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/log5 n) and Θ(k logn), 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 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.
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/log5 n) and Θ(k logn), 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 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.
We develop a novel and powerful technique for communication lower bounds, the pattern matrix method. Specifically, fix an arbitrary function f:{0,1}^n->{0,1} and let A_f be the matrix whose columns are each an application of f to some subset of the variables x_1,x_2,...,x_{4n}. We prove that A_f has bounded-error communication complexity Omega(d), where d is the approximate degree of f. This result remains valid in the quantum model, regardless of prior entanglement. In particular, it gives a new and simple proof of Razborov's breakthrough quantum lower bounds for disjointness and other symmetric predicates. We further characterize the discrepancy, approximate rank, and approximate trace norm of A_f in terms of well-studied analytic properties of f, broadly generalizing several recent results on small-bias communication and agnostic learning. The method of this paper has recently enabled important progress in multiparty communication complexity.
We develop a novel technique for communication lower bounds, the pattern matrix method. Specifically, fix an arbitrary function f:{0,1}n→{0,1}, and let Af be the matrix whose columns are each an application of f to some subset of the variables x1,x2,…,x4n. We prove that Af has bounded-error communication complexity Ω(d), where d is the approximate degree of f. This result remains valid in the quantum model, regardless of prior entanglement. In particular, it gives a new and simple proof of Razborov's breakthrough quantum lower bounds for disjointness and other symmetric predicates. We further characterize the discrepancy, approximate rank, and approximate trace norm of Af in terms of well-studied analytic properties of f, broadly generalizing several recent results on small-bias communication and agnostic learning. The method of this paper has also enabled important progress in multiparty communication complexity.
In this paper, we focus on the quantum communication complexity of functions of the form f∘G=f(G(X1,Y1),…,G(Xn,Yn)) where $f: \Bset^n \to \Bset$ is a symmetric function, $G: \Bset^j \times \Bset^k \to \Bset$ is any function and Alice (resp. Bob) is given (Xi)i∈[n] (resp. (Yi)i∈[n]). Recently, Chakraborty et al. [STACS 2022] showed that the quantum communication complexity of f∘G is $O(Q(f)\QCCEX(G))$ when the parties are allowed to use shared entanglement, where Q(f) is the query complexity of f and $\QCCEX(G)$ is the exact communication complexity of G. }{In this paper, we first show that the same statement holds \emph{without both shared entanglement and shared randomness}, which generalizes their result. Based on the improved result, we next show tight upper bounds on f∘AND2 for any symmetric function f (where $\textrm{AND}_2 : \Bset \times \Bset \to \Bset$ denotes the 2-bit AND function) in both models: with shared entanglement and without shared entanglement. This matches the well-known lower bound by Razborov~[Izv. Math. 67(1) 145, 2003] when shared entanglement is allowed and improves Razborov's bound when shared entanglement is not allowed.
We use a technique of “lifting” functions introduced by Krause and Pudlak [Theor. Comput. Sci., 1997], to amplify degree-hardness measures of a function to corresponding monomial-hardness properties of the lifted function. We then show that any symmetric function F projects onto a “lift” of another suitable symmetric function f . These two key results enable us to prove several results on the complexity of symmetric functions in various models, as given below: 1. We provide a characterization of the approximate spectral norm of symmetric functions in terms of the spectrum of the underlying predicate, affirming a conjecture of Ada et al. [APPROX-RANDOM, 2012] which has several consequences. 2. We characterize symmetric functions computable by quasi-polynomial sized Threshold of Parity circuits. 3. We show that the approximate spectral norm of a symmetric function f characterizes the (quantum and classical) bounded error communication complexity of f o XOR. 4. Finally, we characterize the weakly-unbounded error communication complexity of symmetric XOR functions, resolving a weak form of a conjecture by Shi and Zhang [Quantum Information & Computation, 2009]
A central goal of theoretical computer science is to characterize the limits of efficient computation in a variety of models. We pursue this research objective in the contexts of communication complexity and computational learning theory. In the former case, one seeks to understand which distributed computations require a significant amount of communication among the parties involved. In the latter case, one aims to rigorously explain why computers cannot master some prediction tasks or learn from past experience. While communication and learning may seem to have little in common, they turn out to be closely related, and much insight into both can be gained by studying them jointly. Such is the approach pursued in this thesis. We answer several fundamental questions in communication complexity and learning theory and in so doing discover new relations between the two topics. A consistent theme in our work is the use of analytic methods to solve the problems at hand, such as approximation theory, Fourier analysis, matrix analysis, and duality. (1) We contribute a novel technique, the pattern matrix method , for proving lower bounds on communication. Using our method, we solve an open problem due to Krause and Pudlak (1997) on the comparative power of two well-studied circuit classes: majority circuits and constant-depth AND/OR/NOT circuits. Next, we prove that the pattern matrix method applies not only to classical communication but also to the more powerful quantum model. In particular, we contribute lower bounds for a new class of quantum communication problems, broadly subsuming the celebrated work by Razborov (2002) who used different techniques. In addition, our method has enabled considerable progress by a number of researchers in the area of multiparty communication. (2) Second, we study unbounded-error communication, a natural model with applications to matrix analysis, circuit complexity, and learning. We obtain essentially optimal lower bounds for all symmetric functions, giving the first strong results for unbounded-error communication in years. Next, we resolve a longstanding open problem due to Babai, Frankl, and Simon (1986) on the comparative power of unbounded-error communication and alternation, showing that Scc2n UPPCC: The latter result also yields an unconditional, exponential lower bound for learning DNF formulas by a large class of algorithms, which explains why this central problem in computational learning theory remains open after more than 20 years of research. (3) We establish the computational intractability of learning intersections of halfspaces, a major unresolved challenge in computational learning theory. Specifically, we obtain the first exponential, near-optimal lower bounds for the learning complexity of this problem in Kearns’ statistical query model, Valiant’s PAC model (under standard cryptographic assumptions), and various analytic models. We also prove that the intersection of even two half-spaces on {0, 1}n cannot be sign-represented by a polynomial of degree less than Qpnp , which is an exponential improvement on previous lower bounds and solves an open problem due to Klivans (2002). (4) We fully determine the relations and gaps among three key complexity measures of a communication problem: product discrepancy, sign-rank, and discrepancy. As an application, we solve an open problem due to Kushilevitz and Nisan (1997) on distributional complexity under product versus nonproduct distributions, as well as separate the communication classes PPCC and UPP CC due to Babai, Frankl, and Simon (1986). We give interpretations of ourresults in purely learning-theoretic terms.
In this article we establish new bounds on the quantum communication complexity of distributed problems. Specifically, we consider the amount of communication that is required to transform a bipartite state into another, typically more entangled, state. We obtain lower bounds in this setting by studying the Renyi entropy of the marginal density matrices of the distributed system. The communication bounds on quantum state transformations also imply lower bounds for the model of communication complexity where the task consists of the the distributed evaluation of a function f(x,y). Our approach encapsulates several known lower bound methods that use the log-rank or the von Neumann entropy of the density matrices involved. The technique is also effective for proving lower bounds on problems involving a promise or for which the "hard" distributions of inputs are correlated. As examples, we show how to prove a nearly tight bound on the bounded-error quantum communication complexity of the inner product function in the presence of unlimited amounts of EPR-type entanglement and a similarly strong bound on the complexity of the shifted quadratic character problem.