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arXiv:1501.03379

Variance Reduction for QMC in Reproducing Kernel Hilbert Spaces

Chris J. Oates, Mark Girolami

Published Jan 14, 2015• 1 citation
arXivHTMLPDF

Abstract

Quasi-Monte Carlo (QMC) methods are gaining in popularity in the machine learning community due to the increasingly challenging nature of numerical integrals that are routinely encountered in contemporary applications. For integrands that are $\alpha$-times differentiable, an $\alpha$-optimal QMC algorithm converges at a rate $O(N^{-\alpha-\frac{1}{2}+\epsilon})$ for any $\epsilon>0$ and it is known that this rate is best-possible. However in many applications it can happen that either the value of $\alpha$ is unknown or a rate-optimal QMC algorithm is unavailable. How can we perform efficient numerical integration in such circumstances? A direct approach is to employ $\alpha_L$-optimal QMC where the lower bound $\alpha_L \leq \alpha$ is known, but when $\alpha_L < \alpha$ this does not exploit the full power of QMC. In this paper we show that if an upper bound $\alpha \leq \alpha_U$ is also available, then the direct approach can be accelerated by a factor $O(N^{-(\alpha - \alpha_L)/d})$ where $d$ is the dimension of the integral. Such variance reduction methods are likely to become practically important with the increasing adoption of QMC algorithms.

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Abstract This paper is a contemporary review of quasi-Monte Carlo (QMC) methods, that is, equal-weight rules for the approximate evaluation of high-dimensional integrals over the unit cube [0,1] s . It first introduces the by-now standard setting of weighted Hilbert spaces of functions with square-integrable mixed first derivatives, and then indicates alternative settings, such as non-Hilbert spaces, that can sometimes be more suitable. Original contributions include the extension of the fast component-by-component (CBC) construction of lattice rules that achieve the optimal convergence order (a rate of almost 1/ N , where N is the number of points, independently of dimension) to so-called “product and order dependent” (POD) weights, as seen in some recent applications. Although the paper has a strong focus on lattice rules, the function space settings are applicable to all QMC methods. Furthermore, the error analysis and construction of lattice rules can be adapted to polynomial lattice rules from the family of digital nets.

