexity of every symmetric function, to within logarithmic factors. Since the unbounded-error model is the most powerful of all standard models, earlie... |

41 |
Probabilistic communication complexity
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(Show Context)
Citation Context ...random walks on Zn 2 , discrete approximation theory, the Fourier transform on Zn 2 , linearprogramming duality, and matrix analysis.s1 Introduction The unbounded-error model, due to Paturi and Simon =-=[27]-=-, is a rich and elegant model of communication. Fix a function f : X × Y → {0, 1}, where X and Y are some finite sets. Alice receives an input x ∈ X, Bob receives y ∈ Y, and their objective is to comp... |

41 | Separating ac 0 from depth-2 majority circuits
- Sherstov
- 2007
(Show Context)
Citation Context ...hreshold circuit of superpolynomial size. Communication complexity has been crucial to the progress on this problem. Using randomized communication complexity, many explicit functions have been found =-=[9, 24, 33, 34]-=- that require depth-2 majority circuits of exponential size. Via the reductions due to Goldman et al. [8], these lower bounds remain valid for the broader class of majority-of-threshold circuits. This... |

39 | The pattern matrix method for lower bounds on quantum communication
- Sherstov
- 2007
(Show Context)
Citation Context ...tence of a matrix P with large entries that leads to a small spectral norm �P ◦ F�. To exhibit P with such properties, we use pattern matrices. These matrices arose in two earlier works by the author =-=[32, 34]-=-, where they proved useful in obtaining strong lower bounds on communication. Their purpose in this paper is to reduce the search for P to a search for a smooth orthogonalizing distribution for the pr... |

36 | On computation and communication with small bias
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(Show Context)
Citation Context ...mple, the well-known disjointness function on n-bit strings has complexity O(log n) in the 9sunbounded-error model and �(n) in the randomized model [11, 29]. Furthermore, explicit functions are known =-=[2,31]-=- with unbounded-error complexity O(log n) that require �( √ n) communication in the randomized model to even achieve advantage 2 −√ n/5 over random guessing. More generally, the unbounded-error comple... |

34 | Extremal Combinatorics with Applications in Computer Science - Jukna - 2001 |

31 | Cryptographic hardness for learning intersections of halfspaces
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- 2006
(Show Context)
Citation Context |

31 | Fourier Analysis for Probabilistic Communication Complexity
- Raz
- 1995
(Show Context)
Citation Context ...ions). Symmetric functions have been a vehicle of progress in the study of communication complexity. An illustrative example is the DISJOINTNESS function, whose study has led to considerable advances =-=[3,13,29,31]-=- in randomized communication complexity. Symmetric functions have also contributed to the progress in quantum 3scommunication complexity, starting with the breakthrough result of Razborov [32] and con... |

29 | The communication complexity of threshold gates
- Nisan
- 1993
(Show Context)
Citation Context ...hreshold circuit of superpolynomial size. Communication complexity has been crucial to the progress on this problem. Using randomized communication complexity, many explicit functions have been found =-=[9, 24, 33, 34]-=- that require depth-2 majority circuits of exponential size. Via the reductions due to Goldman et al. [8], these lower bounds remain valid for the broader class of majority-of-threshold circuits. This... |

29 | Geometrical realization of set systems and probabilistic communication complexity
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- 1985
(Show Context)
Citation Context ...nication. Despite this motivation, progress in understanding unbounded-error complexity has been slow and difficult. Indeed, we are aware of only a few nontrivial results on this subject. Alon et al. =-=[1]-=- obtained strong lower bounds for random functions. In a breakthrough result, Forster [7] proved the first strong lower bound for an explicit function. Forster’s proof has seen several extensions and ... |

28 | Complexity measures of sign matrices
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(Show Context)
Citation Context ...y recently that the first nontrivial lower bound was proved (Forster 2001) on the unbounded-error complexity of an explicit function. Forster’s proof has since seen several extensions and refinements =-=[6, 7, 20]-=-. We are not aware of any other progress on unbounded-error complexity. In this paper, we determine the unbounded-error complexity of a natural class of functions that was beyond the reach of the exis... |

26 | Quantum communication complexity of block-composed functions
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(Show Context)
Citation Context ...xity. Symmetric functions have also contributed to the progress in quantum communication complexity, starting with the breakthrough result of Razborov [32] and continuing with more recent work, e.g., =-=[17, 37, 39]-=-. Our main result settles the unbounded-error complexity of every symmetric function, to within logarithmic factors. The only symmetric function whose 3unbounded-error complexity was known prior to t... |

