Recent Advances in Randomized Quasi-Monte Carlo Methods

L’Ecuyer, Pierre; Lemieux, Christiane

doi:10.1007/0-306-48102-2_20

Cited by 178 publications

(163 citation statements)

References 91 publications

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“…After astonishing results in numerical integration (convergence in 1/n instead of 1/ √ n for numerical integration, within logarithmic factors) and successful experiments in random searchs [20], QR points have been used in several works dealing with evolution strategies, during initialization [13,7] or mutation [2,26,25]. Furthermore, "modern" QR sequences using scrambling [14] have been demonstrated to outperform older QR sequences [26,25]. Hence, following [28,25], this paper will only consider Halton sequences with random scrambling, and this Section will only briefly introduce them -please refer to [20,21] for more details and references.…”

Section: Mathematical Backgroundmentioning

confidence: 99%

A Rigorous Runtime Analysis for Quasi-Random Restarts and Decreasing Stepsize

Schoenauer

Teytaud

2012

Lecture Notes in Computer Science

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Abstract. Multi-Modal Optimization (MMO) is ubiquitous in engineering, machine learning and artificial intelligence applications. Many algorithms have been proposed for multimodal optimization, and many of them are based on restart strategies. However, only few works address the issue of initialization in restarts. Furthermore, very few comparisons have been done, between different MMO algorithms, and against simple baseline methods. This paper proposes an analysis of restart strategies, and provides a restart strategy for any local search algorithm for which theoretical guarantees are derived. This restart strategy is to decrease some 'step-size', rather than to increase the population size, and it uses quasi-random initialization, that leads to a rigorous proof of improvement with respect to random restarts or restarts with constant initial step-size. Furthermore, when this strategy encapsulates a (1+1)-ES with 1/5th adaptation rule, the resulting algorithm outperforms state of the art MMO algorithms while being computationally faster.

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Section: Mathematical Backgroundmentioning

confidence: 99%

A Rigorous Runtime Analysis for Quasi-Random Restarts and Decreasing Stepsize

Schoenauer

Teytaud

2012

Lecture Notes in Computer Science

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“…Hlawka (1971) derives another error bound in which V (f ) is replaced by a different notion of variation, called the mean oscillation of f . In fact, a large family of bounds can be derived by adopting different definitions of discrepancy together with corresponding definitions of function variation (Hickernell, 1998;L'Ecuyer and Lemieux, 2002). These bounds can be used to get asymptotic convergence rates but are rarely convenient for practical error assessment.…”

Section: Quasi-monte Carlo (Qmc)mentioning

confidence: 99%

A Randomized Quasi-Monte Carlo Simulation Method for Markov Chains

L’Ecuyer

Lécot

Tuffin

2008

Operations Research

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Abstract. We introduce and study a randomized quasi-Monte Carlo method for estimating the state distribution at each step of a Markov chain. The number of steps in the chain can be random and unbounded. The method simulates n copies of the chain in parallel, using a (d + 1)-dimensional highly-uniform point set of cardinality n, randomized independently at each step, where d is the number of uniform random numbers required at each transition of the Markov chain. This technique is effective in particular to obtain a low-variance unbiased estimator of the expected total cost up to some random stopping time, when state-dependent costs are paid at each step. It is generally more effective when the state space has a natural order related to the cost function.We provide numerical illustrations where the variance reduction with respect to standard Monte Carlo is substantial. The variance can be reduced by factors of several thousands in some cases. We prove bounds on the convergence rate of the worst-case error and variance for special situations. In line with what is typically observed in randomized quasi-Monte Carlo contexts, our empirical results indicate much better convergence than what these bounds guarantee.

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“…Under simple conditions on the randomization (e.g., one must have E[f (V i )] = µ), the sample mean and sample variance of these m averages are unbiased estimators of the exact mean and variance ofȲ n . Further details on this classical RQMC approach can be found in [5,6,8] and other references given there.…”

Section: Markov Chain Modelmentioning

confidence: 99%

“…Here we give the results for (a) a (d + 1)-dimensional Korobov lattice rule with its first coordinate skipped, randomized by a random shift modulo 1 followed by a baker's transformation [1] (denoted ArrayKorobov ) and (b) the first n points of a Sobol sequence randomized by a left (upper triangular) matrix scrambling followed by a random digital shift [6,9] (denoted Array-Sobol ). For Array-Korobov, the multiplier a of the twodimensional Korobov lattice rule was selected so that n is prime, a is a primitive element modulo n (this requirement is actually not needed), and a/n is close to the golden ratio.…”

Section: An M/m/1 Queue With D =mentioning

confidence: 99%

“…It provides unbiased mean and variance estimators. We have theoretical results on the rate of convergence of the variance of the mean estimator (as n → ∞) only for narrow special cases, but our empirical results with a variety of examples indicate that this variance goes down much faster with the proposed method than for standard Monte Carlo (MC) simulation or for randomized QMC (RQMC) methods that use a single τ d-dimensional point to simulate each sample path of the chain (as in [5,6,7], for example).…”

Section: Introductionmentioning

confidence: 99%

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Randomized Quasi-Monte Carlo Simulation of Markov Chains with an Ordered State Space

L’Ecuyer¹,

Lécot²,

Tuffin³

Monte Carlo and Quasi-Monte Carlo Methods 2004

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Summary. We study a randomized quasi-Monte Carlo method for estimating the state distribution at each step of a Markov chain with totally ordered (discrete or continuous) state space. The number of steps in the chain can be random and unbounded. The method can be used in particular to get a low-variance unbiased estimator of the expected total cost up to some random stopping time, when statedependent costs are paid at each step. We provide numerical illustrations where the variance reduction with respect to standard Monte Carlo is substantial.

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Recent Advances in Randomized Quasi-Monte Carlo Methods

Cited by 178 publications

References 91 publications

A Rigorous Runtime Analysis for Quasi-Random Restarts and Decreasing Stepsize

A Rigorous Runtime Analysis for Quasi-Random Restarts and Decreasing Stepsize

A Randomized Quasi-Monte Carlo Simulation Method for Markov Chains

Randomized Quasi-Monte Carlo Simulation of Markov Chains with an Ordered State Space

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