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Lower bounds in communication complexity and learning theory via analytic methods

Adam R. Klivans, Alexander A. Sherstov

Published Jan 1, 2009• 3 citations
OpenAlex

Abstract

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.

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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.
We develop a novel technique for communication lower bounds, the pattern matrix method. Specifically, fix an arbitrary function f:{0,1}n→{0,1}f: \{0,1\}^n \to\{0,1\}f:{0,1}n→{0,1}, and let AfA_fAf​ be the matrix whose columns are each an application of f to some subset of the variables x1,x2,…,x4n.x_1,x_2,\ldots,x_{4n}.x1​,x2​,…,x4n​. We prove that AfA_fAf​ has bounded-error communication complexity Ω(d),\Omega(d),Ω(d), where d is the approximate degree of f.f.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 AfA_fAf​ in terms of well-studied analytic properties of f,f,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 a breakthrough result, Razborov (2003) gave optimal lower bounds on the communication complexity of every function f of the form f(x,y)=D(|x AND y|) for some D:{0,1,...,n}->{0,1}, in the bounded-error quantum model with and without prior entanglement. This was proved by the multidimensional discrepancy method. We give an entirely different proof of Razborov's result, using the original, one-dimensional discrepancy method. This refutes the commonly held intuition (Razborov 2003) that the original discrepancy method fails for functions such as DISJOINTNESS. More importantly, our communication lower bounds hold for a much broader class of functions for which no methods were available. Namely, fix an arbitrary function f:{0,1}n/4->{0,1} and let A be the Boolean matrix whose columns are each an application of f to some subset of the variables x1,x2,...,xn. We prove that the communication complexity of A in the bounded-error quantum model with and without prior entanglement is Omega(d), where d is the approximate degree of f. From this result, Razborov's lower bounds follow easily. Our result also establishes a large new class of total Boolean functions whose quantum communication complexity (regardless of prior entanglement) is at best polynomially smaller than their classical complexity. Our proof method is a novel combination of two ingredients. The first is a certain equivalence of approximation and orthogonality in Euclidean n-space, which follows by linear-programming duality. The second is a new construction of suitably structured matrices with low spectral norm, the pattern matrices, which we realize using matrix analysis and the Fourier transform over (Z2)n. The method of this paper has recently inspired important progress in multiparty communication complexity.
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 present two results for computational models that allow error probabilities close to 1/2. First, most computational complexity classes have an analogous class in communication complexity. The class PP in fact has two, a version with weakly restricted bias called PPcc, and a version with unrestricted bias called UPPcc. Ever since their introduction by Babai, Frankl, and Simon in 1986, it has been open whether these classes are the same. We show that PPccsubne UPPcc. Our proof combines a query complexity separation due to Beigel with a technique of Razborov that translates the acceptance probability of quantum protocols to polynomials. Second, we study how small the bias of minimal-degree polynomials that sign-represent Boolean functions needs to be. We show that the worst-case bias is at worst double- exponentially small in the sign-degree (which was very recently shown to be optimal by Podolski), while the average- case bias can be made single-exponentially small in the sign-degree (which we show to be close to optimal).
The communication class UPPcc\mathbf{UPP}^{\text{cc}}UPPcc is a communication analog of the Turing Machine complexity class PP\mathbf{PP}PP. It is characterized by a matrix-analytic complexity measure called sign-rank (also called dimension complexity), and is essentially the most powerful communication class against which we know how to prove lower bounds. For a communication problem fff, let f∧ff \wedge ff∧f denote the function that evaluates fff on two disjoint inputs and outputs the AND of the results. We exhibit a communication problem fff with UPP(f)=O(log⁡n)\mathbf{UPP}(f)= O(\log n)UPP(f)=O(logn), and UPP(f∧f)=Θ(log⁡2n)\mathbf{UPP}(f \wedge f) = Θ(\log^2 n)UPP(f∧f)=Θ(log2n). This is the first result showing that UPP\mathbf{UPP}UPP communication complexity can increase by more than a constant factor under intersection. We view this as a first step toward showing that UPPcc\mathbf{UPP}^{\text{cc}}UPPcc, the class of problems with polylogarithmic-cost UPP\mathbf{UPP}UPP communication protocols, is not closed under intersection. Our result shows that the function class consisting of intersections of two majorities on nnn bits has dimension complexity nΩ(log⁡n)n^{Ω(\log n)}nΩ(logn). This matches an upper bound of (Klivans, O'Donnell, and Servedio, FOCS 2002), who used it to give a quasipolynomial time algorithm for PAC learning intersections of polylogarithmically many majorities. Hence, fundamentally new techniques will be needed to learn this class of functions in polynomial time.
Quantum-inspired classical algorithms provide us with a new way to understand the computational power of quantum computers for practically-relevant problems, especially in machine learning. In the past several years, numerous efficient algorithms for various tasks have been found, while an analysis of lower bounds is still missing. Using communication complexity, in this work we propose the first method to study lower bounds for these tasks. We mainly focus on lower bounds for solving linear regressions, supervised clustering, principal component analysis, recommendation systems, and Hamiltonian simulations. For those problems, we prove a quadratic lower bound in terms of the Frobenius norm of the underlying matrix. As quantum algorithms are linear in the Frobenius norm for those problems, our results mean that the quantum-classical separation is at least quadratic. As a generalisation, we extend our method to study lower bounds analysis of quantum query algorithms for matrix-related problems using quantum communication complexity. Some applications are given.