The communication class UPP^{cc} is a communication analog of the Turing Machine complexity class 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 f, let f wedge f denote the function that evaluates f on two disjoint inputs and outputs the AND of the results. We exhibit a communication problem f with UPP^{cc}(f)= O(log n), and UPP^{cc}(f wedge f) = Theta(log^2 n). This is the first result showing that UPP communication complexity can increase by more than a constant factor under intersection. We view this as a first step toward showing that UPP^{cc}, the class of problems with polylogarithmic-cost UPP communication protocols, is not closed under intersection. Our result shows that the function class consisting of intersections of two majorities on n bits has dimension complexity n^{Omega(log n)}. 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.
The communication class UPP^{cc} is a communication analog of the Turing Machine complexity class 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 f, let f wedge f denote the function that evaluates f on two disjoint inputs and outputs the AND of the results. We exhibit a communication problem f with UPP^{cc}(f)= O(log n), and UPP^{cc}(f wedge f) = Theta(log^2 n). This is the first result showing that UPP communication complexity can increase by more than a constant factor under intersection. We view this as a first step toward showing that UPP^{cc}, the class of problems with polylogarithmic-cost UPP communication protocols, is not closed under intersection. Our result shows that the function class consisting of intersections of two majorities on n bits has dimension complexity n^{Omega(log n)}. 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.
The communication class UPP cc is a communication analog of the Turing Machine complexity class 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 f , let f ∧ f denote the function that evaluates f on two disjoint inputs and outputs the AND of the results. We exhibit a communication problem f with UPP cc ( f ) = O (log n ), and UPP cc ( f ∧ f ) = Θ (log 2 n ). This is the first result showing that UPP communication complexity can increase by more than a constant factor under intersection. We view this as a first step toward showing that UPP cc , the class of problems with polylogarithmic-cost UPP communication protocols, is not closed under intersection. Our result shows that the function class consisting of intersections of two majorities on n bits has dimension complexity n Omega Ω(log n ) . 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.
The communication class UPPcc is a communication analog of the Turing Machine complexity class 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 f, let f∧f denote the function that evaluates f on two disjoint inputs and outputs the AND of the results. We exhibit a communication problem f with UPP(f)=O(logn), and UPP(f∧f)=Θ(log2n). This is the first result showing that UPP communication complexity can increase by more than a constant factor under intersection. We view this as a first step toward showing that UPPcc, the class of problems with polylogarithmic-cost UPP communication protocols, is not closed under intersection. Our result shows that the function class consisting of intersections of two majorities on n bits has dimension complexity 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.
UPPcc is Not Closed Under Intersection
The sign rank of a ±1-valued matrix A is the minimum rank of a real matrix that agrees with A entrywise in sign. For every sufficiently large d, we give an explicit matrix A with 2Θ(d2) rows and columns and sign rank Θ(d) such that A∧A has sign rank 2Θ(d). Here, A∧A is the communication matrix of the problem obtained by evaluating A on disjoint inputs and taking the conjunction of the results. As the logarithm of sign rank characterizes unbounded-error communication complexity, this implies that the communication class UPPcc is not closed under intersection. The starting point for our lower bound is Forster's method, which relates the sign rank of a matrix to the average margin of representations of its rows and columns by points and halfspaces. We study a continuous analog for conjunctions of inner-product sign tests on the unit sphere. Rotational symmetry connects this problem to polynomial sign representations, yielding an exponentially small upper bound on the average margin. This bound holds even when sign constraints are relaxed near the sign tests' decision boundary, allowing us to transfer it to explicit finite sign matrices arising from intersections of integer halfspaces.
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 study set-disjointness in a generalized model of randomized two-party communication where the probability of acceptance must be at least α(n) on yes-inputs and at most β(n) on no-inputs, for some functions α(n) > β(n). Our main result is a complete characterization of the private-coin communication complexity of set-disjointness for all functions α and β, and a near-complete characterization for public-coin protocols. In particular, we obtain a simple proof of a theorem of Braverman and Moitra (STOC 2013), who studied the case where α = 1/2+ (n) and β = 1/2− (n). The following contributions play a crucial role in our characterization and are interesting in their own right. (1) We introduce two communication analogues of the classical complexity class that captures small bounded-error computations: we define a “restricted” class SBP (which lies between MA and AM) and an “unrestricted” class USBP. The distinction between them is analogous to the distinction between the well-known communication classes PP and UPP. (2) We show that the SBP communication complexity is precisely captured by the classical corruption lower bound method. This sharpens a theorem of Klauck (CCC 2003). (3) We use information complexity arguments to prove a linear lower bound on the USBP complexity of set-disjointness.
We study set-disjointness in a generalized model of randomized two-party communication where the probability of acceptance must be at least alpha(n) on yes-inputs and at most beta(n) on no-inputs, for some functions alpha(n)>beta(n). Our main result is a complete characterization of the private-coin communication complexity of set-disjointness for all functions alpha and beta, and a near-complete characterization for public-coin protocols. In particular, we obtain a simple proof of a theorem of Braverman and Moitra (STOC 2013), who studied the case where alpha=1/2+epsilon(n) and beta=1/2-epsilon(n). The following contributions play a crucial role in our characterization and are interesting in their own right. (1) We introduce two communication analogues of the classical complexity class that captures small bounded-error computations: we define a "restricted" class SBP (which lies between MA and AM) and an "unrestricted" class USBP. The distinction between them is analogous to the distinction between the well-known communication classes PP and UPP. (2) We show that the SBP communication complexity is precisely captured by the classical corruption lower bound method. This sharpens a theorem of Klauck (CCC 2003). (3) We use information complexity arguments to prove a linear lower bound on the USBP complexity of set-disjointness.