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Quasi-Monte Carlo (QMC) methods are gaining in popularity in the machine learning community due to the increasingly challenging nature of numerical integrals that are routinely encountered in contemporary applications. For integrands that are α\alphaα-times differentiable, an α\alphaα-optimal QMC algorithm converges at a rate O(N−α−12+ϵ)O(N^{-\alpha-\frac{1}{2}+\epsilon})O(N−α−21​+ϵ) for any ϵ>0\epsilon>0ϵ>0 and it is known that this rate is best-possible. However in many applications it can happen that either the value of α\alphaα is unknown or a rate-optimal QMC algorithm is unavailable. How can we perform efficient numerical integration in such circumstances? A direct approach is to employ αL\alpha_LαL​-optimal QMC where the lower bound αL≤α\alpha_L \leq \alphaαL​≤α is known, but when αL<α\alpha_L < \alphaαL​<α this does not exploit the full power of QMC. In this paper we show that if an upper bound α≤αU\alpha \leq \alpha_Uα≤αU​ is also available, then the direct approach can be accelerated by a factor O(N−(α−αL)/d)O(N^{-(\alpha - \alpha_L)/d})O(N−(α−αL​)/d) where ddd is the dimension of the integral. Such variance reduction methods are likely to become practically important with the increasing adoption of QMC algorithms.
This paper is a contemporary review of QMC (‘quasi-Monte Carlo’) methods, that is, equal-weight rules for the approximate evaluation of high-dimensional integrals over the unit cube [0,1] s , where s may be large, or even infinite. After a general introduction, the paper surveys recent developments in lattice methods, digital nets, and related themes. Among those recent developments are methods of construction of both lattices and digital nets, to yield QMC rules that have a prescribed rate of convergence for sufficiently smooth functions, and ideally also guaranteed slow growth (or no growth) of the worst-case error as s increases. A crucial role is played by parameters called ‘weights’, since a careful use of the weight parameters is needed to ensure that the worst-case errors in an appropriately weighted function space are bounded, or grow only slowly, as the dimension s increases. Important tools for the analysis are weighted function spaces, reproducing kernel Hilbert spaces, and discrepancy, all of which are discussed with an appropriate level of detail.
This paper is a contemporary review of quasi-Monte Carlo (QMC) methods, that is, equal-weight rules for the approximate evaluation of high-dimensional integrals over the unit cube [0,1]s[0,1]^s[0,1]s. It first introduces the by-now standard setting of weighted Hilbert spaces of functions with square-integrable mixed first derivatives, and then indicates alternative settings, such as non-Hilbert spaces, that can sometimes be more suitable. Original contributions include the extension of the fast component-by-component (CBC) construction of lattice rules that achieve the optimal convergence order (a rate of almost 1/N1/N1/N, where NNN is the number of points, independently of dimension) to so-called “product and order dependent” (POD) weights, as seen in some recent applications. Although the paper has a strong focus on lattice rules, the function space settings are applicable to all QMC methods. Furthermore, the error analysis and construction of lattice rules can be adapted to polynomial lattice rules from the family of digital nets. doi:10.1017/S1446181112000077
Monte Carlo (MQ) method is a powerful tool to approximate high dimensional integrals. The disadvantage of ordinary MQ is the slow convergence rate by cause of the essential randomness of this method. Computations of this convergence can lead to pure time consuming. Quasi-Monte Carlo (QMC) method yields considerably better results since this method is a deterministic alternative which uses Low-Discrepancy sequences instead of random samples . There are a multiplicity of big open problems in QMC-methods, problems partly arising from applications and partly arising from theory. QMC are developed to integrate over unite cube, where it has much more accuracy than MC for integrands of bounded variation. Integration over more general spaces such as triangles, disks and Cartesian products of such spaces is more challenging for QMC. Nevertheless in real-world applications various problems are defined over such spaces. The aim of this thesis is to provide a survey of a solution of such problems of numerical integration defined over non-cubical spaces. We present QMC and randomised QMC (RQMC) constructions in the triangle with a vanishing discrepancy based on the recent work of Basu 2016. The QMC construction is a version of the Van der Corput-Halton sequences specially made to the unit triangle. The attraction of scrambled net is replication based error estimation for QMC with slightly the same accuracy as QMC, and for smooth enough integrands.
Dimensionally unbounded problems are frequently encountered in practice, such as in simulations of stochastic processes, in particle and light transport problems and in the problems of mathematical finance. This paper considers quasi-Monte Carlo integration algorithms for weighted classes of functions of infinitely many variables, in which the dependence of functions on successive variables is increasingly limited. The dependence is modeled by a sequence of weights. The integrands belong to rather general reproducing kernel Hilbert spaces that can be decomposed as the direct sum of a series of their subspaces, each subspace containing functions of only a finite number of variables. The theory of reproducing kernels is used to derive a quadrature error bound, which is the product of two terms: the generalized discrepancy and the generalized variation. Tractability means that the minimal number of function evaluations needed to reduce the initial integration error by a factor ε \varepsilon is bounded by C ε − p C \varepsilon ^{-p} for some exponent p p and some positive constant C C . The ε \varepsilon -exponent of tractability is defined as the smallest power of ε − 1 \varepsilon ^{-1} in these bounds. It is shown by using Monte Carlo quadrature that the ε \varepsilon -exponent is no greater than 2 for these weighted classes of integrands. Under a somewhat stronger assumption on the weights and for a popular choice of the reproducing kernel it is shown constructively using the Halton sequence that the ε \varepsilon -exponent of tractability is 1, which implies that infinite dimensional integration is no harder than one-dimensional integration.
The Monte Carlo method uses a random sample of size n to approximate the value of an s-dimensional integral and has a convergence rate of O(n−1/2). By using lowdiscrepancy sampling, the quasi-Monte Carlo (QMC) method achieves a convergence rate of O(n−1(log n)), which is asymptotically faster than that of the Monte Carlo method. However, due to the dependence of the QMC error bound on the dimension s, this result doesn’t hold at practical values of n for even relatively small values of s. Still, many empirical studies have shown that QMC can outperform Monte Carlo integration in highdimensional problems at practical values of n. Empirical studies are often conducted by researchers to investigate the behaviour of QMC methods since the QMC error bound suffers from several practical limitations. This paper presents a Matlab program that implements some of the more commonly used numerical tests and provides a simple graphical summary of the results.
Abstract This paper is a contemporary review of quasi-Monte Carlo (QMC) methods, that is, equal-weight rules for the approximate evaluation of high-dimensional integrals over the unit cube [0,1] s . It first introduces the by-now standard setting of weighted Hilbert spaces of functions with square-integrable mixed first derivatives, and then indicates alternative settings, such as non-Hilbert spaces, that can sometimes be more suitable. Original contributions include the extension of the fast component-by-component (CBC) construction of lattice rules that achieve the optimal convergence order (a rate of almost 1/ N , where N is the number of points, independently of dimension) to so-called “product and order dependent” (POD) weights, as seen in some recent applications. Although the paper has a strong focus on lattice rules, the function space settings are applicable to all QMC methods. Furthermore, the error analysis and construction of lattice rules can be adapted to polynomial lattice rules from the family of digital nets.