Randomized Quasi-Monte Carlo: An Introduction for Practitioners

L’Ecuyer, Pierre

doi:10.1007/978-3-319-91436-7_2

“…Randomized quasi-Monte Carlo (RQMC) sampling is a popular method to randomize deterministic point sets; see [14] for an excellent introduction. Clever constructions of deterministic point sets, so called quasi-Monte Carlo (QMC) sampling can significantly improve the asymptotic order of integration errors when compared to classical Monte Carlo sampling.…”

Section: Setting and Main Questionsmentioning

confidence: 99%

“…where (14) and N ≥ 2 was used. This confirms the general result that equivolume stratification is always strictly better than Monte Carlo sampling.…”

Section: Generalized To Arbitrary D ≥ 2 By Puttingmentioning

confidence: 99%

Discrepancy of stratified samples from partitions of the unit cube

Kiderlen

¹

,

Pausinger

²

2021

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We extend the notion of jittered sampling to arbitrary partitions and study the discrepancy of the related point sets. Let $${\varvec{\Omega }}=(\Omega _1,\ldots ,\Omega _N)$$ Ω = ( Ω 1 , … , Ω N ) be a partition of $$[0,1]^d$$ [ 0 , 1 ] d and let the ith point in $${{\mathcal {P}}}$$ P be chosen uniformly in the ith set of the partition (and stochastically independent of the other points), $$i=1,\ldots ,N$$ i = 1 , … , N . For the study of such sets we introduce the concept of a uniformly distributed triangular array and compare this notion to related notions in the literature. We prove that the expected $${{{\mathcal {L}}}_p}$$ L p -discrepancy, $${{\mathbb {E}}}{{{\mathcal {L}}}_p}({{\mathcal {P}}}_{\varvec{\Omega }})^p$$ E L p ( P Ω ) p , of a point set $${{\mathcal {P}}}_{\varvec{\Omega }}$$ P Ω generated from any equivolume partition $${\varvec{\Omega }}$$ Ω is always strictly smaller than the expected $${{{\mathcal {L}}}_p}$$ L p -discrepancy of a set of N uniform random samples for $$p>1$$ p > 1 . For fixed N we consider classes of stratified samples based on equivolume partitions of the unit cube into convex sets or into sets with a uniform positive lower bound on their reach. It is shown that these classes contain at least one minimizer of the expected $${{{\mathcal {L}}}_p}$$ L p -discrepancy. We illustrate our results with explicit constructions for small N. In addition, we present a family of partitions that seems to improve the expected discrepancy of Monte Carlo sampling by a factor of 2 for every N.

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“…Unlike the previous class, the approaches of this class, since they do not necessarily know the network conditions that users may face, must widely explore the different possible network conditions. Several methods have been proposed to effectively explore the very wide space of possible conditions, such as the quasi-Monte Carlo method [18], the active learning method [19] or the Fourier Amplitude Sensitivity Test (FAST) [20] that we are exploring in this paper.…”

Section: B Related Workmentioning

confidence: 99%

“…Although this technique does not consider the properties of the network model in question here, it allows exploring the space of possible network configurations without missing important points. Indeed, unlike a generation of samples based on the Monte Carlo method [18], this technique allows to cover the space of network configurations efficiently by avoiding repetitions, thanks to the variation of frequencies. The partial variances obtained in this way make it possible to see the contribution of network metrics to the overall variation measured.…”

Section: B Sensitivity Analysismentioning

confidence: 99%

When Deep Learning meets Web Measurements to infer Network Performance

Taibi

¹

,

Hadjadj‐Aoul

²

,

Barakat³

2020

2020 IEEE 17th Annual Consumer Communications &Amp; Networking Conference (CCNC)

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Web browsing remains one of the dominant applications of the internet, so inferring network performance becomes crucial for both users and providers (access and content) so as to be able to identify the root cause of any service degradation. Recent works have proposed several network troubleshooting tools, e.g, NDT, MobiPerf, SpeedTest, Fathom. Yet, these tools are either computationally expensive, less generic or greedy in terms of data consumption. The main purpose of this work is to leverage passive measurements freely available in the browser and machine learning techniques (ML) to infer network performance (e.g., delay, bandwidth and loss rate) without the addition of new measurement overhead. To enable this inference, we propose a framework based on extensive controlled experiments where network configurations are artificially varied and the Web is browsed, then ML is applied to build models that estimate the underlying network performance. In particular, we contrast classical ML techniques (such as random forest) to deep learning models trained using fully connected neural networks and convolutional neural networks (CNN). Results of our experiments show that neural networks have a higher accuracy compared to classical ML approaches. Furthermore, the model accuracy improves considerably using CNN.

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“…A low-discrepancy sequence It is difficult to analyze the accuracy of the approximation by QMC in practice as the points are regular. Therefore, randomized QMC (RQMC) can be used so that every element of the sequence is uniformly distributed over the unit cub but still has a low-discrepancy property [35][36][37]. Figure 1 shows the first 200 samples of the sequences for RQMC sampling, pseudo-random sampling, and the corresponding Gaussian sampling with a Sobol sequence.…”

Section: Sequential Quasi-monte Carlo Algorithmmentioning

confidence: 99%

Variance Reduction of Sequential Monte Carlo Approach for GNSS Phase Bias Estimation

Tian

¹

,

Ge

²

,

Neitzel

³

2020

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Global navigation satellite systems (GNSS) are an important tool for positioning, navigation, and timing (PNT) services. The fast and high-precision GNSS data processing relies on reliable integer ambiguity fixing, whose performance depends on phase bias estimation. However, the mathematic model of GNSS phase bias estimation encounters the rank-deficiency problem, making bias estimation a difficult task. Combining the Monte-Carlo-based methods and GNSS data processing procedure can overcome the problem and provide fast-converging bias estimates. The variance reduction of the estimation algorithm has the potential to improve the accuracy of the estimates and is meaningful for precise and efficient PNT services. In this paper, firstly, we present the difficulty in phase bias estimation and introduce the sequential quasi-Monte Carlo (SQMC) method, then develop the SQMC-based GNSS phase bias estimation algorithm, and investigate the effects of the low-discrepancy sequence on variance reduction. Experiments with practical data show that the low-discrepancy sequence in the algorithm can significantly reduce the standard deviation of the estimates and shorten the convergence time of the filtering.

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Randomized Quasi-Monte Carlo: An Introduction for Practitioners

Cited by 41 publications

References 64 publications

Discrepancy of stratified samples from partitions of the unit cube

Discrepancy of stratified samples from partitions of the unit cube

When Deep Learning meets Web Measurements to infer Network Performance

Variance Reduction of Sequential Monte Carlo Approach for GNSS Phase Bias Estimation

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