Super-polynomial accuracy of one dimensional randomized nets using the median of means

Abstract

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 onfor any c < 3 log(2)/π 2 ≈ 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 → ∞.We include results for finite precision estimates and some non-asymptotic comparisons to taking the mean of 2r − 1 independent draws.

Confidence Intervals for Randomized Quasi-Monte Carlo Estimators

“…It is shown for the Sob-DS method, but that skewed pattern appeared for other methods too and not just for small s and k. The upper right entry shows the spike plus outliers distribution mentioned earlier. It also arises for nearly additive MC2 at somewhat higher values of k for each s. The phenomenon was pointed out in [35] in one dimension and in [36] for multiple dimensions which includes some numerical illustrations. We expect coverage length to be quite variable in settings like this because, for example, R = 10 replicates might not include any of the outliers.…”

“…A similar phenomenon of symmetric but non-Gaussian limiting distributions arises in a strategy that recycles physically generated random numbers [32]. Paper [36] show that the RQMC error in Sob-LMS is a randomly weighted sum of the average of some Walsh functions. Those Walsh function averages have a symmetric distribution, but because the random weights are dependent, it does not automatically mean that the RQMC error has a symmetric distribution.…”

“…Of those 81 cases, 80 of them were for Sob-LMS and one was for Sob-NUS. For Sob-MLS we know that the underlying kurtosis of the RQMC points diverges to infinity as n increases from results in [35].…”

Myths of Computer Graphics

“…As an alternative, already superimposing four instances of the random rank-1 lattice sequence in a pixel reliably hides the outliers, while remaining competitive in terms of performance. This favorably relates to removing outliers by a median [9,39,8].…”

“…Hence, instead of searching for better number-theoretic constructions, we may take advantage of the fact that many sets of random parameters are good and single out the bad ones. One such approach is to use the median of a number of randomized quasi-Monte Carlo integrations [9] that inspired investigations using random generator matrices [39] and random generator vectors [8].…”