We study a random sampling technique to approximate integrals ∫[0, 1]s f(x) dx by averaging the function at some sampling points. We focus on cases where the integrand is smooth, which is a problem which occurs in statistics. The convergence rate of the approximation error depends on the smoothness of the function f and the sampling technique. For instance, Monte Carlo (MC) sampling yields a convergence of the root mean square error (RMSE) of order N−1/2 (where N is the number of samples) for functions f with finite variance. Randomized QMC (RQMC), a combination of MC and quasi-Monte Carlo (QMC), achieves a RMSE of order N−3/2+ε under the stronger assumption that the integrand has bounded variation. A combination of RQMC with local antithetic sampling achieves a convergence of the RMSE of order N−3/2−1/s+ε (where s ≥ 1 is the dimension) for functions with mixed partial derivatives up to order two. Additional smoothness of the integrand does not improve the rate of convergence of these algorithms in general. On the other hand, it is known that without additional smoothness of the integrand it is not possible to improve the convergence rate. This paper introduces a new RQMC algorithm, for which we prove that it achieves a convergence of the root mean square error (RMSE) of order N−α−1/2+ε provided the integrand satisfies the strong assumption that it has square integrable partial mixed derivatives up to order α > 1 in each variable. Known lower bounds on the RMSE show that this rate of convergence cannot be improved in general for integrands with this smoothness. We provide numerical examples for which the RMSE converges approximately with order N−5/2 and N−7/2, in accordance with the theoretical upper bound.
We study a random sampling technique to approximate integrals ∫[0, 1]s f(x) dx by averaging the function at some sampling points. We focus on cases where the integrand is smooth, which is a problem which occurs in statistics. The convergence rate of the approximation error depends on the smoothness of the function f and the sampling technique. For instance, Monte Carlo (MC) sampling yields a convergence of the root mean square error (RMSE) of order N−1/2 (where N is the number of samples) for functions f with finite variance. Randomized QMC (RQMC), a combination of MC and quasi-Monte Carlo (QMC), achieves a RMSE of order N−3/2+ε under the stronger assumption that the integrand has bounded variation. A combination of RQMC with local antithetic sampling achieves a convergence of the RMSE of order N−3/2−1/s+ε (where s ≥ 1 is the dimension) for functions with mixed partial derivatives up to order two. Additional smoothness of the integrand does not improve the rate of convergence of these algorithms in general. On the other hand, it is known that without additional smoothness of the integrand it is not possible to improve the convergence rate. This paper introduces a new RQMC algorithm, for which we prove that it achieves a convergence of the root mean square error (RMSE) of order N−α−1/2+ε provided the integrand satisfies the strong assumption that it has square integrable partial mixed derivatives up to order α > 1 in each variable. Known lower bounds on the RMSE show that this rate of convergence cannot be improved in general for integrands with this smoothness. We provide numerical examples for which the RMSE converges approximately with order N−5/2 and N−7/2, in accordance with the theoretical upper bound.
This paper studies the convergence rate of randomized quasi--Monte Carlo (RQMC) for discontinuous functions, which are often of infinite variation in the sense of Hardy and Krause. It was previously known that the root mean square error (RMSE) of RQMC is only o(n−1/2) for discontinuous functions. For certain discontinuous functions in d dimensions, we prove that the RMSE of RQMC is O(n−1/2−1/(4d−2)+ϵ) for any ϵ>0 and arbitrary n. If some discontinuity boundaries are parallel to some coordinate axes, the rate can be improved to O(n−1/2−1/(4du−2)+ϵ), where du denotes the so-called irregular dimension, that is, the number of axes which are not parallel to the discontinuity boundaries. Moreover, this paper shows that the RMSE is O(n−1/2−1/(2d)) for certain indicator functions.
