Zero-variance Importance Sampling Research Articles

Double-loop importance sampling for McKean–Vlasov stochastic differential equation

This paper investigates Monte Carlo (MC) methods to estimate probabilities of rare events associated with the solution to the d-dimensional McKean–Vlasov stochastic differential equation (MV-SDE). MV-SDEs are usually approximated using a stochastic interacting P-particle system, which is a set of P coupled d-dimensional stochastic differential equations (SDEs). Importance sampling (IS) is a common technique for reducing high relative variance of MC estimators of rare-event probabilities. We first derive a zero-variance IS change of measure for the quantity of interest by using stochastic optimal control theory. However, when this change of measure is applied to stochastic particle systems, it yields a P×d\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P \ imes d$$\\end{document}-dimensional partial differential control equation (PDE), which is computationally expensive to solve. To address this issue, we use the decoupling approach introduced in (dos Reis et al. 2023), generating a d-dimensional control PDE for a zero-variance estimator of the decoupled SDE. Based on this approach, we develop a computationally efficient double loop MC (DLMC) estimator. We conduct a comprehensive numerical error and work analysis of the DLMC estimator. As a result, we show optimal complexity of OTOLr-4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {O}\\left( \ extrm{TOL}_{\ extrm{r}}^{-4}\\right) $$\\end{document} with a significantly reduced constant to achieve a prescribed relative error tolerance TOLr\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{TOL}_{\ extrm{r}}$$\\end{document}. Subsequently, we propose an adaptive DLMC method combined with IS to numerically estimate rare-event probabilities, substantially reducing relative variance and computational runtimes required to achieve a given TOLr\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{TOL}_{\ extrm{r}}$$\\end{document} compared with standard MC estimators in the absence of IS. Numerical experiments are performed on the Kuramoto model from statistical physics.

A Koopman framework for rare event simulation in stochastic differential equations

We exploit the relationship between the stochastic Koopman operator and the Kolmogorov backward equation to construct importance sampling schemes for stochastic differential equations. Specifically, we propose using eigenfunctions of the stochastic Koopman operator to approximate the Doob transform for an observable of interest (e.g., associated with a rare event) which in turn yields an approximation of the corresponding zero-variance importance sampling estimator. Our approach is broadly applicable and systematic, treating non-normal systems, non-gradient systems, and systems with oscillatory dynamics or rank-deficient noise in a common framework. In nonlinear settings where the stochastic Koopman eigenfunctions cannot be derived analytically, we use dynamic mode decomposition (DMD) methods to approximate them numerically, but the framework is agnostic to the particular numerical method employed. Numerical experiments demonstrate that even coarse approximations of a few eigenfunctions, where the latter are built from non-rare trajectories, can produce effective importance sampling schemes for rare events.

A general purpose strategy for realizing the zero-variance importance sampling and calculating the unknown integration constant

A general purpose strategy is proposed to realize asymptotically the zero-variance importance sampling. The unknown integration constant can also be calculated simultaneously. This strategy can sample efficiently from multi-dimensional zero-variance importance function which is multi-modal by particular Markov Chain random walk. Sampling from this kind of distribution has been a challenge for a long time. Moreover, by using the probability density function reconstruction method, the unknown integration constant can be estimated. This feature is absent in traditional Markov Chain Monte Carlo method. Some multi-dimensional integrals are analyzed carefully. The results show this strategy is efficient.

Importance sampling in stochastic optimization: An application to intertemporal portfolio choice

In this paper, we propose an approach to construct an analytical approximation of the zero-variance importance sampling distribution. We show specifically how this can be designed for the classic intertemporal portfolio choice problem with proportional transaction costs and constant relative risk aversion preferences. We compare the method to standard variance reduction techniques in single-period optimization and multi-stage stochastic programming formulations of the problem. The numerical experiments show that the method produces significant improvements in solution quality. In the single-period setting, the number of scenarios can be reduced by a factor of 400 with maintained solution quality compared to the best standard method; Latin hypercube sampling. Using importance sampling in multi-stage formulations, the gaps between lower and upper bound estimates are reduced by a factor of 26-500 with maintained scenario tree size. On a higher level, we consider analytical approximations of the zero-variance importance sampling distribution to be a promising method to improve solution quality in stochastic optimization.

An Adaptive Metamodel-Based Subset Importance Sampling approach for the assessment of the functional failure probability of a thermal-hydraulic passive system

An Adaptive Metamodel-Based Subset Importance Sampling (AM-SIS) approach, previously developed by the authors, is here employed to assess the (small) functional failure probability of a thermal-hydraulic (T-H) nuclear passive safety system. The approach relies on an iterative Importance Sampling (IS) scheme that efficiently couples the powerful characteristics of Subset Simulation (SS) and fast-running Artificial Neural Networks (ANNs). In particular, SS and ANNs are intelligently employed to build and progressively refine a fully nonparametric estimator of the ideal, zero-variance Importance Sampling Density (ISD), in order to: (i) increase the robustness of the failure probability estimates and (ii) decrease the number of expensive T-H simulations needed (together with the related computational burden). The performance of the approach is thoroughly compared to that of other efficient Monte Carlo (MC) techniques on a case study involving an advanced nuclear reactor cooled by helium.

