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The geometry of logconcave functions and an O∗(n³) sampling algorithm
"... The class of logconcave functions in Rn is a common generalization of Gaussians and of indicator functions of convex sets. Motivated by the problem of sampling from a logconcave density function, we study their geometry and introduce a technique for “smoothing” them out. This leads to an efficient s ..."
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Cited by 35 (13 self)
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The class of logconcave functions in Rn is a common generalization of Gaussians and of indicator functions of convex sets. Motivated by the problem of sampling from a logconcave density function, we study their geometry and introduce a technique for “smoothing” them out. This leads to an efficient sampling algorithm (by a random walk) with no assumptions on the local smoothness of the density function. After appropriate preprocessing, the algorithm produces a point from approximately the right distribution in time O∗(n^4), and in amortized time O∗(n³) if many sample points are needed (where the asterisk indicates that dependence on the error parameter and factors of log n are not shown).
Efficient algorithms for universal portfolios
 Proceedings of the 41st Annual Symposium on the Foundations of Computer Science
, 2000
"... A constant rebalanced portfolio is an investment strategy that keeps the same distribution of wealth among a set of stocks from day to day. There has been much work on Cover's Universal algorithm, which is competitive with the best constant rebalanced portfolio determined in hindsight (3, 9, 2, 8, 1 ..."
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Cited by 32 (9 self)
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A constant rebalanced portfolio is an investment strategy that keeps the same distribution of wealth among a set of stocks from day to day. There has been much work on Cover's Universal algorithm, which is competitive with the best constant rebalanced portfolio determined in hindsight (3, 9, 2, 8, 16, 4, 5, 6). While this algorithm has good performance guarantees, all known implementations are exponential in the number of stocks, restricting the number of stocks used in experiments (9, 4, 2, 5, 6). We present an efficient implementation of the Universal algorithm that is based on nonuniform random walks that are rapidly mixing (1, 14, 7). This same implementation also works for nonfinancial applications of the Universal algorithm, such as data compression (6) and language modeling (11).
Mathematical foundations of the Markov chain Monte Carlo method
 in Probabilistic Methods for Algorithmic Discrete Mathematics
, 1998
"... 7.2 was jointly undertaken with Vivek Gore, and is published here for the first time. I also thank an anonymous referee for carefully reading and providing helpful comments on a draft of this chapter. 1. Introduction The classical Monte Carlo method is an approach to estimating quantities that a ..."
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Cited by 30 (1 self)
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7.2 was jointly undertaken with Vivek Gore, and is published here for the first time. I also thank an anonymous referee for carefully reading and providing helpful comments on a draft of this chapter. 1. Introduction The classical Monte Carlo method is an approach to estimating quantities that are hard to compute exactly. The quantity z of interest is expressed as the expectation z = ExpZ of a random variable (r.v.) Z for which some efficient sampling procedure is available. By taking the mean of some sufficiently large set of independent samples of Z, one may obtain an approximation to z. For example, suppose S = \Phi (x; y) 2 [0; 1] 2 : p i (x; y) 0; for all i \Psi<F12
On The Complexity Of Computing Mixed Volumes
 SIAM J. Comput
, 1998
"... . This paper gives various (positive and negative) results on the complexity of the problem of computing and approximating mixed volumes of polytopes and more general convex bodies in arbitrary dimension. On the negative side, we present several #Phardness results that focus on the di#erence of com ..."
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Cited by 30 (1 self)
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. This paper gives various (positive and negative) results on the complexity of the problem of computing and approximating mixed volumes of polytopes and more general convex bodies in arbitrary dimension. On the negative side, we present several #Phardness results that focus on the di#erence of computing mixed volumes versus computing the volume of polytopes. We show that computing the volume of zonotopes is #Phard (while each corresponding mixed volume can be computed easily) but also give examples showing that computing mixed volumes is hard even when computing the volume is easy. On the positive side, we derive a randomized algorithm for computing the mixed volumes V ( m 1 z } { K 1 , . . . , K 1 , m 2 z } { K 2 , . . . , K 2 , . . . , ms z } { Ks , . . . , Ks ) of wellpresented convex bodies K 1 , . . . , Ks , where m 1 , . . . , ms # N 0 and m 1 # n  #(n) with #(n) = o( log n log log n ). The algorithm is an interpolation method based on polynomialtime ra...
Random Walks And An O*(n⁵) Volume Algorithm For Convex Bodies
, 1996
"... Given a high dimensional convex body K ` IR n by a separation oracle, we can approximate its volume with relative error ", using O (n⁵) oracle calls. Our algorithm also brings the body into isotropic position. As all previous randomized volume algorithms, we use "rounding" followed by a multiph ..."
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Cited by 28 (4 self)
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Given a high dimensional convex body K ` IR n by a separation oracle, we can approximate its volume with relative error ", using O (n⁵) oracle calls. Our algorithm also brings the body into isotropic position. As all previous randomized volume algorithms, we use "rounding" followed by a multiphase MonteCarlo (product estimator) technique. Both parts rely on sampling (generating random points in K), which is done by random walk. Our algorithm introduces three new ideas: ffl the use of the isotropic position (or at least an approximation of it) for rounding, ffl the separation of global obstructions (diameter) and local obstructions (boundary problems) for fast mixing, and ffl a stepwise interlacing of rounding and sampling.
Geometric random walks: a survey
 Combinatorial and Computational Geometry
, 2005
"... Abstract. The developing theory of geometric random walks is outlined here. Three aspects —general methods for estimating convergence (the “mixing ” rate), isoperimetric inequalities in R n and their intimate connection to random walks, and algorithms for fundamental problems (volume computation and ..."
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Cited by 28 (4 self)
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Abstract. The developing theory of geometric random walks is outlined here. Three aspects —general methods for estimating convergence (the “mixing ” rate), isoperimetric inequalities in R n and their intimate connection to random walks, and algorithms for fundamental problems (volume computation and convex optimization) that are based on sampling by random walks —are discussed. 1.
A mildly exponential time algorithm for approximating the number of solutions to a multidimensional knapsack problem
, 2006
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Fast algorithms for logconcave functions: sampling, rounding, integration and optimization
 Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science
, 2006
"... We prove that the hitandrun random walk is rapidly mixing for an arbitrary logconcave distribution starting from any point in the support. This extends the work of [26], where this was shown for an important special case, and settles the main conjecture formulated there. From this result, we deriv ..."
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Cited by 22 (5 self)
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We prove that the hitandrun random walk is rapidly mixing for an arbitrary logconcave distribution starting from any point in the support. This extends the work of [26], where this was shown for an important special case, and settles the main conjecture formulated there. From this result, we derive asymptotically faster algorithms in the general oracle model for sampling, rounding, integration and maximization of logconcave functions, improving or generalizing the main results of [24, 25, 1] and [16] respectively. The algorithms for integration and optimization both use sampling and are surprisingly similar.
Approximating the volume of unions and intersections of highdimensional geometric objects
, 2008
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Simulated Annealing for Convex Optimization
 Mathematics of Operations Research
, 2004
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