Super-polynomial accuracy of multidimensional randomized nets using the median-of-means
Super-polynomial accuracy of multidimensional randomized nets using the median-of-means
复制标题
使用均值中位数的多维随机网络的超多项式精度
DOI：
10.48550/arxiv.2208.05078
复制
发表时间：
2022
期刊：
影响因子：
--
通讯作者：
A. Owen
中科院分区：
文献类型：
--
作者：
Z. Pan;A. Owen
我们研究了$ f $超过$ [0,1]^s $的近似整合，该$基于$ 2R-1 $ $的积分估计值，该估计来自于独立的随机$（t，m，s）$ - 基础中的nets $ 2 $。网络由Matousek的随机线性争夺随机分配，并具有数字偏移。如果$ f $超过$ [0,1]^s $，那么任何一个随机网的估算的概率都有大于$ 2^{ - cm^2/s} $ times的错误，根据$ f $对于任何$ c <3 \ log（2）/\ pi^2 \大约0.21 $，IS $ O（1/\ sqrt {m}）$。结果，对于$ n = 2^m $ function评估，这些炒网的分布的中位数为$ o（n^{ - c \ log（n）/s}）$。 $ 2R-1 $独立抽奖的样本中位数也达到了这一速率，只要$ r/m^2 $从零以$ m \至\ infty $限制。我们包括有限精度估计值的结果，以及对$ 2R-1 $独立抽奖的平均值进行一些非反应比较。
We study approximate integration of a function $f$ over $[0,1]^s$ based on taking the median of $2r-1$ integral estimates derived from independently randomized $(t,m,s)$-nets in base $2$. The nets are randomized by Matousek's random linear scramble with a digital shift. If $f$ is analytic over $[0,1]^s$, then the probability that any one randomized net's estimate has an error larger than $2^{-cm^2/s}$ times a quantity depending on $f$ is $O(1/\sqrt{m})$ for any $c<3\log(2)/\pi^2\approx 0.21$. As a result the median of the distribution of these scrambled nets has an error that is $O(n^{-c\log(n)/s})$ for $n=2^m$ function evaluations. The sample median of $2r-1$ independent draws attains this rate too, so long as $r/m^2$ is bounded away from zero as $m\to\infty$. We include results for finite precision estimates and some non-asymptotic comparisons to taking the mean of $2r-1$ independent draws.
DOI：
--
发表时间：
2022
期刊：
影响因子：
--
作者：
Schmid David;Rosset Denis;Buscemi Francesco;Takashi Goda
通讯作者：
Takashi Goda