A General Importance Sampling Algorithm for Estimating Portfolio Loss Probabilities in Linear Factor Models

This paper develops a novel importance sampling algorithm for estimating the probability of large portfolio losses in the conditional independence framework. In a sense we modify the algorithm of Glasserman and Li (2005) so that it can be applied in a wider variety of models, including the t copula. The basic idea is to apply exponential tilts to (i) the distribution of the natural sufficient statistics of the systematic risk factors and (ii) conditional default probabilities, given the simulated values of the systematic risk factors. Motivated by the cross-entropy method we determine optimal values of the tilt parameters by minimizing the Kullback-Liebler divergence of the candidate family from the ideal (i.e. zero-variance) importance sampling measure, leading to an intuitive moment-matching characterization. Finally, by exploiting the large portfolio approximation to the portfolio loss, we develop efficient and accurate approximations to the requisite moments. In the context of the Gaussian copula the proposed algorithm offers comparable performance to the Glasserman and Li (2005) algorithm, which is known to be asymptotically optimal. In the context of the t copula the proposed algorithm offers comparable performance for substantially less computational time than the Chan and Kroese (2010) algorithm, which also has good asymptotic properties.

Automated State-Dependent Importance Sampling for Markov Jump Processes via Sampling from the Zero-Variance Distribution

Many complex systems can be modeled via Markov jump processes. Applications include chemical reactions, population dynamics, and telecommunication networks. Rare-event estimation for such models can be difficult and is often computationally expensive, because typically many (or very long) paths of the Markov jump process need to be simulated in order to observe the rare event. We present a state-dependent importance sampling approach to this problem that is adaptive and uses Markov chain Monte Carlo to sample from the zero-variance importance sampling distribution. The method is applicable to a wide range of Markov jump processes and achieves high accuracy, while requiring only a small sample to obtain the importance parameters. We demonstrate its efficiency through benchmark examples in queueing theory and stochastic chemical kinetics.

Automated State-Dependent Importance Sampling for Markov Jump Processes via Sampling from the Zero-Variance Distribution

Many complex systems can be modeled via Markov jump processes. Applications include chemical reactions, population dynamics, and telecommunication networks. Rare-event estimation for such models can be difficult and is often computationally expensive, because typically many (or very long) paths of the Markov jump process need to be simulated in order to observe the rare event. We present a state-dependent importance sampling approach to this problem that is adaptive and uses Markov chain Monte Carlo to sample from the zero-variance importance sampling distribution. The method is applicable to a wide range of Markov jump processes and achieves high accuracy, while requiring only a small sample to obtain the importance parameters. We demonstrate its efficiency through benchmark examples in queueing theory and stochastic chemical kinetics.

An adaptive zero-variance importance sampling approximation for static network dependability evaluation

We propose an adaptive parameterized method to approximate the zero-variance change of measure for the evaluation of static network reliability models, with links subject to failures. The method uses two rough approximations of the unreliability function, conditional on the states of any subset of links being fixed. One of these approximations, based on mincuts, under-estimates the true unknown unreliability, whereas the other one, based on minpaths, over-estimates it. Our proposed change of measure takes a convex linear combination of the two, estimates the optimal (graph-dependent) coefficient in this combination from pilot runs, and uses the resulting conditional unreliability approximation at each step of a dynamic importance sampling algorithm. This new scheme is more general and more flexible than a previously proposed zero-variance approximation scheme, based on mincuts only, and which was shown to be robust asymptotically when unreliabilities of individual links decrease toward zero. Our numerical examples show that the new scheme is often more efficient when low unreliability comes from a large number of possible paths connecting the considered nodes rather than from small failure probabilities of the links.

Efficient Simulation of Large Deviation Events for Sums of Random Vectors Using Saddle-Point Representations

We consider the problem of efficient simulation estimation of the density function at the tails, and the probability of large deviations for a sum of independent, identically distributed (i.i.d.), light-tailed, and nonlattice random vectors. The latter problem besides being of independent interest, also forms a building block for more complex rare event problems that arise, for instance, in queueing and financial credit risk modeling. It has been extensively studied in the literature where state-independent, exponential-twisting-based importance sampling has been shown to be asymptotically efficient and a more nuanced state-dependent exponential twisting has been shown to have a stronger bounded relative error property. We exploit the saddle-point-based representations that exist for these rare quantities, which rely on inverting the characteristic functions of the underlying random vectors. These representations reduce the rare event estimation problem to evaluating certain integrals, which may via importance sampling be represented as expectations. Furthermore, it is easy to identify and approximate the zero-variance importance sampling distribution to estimate these integrals. We identify such importance sampling measures and show that they possess the asymptotically vanishing relative error property that is stronger than the bounded relative error property. To illustrate the broader applicability of the proposed methodology, we extend it to develop an asymptotically vanishing relative error estimator for the practically important expected overshoot of sums of i.i.d. random variables.

