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19
The Markov Chain Monte Carlo method: an approach to approximate counting and integration
, 1996
"... In the area of statistical physics, Monte Carlo algorithms based on Markov chain simulation have been in use for many years. The validity of these algorithms depends crucially on the rate of convergence to equilibrium of the Markov chain being simulated. Unfortunately, the classical theory of stocha ..."
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Cited by 234 (13 self)
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In the area of statistical physics, Monte Carlo algorithms based on Markov chain simulation have been in use for many years. The validity of these algorithms depends crucially on the rate of convergence to equilibrium of the Markov chain being simulated. Unfortunately, the classical theory of stochastic processes hardly touches on the sort of nonasymptotic analysis required in this application. As a consequence, it had previously not been possible to make useful, mathematically rigorous statements about the quality of the estimates obtained. Within the last ten years, analytical tools have been devised with the aim of correcting this deficiency. As well as permitting the analysis of Monte Carlo algorithms for classical problems in statistical physics, the introduction of these tools has spurred the development of new approximation algorithms for a wider class of problems in combinatorial enumeration and optimization. The “Markov chain Monte Carlo ” method has been applied to a variety of such problems, and often provides the only known efficient (i.e., polynomial time) solution technique.
An Algorithmic Theory of Lattice Points in Polyhedra
, 1999
"... We discuss topics related to lattice points in rational polyhedra, including efficient enumeration of lattice points, “short” generating functions for lattice points in rational polyhedra, relations to classical and higherdimensional Dedekind sums, complexity of the Presburger arithmetic, efficien ..."
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Cited by 91 (6 self)
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We discuss topics related to lattice points in rational polyhedra, including efficient enumeration of lattice points, “short” generating functions for lattice points in rational polyhedra, relations to classical and higherdimensional Dedekind sums, complexity of the Presburger arithmetic, efficient computations with rational functions, and others. Although the main slant is algorithmic, structural results are discussed, such as relations to the general theory of valuations on polyhedra and connections with the theory of toric varieties. The paper surveys known results and presents some new results and connections.
An Optimal Algorithm for Monte Carlo Estimation
, 1995
"... A typical approach to estimate an unknown quantity is to design an experiment that produces a random variable Z distributed in [0; 1] with E[Z] = , run this experiment independently a number of times and use the average of the outcomes as the estimate. In this paper, we consider the case when no a ..."
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Cited by 53 (4 self)
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A typical approach to estimate an unknown quantity is to design an experiment that produces a random variable Z distributed in [0; 1] with E[Z] = , run this experiment independently a number of times and use the average of the outcomes as the estimate. In this paper, we consider the case when no a priori information about Z is known except that is distributed in [0; 1]. We describe an approximation algorithm AA which, given ffl and ffi, when running independent experiments with respect to any Z, produces an estimate that is within a factor 1 + ffl of with probability at least 1 \Gamma ffi. We prove that the expected number of experiments run by AA (which depends on Z) is optimal to within a constant factor for every Z. An announcement of these results appears in P. Dagum, D. Karp, M. Luby, S. Ross, "An optimal algorithm for MonteCarlo Estimation (extended abstract)", Proceedings of the Thirtysixth IEEE Symposium on Foundations of Computer Science, 1995, pp. 142149 [3]. Section ...
Random Walks on Truncated Cubes and Sampling 01 Knapsack Solutions
 in Proc. 40th IEEE Symp. on Foundations of Computer Science
, 2002
"... We solve an open problem concerning the mixing time of symmetric random walk on the n dimensional cube truncated by a hyperplane, showing that it is polynomial in n. As a consequence, we obtain a fullypolynomial randomized approximation scheme for counting the feasible solutions of a 01 knapsa ..."
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Cited by 49 (1 self)
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We solve an open problem concerning the mixing time of symmetric random walk on the n dimensional cube truncated by a hyperplane, showing that it is polynomial in n. As a consequence, we obtain a fullypolynomial randomized approximation scheme for counting the feasible solutions of a 01 knapsack problem. The results extend to the case of any xed number of hyperplanes.
Markov Chains and Polynomial time Algorithms
, 1994
"... This paper outlines the use of rapidly mixing Markov Chains in randomized polynomial time algorithms to solve approximately certain counting problems. They fall into two classes: combinatorial problems like counting the number of perfect matchings in certain graphs and geometric ones like computing ..."
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Cited by 48 (0 self)
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This paper outlines the use of rapidly mixing Markov Chains in randomized polynomial time algorithms to solve approximately certain counting problems. They fall into two classes: combinatorial problems like counting the number of perfect matchings in certain graphs and geometric ones like computing the volumes of convex sets.
Every linear threshold function has a lowweight approximator
 In Proceedings of the 21st Conference on Computational Complexity (CCC
, 2006
"... 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 ..."
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Cited by 20 (6 self)
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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 zeroone 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.
