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Added by
Aleksei Sorokin
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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Practitioners wishing to experience the efficiency gains from using low discrepancy sequences need correct, well-written software. This article, based on our MCQMC 2020 tutorial, describes some of the better quasi-Monte Carlo (QMC) software available. We highlight the key software components required to approximate multivariate integrals or expectations of functions of vector random variables by QMC. We have combined these components in QMCPy, a Python open source library, which we hope will draw the support of the QMC community. Here we introduce QMCPy.
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Monte Carlo planners can often return sub-optimal actions, even if they are guaranteed to converge in the limit of infinite samples. Known asymptotic regret bounds do not provide any way to measure confidence of a recommended action at the conclusion of search. In this work, we prove bounds on the sub-optimality of Monte Carlo estimates for non-stationary bandits and Markov decision processes. These bounds can be directly computed at the conclusion of the search and do not require knowledge of the true action-value. The presented bound holds for general Monte Carlo solvers meeting mild convergence conditions. We empirically test the tightness of the bounds through experiments on a multi-armed bandit and a discrete Markov decision process for both a simple solver and Monte Carlo tree search.
Generative adversarial networks (GANs) have shown promising results when applied on partial differential equations and financial time series generation. We investigate if GANs can also be used to approximate one-dimensional Ito stochastic differential equations (SDEs). We propose a scheme that approximates the path-wise conditional distribution of SDEs for large time steps. Standard GANs are only able to approximate processes in distribution, yielding a weak approximation to the SDE. A conditional GAN architecture is proposed that enables strong approximation. We inform the discriminator of this GAN with the map between the prior input to the generator and the corresponding output samples, i.e. we introduce a `supervised GAN. We compare the input-output map obtained with the standard GAN and supervised GAN and show experimentally that the standard GAN may fail to provide a path-wise approximation. The GAN is trained on a dataset obtained with exact simulation. The architecture was tested on geometric Brownian motion (GBM) and the Cox-Ingersoll-Ross (CIR) process. The supervised GAN outperformed the Euler and Milstein schemes in strong error on a discretisation with large time steps. It also outperformed the standard conditional GAN when approximating the conditional distribution. We also demonstrate how standard GANs may give rise to non-parsimonious input-output maps that are sensitive to perturbations, which motivates the need for constraints and regularisation on GAN generators.
This paper continues our treatment of the Neutron Transport Equation (NTE) building on the work in [arXiv:1809.00827v2], [arXiv:1810.01779v4] and [arXiv:1901.00220v3], which describes the flux of neutrons through inhomogeneous fissile medium. Our aim is to analyse existing and novel Monte-Carlo (MC) algorithms, aimed at simulating the lead eigenvalue associated with the underlying model. This quantity is of principal importance in the nuclear regulatory industry for which the NTE must be solved on complicated inhomogenous domains corresponding to nuclear reactor cores, irradiative hospital equipment, food irradiation equipment and so on. We include a complexity analysis of such MC algorithms, noting that no such undertaking has previously appeared in the literature. The new MC algorithms offer a variety of advantages and disadvantages of accuracy vs cost, as well as the possibility of more convenient
This position paper summarizes a recently developed research program focused on inference in the context of data centric science and engineering applications, and forecasts its trajectory forward over the next decade. Often one endeavours in this context to learn complex systems in order to make more informed predictions and high stakes decisions under uncertainty. Some key challenges which must be met in this context are robustness, generalizability, and interpretability. The Bayesian framework addresses these three challenges, while bringing with it a fourth, undesirable feature: it is typically far more expensive than its deterministic counterparts. In the 21st century, and increasingly over the past decade, a growing number of methods have emerged which allow one to leverage cheap low-fidelity models in order to precondition algorithms for performing inference with more expensive models and make Bayesian inference tractable in the context of high-dimensional and expensive models. Notable examples are multilevel Monte Carlo (MLMC), multi-index Monte Carlo (MIMC), and their randomized counterparts (rMLMC), which are able to provably achieve a dimension-independent (including $infty-$dimension) canonical complexity rate with respect to mean squared error (MSE) of $1/$MSE. Some parallelizability is typically lost in an inference context, but recently this has been largely recovered via novel double randomization approaches. Such an approach delivers i.i.d. samples of quantities of interest which are unbiased with respect to the infinite resolution target distribution. Over the coming decade, this family of algorithms has the potential to transform data centric science and engineering, as well as classical machine learning applications such as deep learning, by scaling up and scaling out fully Bayesian inference.
We introduce an ensemble Markov chain Monte Carlo approach to sampling from a probability density with known likelihood. This method upgrades an underlying Markov chain by allowing an ensemble of such chains to interact via a process in which one chains state is cloned as anothers is deleted. This effective teleportation of states can overcome issues of metastability in the underlying chain, as the scheme enjoys rapid mixing once the modes of the target density have been populated. We derive a mean-field limit for the evolution of the ensemble. We analyze the global and local convergence of this mean-field limit, showing asymptotic convergence independent of the spectral gap of the underlying Markov chain, and moreover we interpret the limiting evolution as a gradient flow. We explain how interaction can be applied selectively to a subset of state variables in order to maintain advantage on very high-dimensional problems. Finally we present the application of our methodology to Bayesian hyperparameter estimation for Gaussian process regression.
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