We prove that a single threshold gate with arbitrary weights can be simulated by an explicit polynomial-size depth 2 majority circuit. In general we show that a depth d threshold circuit can be simulated uniformly by a majority circuit of depth d + 1. Goldmann, Hastad, and Razborov showed in [10] that a non-uniform simulation exists. Our construction answers two open questions posed in [10]: we give an explicit construction whereas [10] uses a randomized existence argument, and we show that such a simulation is possible even if the depth d grows with the number of variables n (the simulation in [10] gives polynomial-size circuits only when d is constant). 1 A preliminary version of this paper appeared in Proc. 25th ACM STOC (1993), pp. 551--560. 2 Laboratory for Computer Science, MIT, Cambridge MA 02139, Email: migo@theory.lcs.mit.edu. This author 's work was done at Royal Institute of Technology in Stockholm, and while visiting the University of Bonn 3 Department of Com...

. We survey some of the central results in the complexity theory of discrete neural networks, with pointers to the literature. Our main emphasis is on the computational power of various acyclic and cyclic network models, but we also discuss briefly the complexity aspects of synthesizing networks from examples of their behavior. CR Classification: F.1.1 [Computation by Abstract Devices]: Models of Computation---neural networks, circuits; F.1.3 [Computation by Abstract Devices ]: Complexity Classes---complexity hierarchies Key words: Neural networks, computational complexity, threshold circuits, associative memory 1. Introduction The currently again very active field of computation by "neural" networks has opened up a wealth of fascinating research topics in the computational complexity analysis of the models considered. While much of the general appeal of the field stems not so much from new computational possibilities, but from the possibility of "learning", or synthesizing networks...

In this paper we analyze the PAC learning abilities of several simple iterative algorithms for learning linear threshold functions, obtaining both positive and negative results. We show that Littlestone’s Winnow algorithm is not an efficient PAC learning algorithm for the class of positive linear threshold functions. We also prove that the Perceptron algorithm cannot efficiently learn the unrestricted class of linear threshold functions even under the uniform distribution on boolean examples. However, we show that the Perceptron algorithm can efficiently PAC learn the class of nested functions (a concept class known to be hard for Perceptron under arbitrary distributions) under the uniform distribution on boolean examples. Finally, we give a very simple Perceptron-like algorithm for learning origin-centered halfspaces under the uniform distribution on the unit sphere in R^n. Unlike the Perceptron algorithm, which cannot learn in the presence of classification noise, the new algorithm can learn in the presence of monotonic noise (a generalization of classification noise). The new algorithm is significantly faster than previous algorithms in both the classification and monotonic noise settings.

A halfspace matrix is a Boolean 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 demonstrate the potential of halfspace matrices as tools to answer nontrivial open questions. 1. (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 [25]. 2. (Complexity of sign matrices) We construct a matrix A ∈ {−1,1} N×NlogN with dimension complexity logN but margin complexity Ω(N 1/4 / √ logN). This gap is an exponential improvement over previous work. As an application to circuit complexity, we prove an Ω(2n/4 /(d √ n)) circuit lower bound for computing halfspaces by a majority of an arbitrary set of d gates. This complements a result of Goldmann, H˚astad, and Razborov [15]. In addition, we prove new results on the complexity measures of sign matrices, complementing recent work by Linial et al. [27–29]. 3. (Learning theory) We give a short and simple proof that the statistical-query (SQ) dimension of halfspaces in n dimensions is less than 2(n + 1) 2 under all distributions (with n + 1 being a trivial lower bound). This improves on the n O(1) estimate from the fundamental paper of Blum et al. [5]. Finally, we motivate our learning-theoretic result for the complexity community by showing that SQ dimension estimates for natural classes of Boolean functions can resolve major open problems in complexity theory. Specifically, we show that an exp(2 (logn)o(1) ) upper bound on the SQ dimension of AC 0 would imply an explicit language in PSPACE cc \ PH cc. 1