Sampling Contingency Tables
 Random Structures & Algorithms
, 1995
"... this paper, discuss our work counting 4 \Theta 4 contingency tables. 5 4 Sampling Contingency tables : Reduction to continuous sampling This section reduces the problem of sampling from the discrete set of contingency tables to the problem of sampling with nearuniform density from a contingency po ..."
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Cited by 15 (6 self)
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this paper, discuss our work counting 4 \Theta 4 contingency tables. 5 4 Sampling Contingency tables : Reduction to continuous sampling This section reduces the problem of sampling from the discrete set of contingency tables to the problem of sampling with nearuniform density from a contingency polytope. To this end, we first take a natural basis for the lattice of all integer points in
Efficiently approximating weighted sums with exponentially many terms
 In Proceedings of the Fourteenth Annual Conference on Computational Learning Theory
, 2001
"... Multiplicative weightupdate algorithms such as Winnow and Weighted Majority have been studied extensively due to their online mistake bounds ’ logarithmic dependence on N, the total number of inputs, which allows them to be applied to problems where N is exponential. However, a large N requires te ..."
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Cited by 11 (3 self)
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Multiplicative weightupdate algorithms such as Winnow and Weighted Majority have been studied extensively due to their online mistake bounds ’ logarithmic dependence on N, the total number of inputs, which allows them to be applied to problems where N is exponential. However, a large N requires techniques to efficiently compute the weighted sums of inputs to these algorithms. In special cases, the weighted sum can be exactly computed efficiently, but for numerous problems such an approach seems infeasible. Thus we explore applications of Markov chain Monte Carlo (MCMC) methods to estimate the total weight. Our methods are very general and applicable to any representation of a learning problem for which the inputs to a linear learning algorithm can be represented as states in a completely connected, untruncated Markov chain. We give theoretical worstcase guarantees on our technique and then apply it to two problems: learning DNF formulas using Winnow, and pruning classifier ensembles using Weighted Majority. We then present empirical results on simulated data indicating that in practice, the time complexity is much better than what is implied by our worstcase theoretical analysis.
Sampling lattice points
 In Proceedings of the TwentyNinth Annual ACM Symposium on Theory of Computing
, 1997
"... Abstract When is the volume of a convex polytope in Rn close to the number of lattice points in the polytope? We show that if the polytope contains a ball of radius nplog m, where m is the number of facets, then the volume approximates the number of lattice points to within a constant factor. This g ..."
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Cited by 11 (4 self)
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Abstract When is the volume of a convex polytope in Rn close to the number of lattice points in the polytope? We show that if the polytope contains a ball of radius nplog m, where m is the number of facets, then the volume approximates the number of lattice points to within a constant factor. This general condition is then specialized to derive polynomial time sampling and counting algorithms for various combinatorial problems whose solutions can be viewed as lattice points of convex polytopes. We also show, via tight examples, that our condition is essentially the best possible. 1 Introduction In this paper we consider the problem of counting approximately the number of lattice points in an n\Gamma dimensional polytope of the form P = fx 2 Rn: Ax ^ bg; where A is an m \Theta n matrix of nonnegative reals and b is an m\Gamma vector of nonnegative reals. Letting Zn denote the set of integer points (points with all integer coordinates), we are interested in the problem of estimating jP " Znj given A; b. Closely related to it is the problem of sampling nearly uniformly from the set P " Zn [10].
A bound on the precision required to estimate a boolean perceptron from its average satisfying assignment
 SIAM JOURNAL ON DISCRETE MATHEMATICS
, 2006
"... A boolean perceptron is a linear threshold function over the discrete boolean domain f0; 1g n That is, it maps any binary vector to 0 or 1 depending on whether the vector's components satisfy some linear inequality. In 1961, Chow showed that any boolean perceptron is determined by the average or &q ..."
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Cited by 9 (0 self)
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A boolean perceptron is a linear threshold function over the discrete boolean domain f0; 1g n That is, it maps any binary vector to 0 or 1 depending on whether the vector's components satisfy some linear inequality. In 1961, Chow showed that any boolean perceptron is determined by the average or "center of gravity " of its "true " vectors (those that are mapped to 1), together with the total number of true vectors. Moreover, these quantities distinguish the function from any other boolean function, not just other boolean perceptrons. In this paper we go further, by identifying a lower bound on the Euclidean distance between the average satisfying assignment of a boolean perceptron, and the average satisfying assignment of a boolean function that disagrees with that boolean perceptron on a fraction ffl of the input vectors. The distance between the two means is shown to be at least (ffl=n) O(log(n=ffl) log(1=ffl)) This is motivated by the statistical question of whether an empirical estimate of this average allows us to recover a good approximation to the perceptron. Our result provides a mildly superpolynomial upper bound on the growth rate of the sample size required to learn boolean perceptrons in the "restricted focus of attention " setting. In the process we also find some interesting geometrical properties of the vertices of the unit hypercube.