## dropping a point

“A discussion about whether to drop the initial point came up in the plenary tutorial of Fred Hickernell at MCQMC 2020 about QMCPy software for QMC. The issue has been discussed by the pytorch community , and the scipy community, which are both incorporating QMC methods.”

**A**rt Owen recently arXived a paper entitled On dropping the first Sobol’ point in which he examines the impact of a common practice consisting in skipping the first point of a Sobol’ sequence when using quasi-Monte Carlo. By analogy with the burn-in practice for MCMC that aims at eliminating the biais due to the choice of the starting value. Art’s paper shows that by skipping just this one point the rate of convergence of some QMC estimates may drop by a factor, bringing the rate back to Monte Carlo values! As this applies to randomised scrambled Sobol sequences, this is quite amazing. The explanation centers on the suppression leaving one region of the hypercube unexplored, with an O(n⁻¹) error ensuing.

The above picture from the paper makes the case in a most obvious way: the mean squared error is not decreasing at the same rate for the no-drop and one-drop versions, since they are -3/2 and -1, respectively. The paper further “recommends against using roundnumber sample sizes and thinning QMC points.” Conclusion: QMC is not MC!

September 8, 2020 at 3:05 am

The effect is even bigger than that. It is actually the RMSE that changes from n^-3/2 to n^-1, so the MSE goes from n^-3 to n^-2, when just one point is skipped. That example function is additive. Other problems will show other rates, but I expect a large benefit from not skipping this point to be quite common, especially for functions with a large additive component.

There have been tremendous new efforts recently making scrambled Sobol’ points more widely available: Pamphile Roy for scipy, Max Balandat for pytorch, Fred Hickernell & Alesei Sorokin for QMCPy. Much of it, like the paper you link, had a major impetus from Fred’s plenary tutorial on software for QMC.