We study the numerical approximation of high-dimensional integrals using randomized quasi-Monte Carlo (RQMC) methods, with a focus on scrambled Sobol' sequences. While asymptotically faster than Monte Carlo, classical RQMC suffers from error bounds that grow exponentially in the dimension s, and its root mean squared error (RMSE) convergence rate is generally no better than O(N−3/2). To overcome these limitations, we combine two recent developments: Suzuki's coarse scrambling and the median trick. Coarse scrambling randomizes Sobol' sequences according to their generating base polynomials and significantly reduces the maximal gain coefficient. By appropriately choosing the base polynomials, we show that the coefficient can be made uniformly bounded in s, yielding an O(N−1/2) RMSE for L2 integrands with a dimension-independent constant. The median trick then enables near-optimal convergence for function classes beyond L2: for Lp integrands with p∈(1,2), the median of independent coarsely scrambled estimates achieves an error of O(N−1+1/p) with high probability; for integrands in the Haar wavelet space Hwav,α,s,p,q with αp:=α−(1/p−1/2)+>0, we prove a high-probability error bound of O(N−αp−1/2+ε) for any ε>0; the bound improves to O(N−r−αp−1/2+ε) when the integrand has dominating mixed derivatives of order r≥1 that belong to Hwav,α,s,p,q. The latter two bounds are uniform in s under suitable conditions on the ANOVA components of the integrands. Numerical experiments confirm the predicted convergence rates and demonstrate the robustness of the proposed approach.
Randomized quasi-Monte Carlo (RQMC) methods estimate the expectation of a random variable by the average of n dependent realizations of it. In general, due to the strong dependence, the estimation error may not obey a central limit theorem. Analysis of RQMC methods have so far focused mostly on the convergence rates of asymptotic worst-case error bounds and variance bounds, when n ¿ ¿, but little is known about the limiting distribution of the error. Here we examine this limiting distribution for the special case of a randomly-shifted lattice rule, when the integrand is smooth. We start with simple one-dimensional functions, where we show that the limiting distribution is uniform over a bounded interval if the integrand is non-periodic, and has a square root form over a bounded interval if the integrand is periodic. In higher dimensions, for linear functions, the distribution function of the properly standardized error converges to a spline of degree equal to the dimension.
We study the sample median of independently generated quasi-Monte Carlo estimators based on randomized digital nets and prove it approximates the target integral value at almost the optimal convergence rate for various function spaces. In contrast to previous methods, the algorithm does not require a priori knowledge of underlying function spaces or even an input of pre-designed ( t , m , s ) (t,m,s) -digital nets, and is therefore easier to implement. This study provides further evidence that some types of randomized quasi-Monte Carlo estimators are heavy-tailed when applied to smooth integrands and taking the median can significantly improve the error by filtering out the outliers.
Randomized quasi-Monte Carlo (RQMC) methods estimate the expectation of a random variable by the average of n dependent realizations of it. In general, due to the strong dependence, the estimation error may not obey a central limit theorem. Analysis of RQMC methods have so far focused mostly on the convergence rates of asymptotic worst-case error bounds and variance bounds, when n → ∞, but little is known about the limiting distribution of the error. Here we examine this limiting distribution for the special case of a randomly-shifted lattice rule, when the integrand is smooth. We start with simple one-dimensional functions, where we show that the limiting distribution is uniform over a bounded interval if the integrand is non-periodic, and has a square root form over a bounded interval if the integrand is periodic. In higher dimensions, for linear functions, the distribution function of the properly standardized error converges to a spline of degree equal to the dimension.
Abstract. We study quasi-Monte Carlo (QMC) integration over the multidimensional unit cube in several weighted function spaces with different smoothness classes. We consider approximating the integral of a function by the median of several integral estimates under independent and random choices of the underlying QMC point sets (either linearly scrambled digital nets or infinite-precision polynomial lattice point sets). Even though our approach does not require any information on the smoothness and weights of a target function space as an input, we can prove a probabilistic upper bound on the worst-case error for the respective weighted function space, where the failure probability converges to 0 exponentially fast as the number of estimates increases. Our obtained rates of convergence are nearly optimal for function spaces with finite smoothness, and we can attain a dimension-independent super-polynomial convergence for a class of infinitely differentiable functions. This implies that our median-based QMC rule is universal in the sense that it does not need to be adjusted to the smoothness and the weights of the function spaces and yet exhibits the nearly optimal rate of convergence. Numerical experiments support our theoretical results.