On the distribution of scrambled $(0,m,s)$-nets over unanchored boxes

随机网局部对偶抽样

Local antithetic sampling with scrambled nets

我们考虑的问题，计算近似的积分$I=\int_{[0,1]^d}f(x) dx$。蒙特卡罗（MC）抽样通常会从$n$独立随机函数评估中获得$O(n^{-1/2})$的均方误差（RMSE）。相比之下，使用仔细等间距的评估点的准蒙特卡罗（QMC）采样可以针对任何$\varepsilon>0$获得速率$O(n^{-1+\varepsilon})$，并且随机化QMC（RQMC）可以获得RMSE $O(n^{-3/2+\varepsilon})$，两者都在$f$上的温和条件下。经典的MC方差缩减方法可以适用于QMC。发表的结果结合QMC与重要性抽样和控制变量发现值得改进，但没有改变的错误率。本文对经典的对偶抽样方差缩减方法进行了推广，并将其与RQMC相结合。一个这样的方法被证明带来了适度的改善RMSE率，达到$O(n^{-3/2-1/d+\varepsilon})$的任何$\varepsilon>0$，足够平滑的$f$。

基于任意尺寸的随机数网络的积分方法

On Integration Methods Based on Scrambled Nets of Arbitrary Size

本文研究了使用随机数网络改进积分方法的收敛性，推导了方差界并证明了顺序拟蒙特卡洛方法在任意N下的收敛性。

The nonzero gain coefficients of Sobol's sequences are always powers of two

When a plain Monte Carlo estimate on $n$ samples has variance $σ^2/n$, then scrambled digital nets attain a variance that is $o(1/n)$ as $n\to\infty$. For finite $n$ and an adversarially selected integrand, the variance of a scrambled $(t,m,s)$-net can be at most $Γσ^2/n$ for a maximal gain coefficient $Γ<\infty$. The most widely used digital nets and sequences are those of Sobol'. It was previously known that $Γ\leqslant 2^t3^s$ for Sobol' points as well as Niederreiter-Xing points. In this paper we study nets in base $2$. We show that $Γ\leqslant2^{t+s-1}$ for nets. This bound is a simple, b…

使用中位数-均值方法的随机网络在维度无关收敛率

Dimension-independent convergence rates of randomized nets using median-of-means

该研究通过中位数-均值方法，在不依赖维度的情况下实现了高维积分的更优收敛速率。

置乱 Halton 点的增益系数

Gain coefficients for scrambled Halton points

本文研究置乱 Halton 点在随机化拟蒙特卡洛中的方差增益，证明其增益系数具有 $O(\log d)$ 的显式上界，并给出 $6\leqslant d\leqslant10^6$ 时的具体界。

A strong law of large numbers for scrambled net integration

This article provides a strong law of large numbers for integration on digital nets randomized by a nested uniform scramble. The motivating problem is optimization over some variables of an integral over others, arising in Bayesian optimization. This strong law requires that the integrand have a finite moment of order $p$ for some $p>1$. Previously known results implied a strong law only for Riemann integrable functions. Previous general weak laws of large numbers for scrambled nets require a square integrable integrand. We generalize from $L^2$ to $L^p$ for $p>1$ via the Riesz-Thorin interpol…