Random numbers are the nuts and bolts of simulation. Typically, all the randomness required by the model is simulated by a random number generator whose output is assumed to be a sequence of independent and identically distributed (IID) U(0, 1) random variables (i.e., continuous random variables distributed uniformly over the interval

Abstract. We introduce and study a randomized quasi-Monte Carlo method for estimating the state distribution at each step of a Markov chain. The number of steps in the chain can be random and unbounded. The method simulates n copies of the chain in parallel, using a (d + 1)-dimensional highly-uniform point set of cardinality n, randomized independently at each step, where d is the number of uniform random numbers required at each transition of the Markov chain. This technique is effective in particular to obtain a low-variance unbiased estimator of the expected total cost up to some random stopping time, when state-dependent costs are paid at each step. It is generally more effective when the state space has a natural order related to the cost function. We provide numerical illustrations where the variance reduction with respect to standard Monte Carlo is substantial. The variance can be reduced by factors of several thousands in some cases. We prove bounds on the convergence rate of the worst-case error and variance for special situations. In line with what is typically observed in randomized quasi-Monte Carlo contexts, our empirical results indicate much better convergence than what these bounds guarantee.

We introduce SSJ, an organized set of software tools implemented in the Java programming language and offering general-purpose facilities for stochastic simulation programming. It supports the event view, process view, continuous simulation, and arbitrary mixtures of these. Performance, flexibility, and extensibility were key criteria in its design and implementation. We illustrate its use by simple examples and discuss how we dealt with some performance issues in the implementation.

Lattice rules are quasi-Monte Carlo methods for estimating large-dimensional integrals over the unit hypercube. In this paper, after briefly reviewing key ideas of quasi-Monte Carlo methods, we give an overview of recent results, generalize them, and provide several new results, for lattice rules defined in spaces of polynomials and of formal series with coeffocients in a finite ring. We discuss basic properties, implementations, a randomized version, and quality criteria (i.e., measures of uniformity) for selecting the parameters. Two types of polynomial lattice rules are examined: dimensionwise lattices and resolutionwise lattices. These rules turn out to be special cases of digital net constructions, which we reinterpret as yet another type of lattice in a space of formal series. Our development underlines the connections between integration lattices and digital nets.

In this paper we investigate an efficient implementation of the Monte Carlo EM al-gorithm based on Quasi-Monte Carlo sampling. The Monte Carlo EM algorithm is a stochastic version of the deterministic EM (Expectation-Maximization) algorithm in which an intractable E-step is replaced by a Monte Carlo approximation. Quasi-Monte Carlo methods produce deterministic sequences of points that can significantly improve the accuracy of Monte Carlo approximations over purely random sampling. One draw-back to deterministic Quasi-Monte Carlo methods is that it is generally difficult to determine the magnitude of the approximation error. However, in order to implement the Monte Carlo EM algorithm in an automated way, the ability to measure this error is fundamental. Recent developments of randomized Quasi-Monte Carlo methods can overcome this drawback. We investigate the implementation of an automated, data-driven Monte Carlo EM algorithm based on randomized Quasi-Monte Carlo methods. We apply this algorithm to a geostatistical model of online purchases and find that it can significantly decrease the total simulation effort, thus showing great potential for improving upon the efficiency of the classical Monte Carlo EM algorithm. Key words and phrases: Monte Carlo error; low-discrepancy sequence; Halton sequence; EM algo-rithm; geostatistical model.

Several methods for reducing the variance in the context of Monte Carlo simulation are based on correlation induction. This includes antithetic variates, Latin hypercube sampling, and randomized version of quasi-Monte Carlo methods such as lattice rules and digital nets, where the resulting estimators are usually weighted averages of several dependent random variables that can be seen as function evaluations at a finite set of random points in the unit hypercube. In this paper, we consider a setting where these methods can be combined with the use of control variates and we provide conditions under which we can formally prove that the variance is minimized by choosing equal weights and equal control variate coe#cients across the di#erent points of evaluation, regardless of the function (integrand) that is evaluated. 1

In a preliminary part of this paper, we analyze the necessity of randomness in evolution strategies. We conclude to the necessity of ”continuous”-randomness, but with a much more limited use of randomness than what is commonly used in evolution strategies. We then apply these results to CMA-ES, a famous evolution strategy already based on the idea of derandomization, which uses random independent Gaussian mutations. We here replace these random independent Gaussian mutations by a quasi-random sample. The modification is very easy to do, the modified algorithm is computationally more efficient and its convergence is faster in terms of the number of iterates for a given precision.

Quantile estimation has become increasingly important,particularly in the financial industry,where value at risk (VaR) has emerged as a standard measurement tool for controlling portfolio risk. In this paper,we analyze the probability that a simulation-based quantile estimator fails to lie in a prespecified neighborhood of the true quantile. First,we show that this error probability converges to zero exponentially fast with sample size for negatively dependent sampling. Then we consider stratified quantile estimators and show that the error probability for these estimators can be guaranteed to be 0 with sufficiently large,but finite,sample size. These estimators,however,require sample sizes that grow exponentially in the problem dimension. Numerical experiments on a simple VaR example illustrate the potential for variance reduction.

Summary. We construct infinite-dimensional highly-uniform point sets for quasi-Monte Carlo integration. The successive coordinates of each point are determined by a linear recurrence in F2 w, the finite field with 2w elements where w is an integer, and a mapping from this field to the interval [0, 1). One interesting property of these point sets is that almost all of their two-dimensional projections are perfectly equidistributed. We performed searches for specific parameters in terms of different measures of uniformity and different numbers of points. We give a numerical illustration showing that using randomized versions of these point sets in place of independent random points can reduce the variance drastically for certain functions. 1

Abstract: We study active learning as a derandomized form of sampling. We show that full derandomization is not suitable in a robust framework, propose partially derandomized samplings, and develop new active learning methods (i) in which expert knowledge is easy to integrate (ii) with a parameter for the exploration/exploitation dilemma (iii) less randomized than the full-random sampling (yet also not deterministic). Experiments are performed in the case of regression for value-function learning on a continuous domain. Our main results are (i) efficient partially derandomized point sets (ii) moderate-derandomization theorems (iii) experimental evidence of the importance of the frontier (iv) a new regression-specific user-friendly sampling tool lessrobust than blind samplers but that sometimes works very efficiently in large dimensions. All experiments can be reproduced by downloading the source code and running the provided command line. 1