A table of short-period Tausworthe generators for Markov chain quasi-Monte Carlo

Abstract

We consider the problem of estimating expectations by using Markov chain Monte Carlo methods and improving the accuracy by replacing IID uniform random points with quasi-Monte Carlo (QMC) points. Recently, it has been shown that Markov chain QMC remains consistent when the driving sequences are completely uniformly distributed (CUD). However, the definition of CUD sequences is not constructive, so an implementation method using short-period Tausworthe generators (i.e., linear feedback shift register generators over the two-element field) that approximate CUD sequences has been proposed. In this paper, we conduct an exhaustive search of short-period Tausworthe generators for Markov chain QMC in terms of the $t$-value, which is a criterion of uniformity widely used in the study of QMC methods. We provide a parameter table of Tausworthe generators and show the effectiveness in numerical examples using Gibbs sampling.

Introduction

We consider the problem of estimating the expectation $E_{\pi }\left[f\left(\mathbf{X}\right)\right]$ by using Markov chain Monte Carlo (MCMC) methods for a target distribution $\pi$ and some function $f$. For this problem, we want to improve the accuracy by replacing independent and identically distributed (IID) uniform random points with quasi-Monte Carlo (QMC) points. However, typical QMC points (e.g., Sobol’, Faure, and Niederreiter–Xing) are not applicable in general. Motivated by a simulation study by Liao [1], Owen and Tribble [2] and Chen et al. [3] proved that Markov chain QMC remains consistent when the driving sequences are completely uniformly distributed (CUD). Here, a sequence $u_0,u_1,u_2$, $\dots$ $\in [0,1)$ is said to be CUD if overlapping $s$-blocks $(u_i,u_{i+1},\dots ,u_{i+s-1})$ , $i=0,1,2,\dots$, are uniformly distributed for every dimension $s\ge 1$.

Levin [4] provided several constructions for CUD sequences, but they are not convenient to implement. Instead, to construct CUD sequences approximately, Tribble and Owen [5] and Tribble [6] proposed an implementation method using short-period linear congruential and Tausworthe generators (i.e., linear feedback shift register generators over the two-element field ${\mathbb{F}}_2≔\{0,1\}$) that run for the entire period. Chen et al. [7] implemented short-period Tausworthe generators optimized in terms of the equidistribution property, which is a coarse criterion used in the area of pseudorandom number generation (see [8, §8.1] for the complete parameter table). In the theory of $(t,m,s)$-nets and $(t,s)$-sequences, the $t$-value is a central criterion of uniformity. In fact, typical QMC points (e.g., Sobol’, Faure, and Niederreiter–Xing) are optimized in terms of the $t$-value (see [9], [10]).

The aim of this paper is to conduct an exhaustive search of short-period Tausworthe generators for Markov chain QMC in terms of the $t$-value and to provide a parameter table of Tausworthe generators. It is known that Tausworthe generators can be viewed as polynomial Korobov lattice point sets with a denominator polynomial $p\left(x\right)$ and a numerator polynomial $q\left(x\right)$ over ${\mathbb{F}}_2$ (e.g., see [11], [12]). For dimension $s=2$, there is a connection between the $t$-value and continued fraction expansions, that is, the $t$-value is optimal (i.e., the $t$-value is zero) if and only if the partial quotients in the continued fraction of $q\left(x\right)∕p\left(x\right)$ are all of degree one. To satisfy the definition of CUD sequences approximately, we want to search for parameters $(p\left(x\right),q\left(x\right))$ whose $t$-values are optimal for $s=2$ and as small as possible for $s\ge 3$. As a previous study, in 1993, Tezuka and Fushimi [13] proposed an algorithm to search for such parameters using a polynomial analogue of Fibonacci numbers from the viewpoint of continued fraction expansions. Thus, we refine their algorithm on modern computers, and conduct an exhaustive search again. In addition, we report numerical examples using Gibbs sampling in which the resulting QMC point sets perform better than the existing point sets developed by Chen et al. [7].

One might consider searching for parameters $(p\left(x\right),q\left(x\right))$ with $t$-value zero for $s=3$. Kajiura et al. [14] proved that there exists no maximal-period Tausworthe generator with this property.

The remainder of this paper is organized as follows: In Section 2, we briefly recall the definition of CUD sequences, Tausworthe generators, and the $t$-value and equidistribution property. Section 3 is devoted to our main results: we describe an exhaustive search algorithm and provide a table of short-period Tausworthe generators for Markov chain QMC. We also compare our new generators with existing generators developed by Chen et al. [7] in terms of the $t$-value and equidistribution property. In Section 4, we present numerical examples using Gibbs sampling. In Section 5, we conclude this paper.