We survey some of the recent developments on quasi-Monte Carlo (QMC) methods, which, in their basic form, are a deterministic counterpart to the Monte Carlo (MC) method. Our main focus is the applicability of these methods to practical problems that involve the estimation of a high-dimensional integral. We review several QMC constructions and dierent randomizations that have been proposed to provide unbiased estimators and for error estimation. Randomizing QMC methods allows us to view them as variance reduction techniques. New and old results on this topic are used to explain how these methods can improve over the MC method in practice. We also discuss how this methodology can be coupled with clever transformations of the integrand in order to reduce the variance further. Additional topics included in this survey are the description of gures of merit used to measure the quality of the constructions underlying these methods, and other related techniques for multidimensional integration. 1 2 1.

Integration lattices are one of the main types of low discrepancy sets used in quasi-Monte Carlo methods. However, they have the disadvantage of being of fixed size. This article describes the construction of an infinite sequence of points, the first b m of which form a lattice for any non-negative integer m. Thus, if the quadrature error using an initial lattice is too large, the lattice can be extended without discarding the original points. Generating vectors for extensible lattices are found by minimizing a loss function based on some measure of discrepancy or nonuniformity of the lattice. The spectral test used for finding pseudo-random number generators is one important example of such a discrepancy. The performance of the extensible lattices proposed here is compared to that of other methods for some practical quadrature problems.

Abstract. We describe a technique for comparing distributions without the need for density estimation as an intermediate step. Our approach relies on mapping the distributions into a reproducing kernel Hilbert space. Applications of this technique can be found in two-sample tests, which are used for determining whether two sets of observations arise from the same distribution, covariate shift correction, local learning, measures of independence, and density estimation. Kernel methods are widely used in supervised learning [1, 2, 3, 4], however they are much less established in the areas of testing, estimation, and analysis of probability distributions, where information theoretic approaches [5, 6] have long been dominant. Recent examples include [7] in the context of construction of graphical models, [8] in the context of feature extraction, and [9] in the context of independent component analysis. These methods have by and large a common issue: to compute quantities such as the mutual information, entropy, or Kullback-Leibler divergence, we require sophisticated space partitioning and/or

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The general problem of reconstructing an unknown function from a finite collection of samples is considered, in case the position of each input vector in the training set is not fixed beforehand, but is part of the learning process. In particular, the consistency of the Empirical Risk Minimization (ERM) principle is analyzed, when the points in the input space are generated by employing a purely deterministic algorithm (deterministic learning). When the

This paper analyzes some schemes for reducing the computational burden of digital scrambling. Some such schemes have been shown not to affect the mean squared L² discrepancy. This paper shows that some discrepancy-preserving alternative scrambles can change the variance in scrambled net quadrature. Even the rate of convergence can be adversely affected by alternative scramblings. Finally, some alternatives reduce the computational burden and can also be shown to improve the rate of convergence for the variance, at least in dimension 1.

In this paper a comparison is made between four frequently encountered resampling algorithms for particle filters. A theoretical framework is introduced to be able to understand and explain the differences between the resampling algorithms. This facilitates a comparison of the algorithms with respect to their resampling quality and computational complexity. Using extensive Monte Carlo simulations the theoretical results are verified. It is found that systematic resampling is favourable, both in terms of resampling quality and computational complexity. 1.

The general problem of reconstructing an unknown function from a finite collection of samples is considered, in case the position of each input vector in the training set is not fixed beforehand, but is part of the learning process. In particular, the consistency of the Empirical Risk Minimization (ERM) principle is analyzed, when the points in the input space are generated by employing a purely deterministic algorithm (deterministic learning). When the output generation is not subject to noise, classical number-theoretic results, involving discrepancy and variation, allow to establish a su#cient condition for the consistency of the ERM principle. In addition, the adoption of low-discrepancy sequences permits to achieve a learning rate of O(1/L), being L the size of the training set. An extension to the noisy case is discussed.

In this article we investigate Quasi-Monte Carlo methods for multidimensional improper integrals with respect to a measure other than the uniform distribution. Additionally, the integrand is allowed to be unbounded at the lower boundary of the integration domain. We establish convergence of the Quasi-Monte Carlo estimator to the value of the improper integral under conditions involving both the integrand and the sequence used. Furthermore, we suggest a modification of an approach proposed by Hlawka and Mück for the creation of low-discrepancy sequences with regard to a given density, which are suited for singular integrands. Key words: Quasi-Monte Carlo integration, weighted integration, non-uniformly distributed low-discrepancy sequences This paper is devoted to Quasi-Monte Carlo (QMC) techniques for weighted integration problems of the form

There are many examples of option contracts in which the payoff depends on several stochastic variables. These options often can be priced by the valuation of multidimensional integrals. Quasi– Monte Carlo methods are an effective numerical tool for this task. We show that, when the dimensions of the problem are small (say, less than 10), a special type of quasi–Monte Carlo known as the lattice rule method is very efficient. We provide an overview of lattice rules, and we show how to implement this method and demonstrate its efficiency relative to standard Monte Carlo and classical quasi–Monte Carlo. To maximize the efficiency gains, we show how to exploit the regularity of the integrand through a periodization technique. We demonstrate the superior efficiency of the method both in the estimation of prices as well as in the estimation of partial derivatives of these prices (the so-called Greeks). In particular this approach provides good estimates of the second derivative (the gamma) of the price in contrast to traditional Monte Carlo methods, which normally yield poor estimates. Although this method is not new, it appears that the advantages of lattice rules in the context of insurance and finance applications have not been fully appreciated in the literature. 1.