Efficient Simulation of Large Deviation Events for Sums of Random Vectors Using Saddle-Point Representations

We consider the problem of efficient simulation estimation of the density function at the tails, and the probability of large deviations for a sum of independent, identically distributed (i.i.d.), light-tailed, and nonlattice random vectors. The latter problem besides being of independent interest, also forms a building block for more complex rare event problems that arise, for instance, in queueing and financial credit risk modeling. It has been extensively studied in the literature where state-independent, exponential-twisting-based importance sampling has been shown to be asymptotically efficient and a more nuanced state-dependent exponential twisting has been shown to have a stronger bounded relative error property. We exploit the saddle-point-based representations that exist for these rare quantities, which rely on inverting the characteristic functions of the underlying random vectors. These representations reduce the rare event estimation problem to evaluating certain integrals, which may via importance sampling be represented as expectations. Furthermore, it is easy to identify and approximate the zero-variance importance sampling distribution to estimate these integrals. We identify such importance sampling measures and show that they possess the asymptotically vanishing relative error property that is stronger than the bounded relative error property. To illustrate the broader applicability of the proposed methodology, we extend it to develop an asymptotically vanishing relative error estimator for the practically important expected overshoot of sums of i.i.d. random variables.

Efficient rare event simulation for heavy-tailed systems via cross entropy

The cross entropy method is an iterative technique that is used to obtain a low-variance importance sampling (IS) distribution from a given parametric family, which must satisfy two properties. First, subsequent iterations of the parameters must be easily computable and, second, the family should approximate the zero-variance IS distribution. We obtain parametric families for which these two properties are satisfied for a large class of heavy-tailed systems. Our estimators are shown to be strongly efficient in these settings.

Approximate Zero-Variance Importance Sampling for Static Network Reliability Estimation

We propose a new Monte Carlo method, based on dynamic importance sampling, to estimate the probability that a given set of nodes is connected in a graph (or network) where each link is failed with a given probability. The method generates the link states one by one, using a sampling strategy that approximates an ideal zero-variance importance sampling scheme. The approximation is based on minimal cuts in subgraphs. In an asymptotic rare-event regime where failure probability becomes very small, we prove that the relative error of our estimator remains bounded, and even converges to 0 under additional conditions, when the unreliability of individual links converges to 0. The empirical performance of the new sampling scheme is illustrated by examples.

Approximating zero-variance importance sampling in a reliability setting

We consider a class of Markov chain models that includes the highly reliable Markovian systems (HRMS) often used to represent the evolution of multicomponent systems in reliability settings. We are interested in the design of efficient importance sampling (IS) schemes to estimate the reliability of such systems by simulation. For these models, there is in fact a zero-variance IS scheme that can be written exactly in terms of a value function that gives the expected cost-to-go (the exact reliability, in our case) from any state of the chain. This IS scheme is impractical to implement exactly, but it can be approximated by approximating this value function. We examine how this can be effectively used to estimate the reliability of a highly-reliable multicomponent system with Markovian behavior. In our implementation, we start with a simple crude approximation of the value function, we use it in a first-order IS scheme to obtain a better approximation at a few selected states, then we interpolate in between and use this interpolation in our final (second-order) IS scheme. In numerical illustrations, our approach outperforms the popular IS heuristics previously proposed for this class of problems. We also perform an asymptotic analysis in which the HRMS model is parameterized in a standard way by a rarity parameter ε, so that the relative error (or relative variance) of the crude Monte Carlo estimator is unbounded when ε→0. We show that with our approximation, the IS estimator has bounded relative error (BRE) under very mild conditions, and vanishing relative error (VRE), which means that the relative error converges to 0 when ε→0, under slightly stronger conditions.

Exponential convergence of adaptive importance sampling for Markov chains

We consider adaptive importance sampling for a Markov chain with scoring. It is shown that convergence to the zero-variance importance sampling chain for the mean total score occurs exponentially fast under general conditions. These results extend previous work in Kollman (1993) and in Kollman et al. (1999) for finite state spaces.

Exponential convergence of adaptive importance sampling for Markov chains

We consider adaptive importance sampling for a Markov chain with scoring. It is shown that convergence to the zero-variance importance sampling chain for the mean total score occurs exponentially fast under general conditions. These results extend previous work in Kollman (1993) and in Kollman et al. (1999) for finite state spaces.