Given any linear threshold function f on n Boolean variables, we construct a linear threshold function g which disagrees with f on at most an ɛ fraction of inputs and has integer weights each of magnitude at most √ n · 2 Õ(1/ɛ2). We show that the construction is optimal in terms of its dependence on n by proving a lower bound of Ω ( √ n) on the weights required to approximate a particular linear threshold function. We give two applications. The first is a deterministic algorithm for approximately counting the fraction of satisfying assignments to an instance of the zero-one knapsack problem to within an additive ±ɛ. The algorithm runs in time polynomial in n (but exponential in 1/ɛ 2). In our second application, we show that any linear threshold function f is specified to within error ɛ by estimates of its Chow parameters (degree 0 and 1 Fourier coefficients) which are accurate to within an additive ±1/(n · 2 Õ(1/ɛ2)). This is the first such accuracy bound which is inverse polynomial in n (previous work of Goldberg [12] gave a 1/quasipoly(n) bound), and gives the first polynomial bound (in terms of n) on the number of examples required for learning linear threshold functions in the “restricted focus of attention ” framework.

. We survey some of the central results in the complexity theory of discrete neural networks, with pointers to the literature. 1 Introduction The recently revived field of computation by "neural" networks provides the complexity theorist with a wealth of fascinating research topics. While much of the general appeal of the field stems not so much from new computational possibilities, but from the possibility of "learning", or synthesizing networks directly from examples of their desired input-output behavior, it is nevertheless important to pay attention also to the complexity issues: firstly, what kinds of functions are computable by networks of a given type and size, and secondly, what is the complexity of the synthesis problems considered. In fact, inattention to these issues was a significant factor in the demise of the first stage of neural networks research in the late 60's, under the criticism of Minsky and Papert [51]. The intent of this paper is to survey some of the centra...

We show that any distribution on {−1, +1} n that is k-wise independent fools any halfspace (a.k.a. linear threshold function) h: {−1, +1} n → {−1, +1}, i.e., any function of the form h(x) = sign ( ∑n i=1 wixi − θ) where the w1,..., wn, θ are arbitrary real numbers, with error ɛ for k = O(ɛ−2 log 2 (1/ɛ)). Our result is tight up to log(1/ɛ) factors. Using standard constructions of k-wise independent distributions, we obtain the first explicit pseudorandom generators G: {−1, +1} s → {−1, +1} n that fool halfspaces. Specifically, we fool halfspaces with error ɛ and seed length s = k · log n = O(log n · ɛ−2 log 2 (1/ɛ)). Our approach combines classical tools from real approximation theory with structural results on halfspaces by Servedio (Comput. Complexity 2007).

We introduce a new Boolean computing element related to the Boolean version of a neural element. Instead of the sign function in the Boolean neural element (also known as an LT element), it computes an arbitrary (with polynomialy many transitions) Boolean function of the weighted sum of its inputs. We call the new computing element an LTM element, which stands for Linear Threshold with Multiple transitions. The paper consists of the following main contributions related to our study of LTM circuits: (i) the characterization of the computing power of LTM relative to LT circuits, (ii) a proof that the area of the VLSI layout is reduced from O(n 2 ) in LT circuits to O(n) in LTM circuits, for n inputs symmetric Boolean functions, and (iii) the creation of efficient designs of LTM circuits for the addition of a multiple number of integers and the product of two integers. In particular, we show how to compute the addition of m integers with a single layer of LTM elements. Category : The...

Perceptron-like learning rules are known to require exponentially many correction steps in order to identify Boolean threshold functions exactly. We introduce criteria that are weaker than exact identification and investigate whether learning becomes significantly faster if exact identification is replaced by one of these criteria: PAC identification, order identification, and sign identification. PAC identification is based on the learning paradigm introduced by Valiant and known to be easier than exact identification. Order identification uses the fact that each threshold function induces an ordering relation on the input variables which can be represented by weights of linear size. Sign identification is based on a property of threshold functions known as unateness and requires only weights of constant size. We show that Perceptron-like learning rules cannot satisfy these criteria when the number of correction steps is to be bounded by a polynomial. We also present an exponential lo...

We show that any monotone linear threshold function on n Boolean variables can be approximated to within any constant accuracy by a monotone Boolean formula of poly(n) size. 1