[Paper Review] On dropping the first Sobol' point

Art B. Owen|arXiv (Cornell University)|Aug 18, 2020

Mathematical Approximation and IntegrationMathematics20 citations

TL;DR

This paper demonstrates that skipping the first Sobol' point—commonly done to avoid the origin point (0,...,0)—significantly degrades the accuracy of quasi-Monte Carlo (QMC) and randomized QMC (RQMC) integration. The practice destroys the digital net structure and increases the root mean squared error by a factor proportional to √n, making it a detrimental practice despite its intuitive appeal.

ABSTRACT

Quasi-Monte Carlo (QMC) points are a substitute for plain Monte Carlo (MC) points that greatly improve integration accuracy under mild assumptions on the problem. Because QMC can give errors that are $o(1/n)$ as $n o\infty$, changing even one point can change the estimate by an amount much larger than the error would have been and worsen the convergence rate. As a result, certain practices that fit quite naturally and intuitively with MC points are very detrimental to QMC performance. These include thinning, burn-in, and taking sample sizes such as powers of $10$, other than the ones for which the QMC points were designed. This article looks at the effects of a common practice in which one skips the first point of a Sobol' sequence. The retained points ordinarily fail to be a digital net and when scrambling is applied, skipping over the first point can increase the numerical error by a factor proportional to $\sqrt{n}$ where $n$ is the number of function evaluations used.

Motivation & Objective

• To investigate the impact of skipping the first Sobol’ point on QMC and RQMC integration accuracy.
• To demonstrate that skipping the first point disrupts the digital net structure of Sobol’ sequences.
• To show that such skipping increases the root mean squared error (RMSE) by a factor proportional to √n in scrambled Sobol’ sequences.
• To discourage common practices like burn-in and thinning in QMC, which are standard in MCMC but harmful in QMC.
• To advocate for using scrambled Sobol’ points from the start, rather than discarding the initial point, for optimal convergence and error control.

Proposed method

• Analyzes the digital net structure of Sobol’ sequences and shows that removing the first point breaks this property.
• Uses the root mean squared error (RMSE) as the primary metric to evaluate integration accuracy in RQMC.
• Compares the performance of estimators that include the first point (μ̂_x,1) versus those that skip it (μ̂_x,2).
• Employs theoretical analysis and numerical examples to show that skipping the first point leads to a √n increase in error.
• Considers alternative remedies such as shifting all points by 1/(2n) to avoid the origin without breaking the digital net structure.
• Reviews existing software implementations (e.g., in SciPy, PyTorch, MATLAB) that include skip and leap options, highlighting their risks.

Experimental results

Research questions

• RQ1 Does skipping the first Sobol’ point degrade the accuracy of QMC and RQMC integration?
• RQ2 Why does skipping the first point increase the root mean squared error by a factor proportional to √n in scrambled Sobol’ sequences?
• RQ3 How does skipping the first point affect the digital net structure of Sobol’ sequences?
• RQ4 Are common MCMC practices like burn-in and thinning compatible with QMC methods?
• RQ5 What are safer alternatives to skipping the first Sobol’ point that preserve the digital net structure and integration accuracy?

Key findings

• Skipping the first Sobol’ point destroys the digital net structure, which is essential for the low-discrepancy properties of QMC.
• In randomized QMC (RQMC), skipping the first point increases the root mean squared error (RMSE) by a factor proportional to √n.
• The first point of a Sobol’ sequence is (0,0,…,0), and skipping it is a common but harmful practice, especially when transforming to unbounded distributions.
• Using a sample size that is not a power of two—such as 1000 or 10^6—can be significantly less accurate than using a power of two, especially in digital nets.
• Alternative methods like shifting all points by 1/(2n) preserve the digital net structure and avoid the origin without increasing error.
• Software features like 'skip' and 'leap' in QMC libraries (e.g., MATLAB, SciPy) are dangerous and should be avoided unless carefully justified.