22 |
Relations between communication complexity, linear arrangements, and computational complexity
- Forster, Krause, et al.
- 2001
(Show Context)
Citation Context ...nds remain valid for the broader class of majority-of-threshold circuits. This solves an important special case of the general problem. The unbounded-error model solves another important special case =-=[6]-=-: it supplies exponential lower bounds against threshold-of-majority circuits, i.e., circuits with a threshold gate at the top that receives inputs from majority gates. To our knowledge, the unbounded... |

22 | Halfspace matrices
- Sherstov
- 2007
(Show Context)
Citation Context ...mple, the well-known disjointness function on n-bit strings has complexity O(log n) in the 9sunbounded-error model and �(n) in the randomized model [11, 29]. Furthermore, explicit functions are known =-=[2,31]-=- with unbounded-error complexity O(log n) that require �( √ n) communication in the randomized model to even achieve advantage 2 −√ n/5 over random guessing. More generally, the unbounded-error comple... |

21 | The sign-rank of AC 0
- Razborov, Sherstov
- 2008
(Show Context)
Citation Context ...ugh result, Forster [7] proved the first strong lower bound for an explicit function. Forster’s proof has seen several extensions and refinements [8, 9]. Subsequent to our work, Razborov and Sherstov =-=[33]-=- solved a long-standing open problem regarding the comparative power of alternation (the classes �cc 2 and �cc 2 ) and unbounded-error communication, posed by Babai et al. [2]. This paper focuses on s... |

20 |
Learning DNF in time 2
- KLIVANS, SERVEDIO
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(Show Context)
Citation Context ...ctions φ1, . . . , φr : {0, 1} n → R such that every f ∈ C is expressible in the form f (x) ≡ sign(a1φ1(x) + · · · + arφr(x)) for some reals a1, . . . , ar. There is a simple and well-known algorithm =-=[15]-=-, based on linear programming, that PAC learns C in time polynomial in dc(C). To relate this discussion to sign-rank (or equivalently, to unbounded-error complexity), let = [ f (x)] f ∈C, x∈{0,1} n be... |

18 | Quantum Computing and Communication Complexity
- Wolf
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(Show Context)
Citation Context ... ∗ 1/3 ( f ) + log log [|X| + |Y |]� , where Q∗ 1/3 refers to the quantum model with prior entanglement. An identical inequality is clearly valid for the quantum model without prior entanglement. See =-=[3, 19]-=- for rigorous definitions of these various models; our sole intention was to point out that the unbounded-error model is at least as powerful. Unlike other models of communication complexity, the unbo... |

18 | Improved lower bounds on the rigidity of Hadamard matrices
- Kashin, Razborov
- 1998
(Show Context)
Citation Context ...sign-preserving perturbations. The sensitivity of rank is an important and difficult subject in complexity theory. For example, much work has focused on the closely related concept of matrix rigidity =-=[12, 21]-=-. On the surface, unbounded-error complexity and sign-rank seem unrelated. In reality, they are equivalent notions! More specifically, let f : X × Y → {0, 1} be a given function. Consider its communic... |

18 | A lower bound for agnostically learning disjunctions
- Klivans, Sherstov
- 2007
(Show Context)
Citation Context |

17 |
On the smallest possible dimension and the largest possible margin of linear arrangements representing given concept classes
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(Show Context)
Citation Context ...y recently that the first nontrivial lower bound was proved (Forster 2001) on the unbounded-error complexity of an explicit function. Forster’s proof has since seen several extensions and refinements =-=[6, 7, 20]-=-. We are not aware of any other progress on unbounded-error complexity. In this paper, we determine the unbounded-error complexity of a natural class of functions that was beyond the reach of the exis... |

17 | Unconditional lower bounds for learning intersections of halfspaces
- Klivans, Sherstov
(Show Context)
Citation Context |

12 | Improved lower bounds for learning intersections of halfspaces - Klivans, Sherstov - 2006 |

9 |
Bounded-depth formulae over the basis {&, ⊕} and some combinatorial problem
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- 1988
(Show Context)
Citation Context ...er of predicates from D. To this end, we model the probabilistic argument as a random walk on Z n 2 and place a strong upper bound on its mixing time. Our analysis uses a known bound, due to Razborov =-=[30]-=-, on the rate of convergence in terms of the probability of a basis for Zn 2 (see Lemma 3.3). Solution for dense predicates. Using Chebyshev polynomials and the MarkovBernstein inequalities, Paturi [2... |

9 | The pattern matrix method
- Sherstov
(Show Context)
Citation Context ...xity. Symmetric functions have also contributed to the progress in quantum communication complexity, starting with the breakthrough result of Razborov [32] and continuing with more recent work, e.g., =-=[17, 37, 39]-=-. Our main result settles the unbounded-error complexity of every symmetric function, to within logarithmic factors. The only symmetric function whose 3unbounded-error complexity was known prior to t... |

5 | Powering requires threshold depth 3
- Sherstov
(Show Context)
Citation Context |