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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.
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.
Sampling algorithms based on discretizations of Stochastic Differential Equations (SDEs) compose a rich and popular subset of MCMC methods. This work provides a general framework for the non-asymptotic analysis of sampling error in 2-Wasserstein distance, which also leads to a bound of mixing time. The method applies to any consistent discretization of contractive SDEs. When applied to Langevin Monte Carlo algorithm, it establishes $tilde{mathcal{O}}left( frac{sqrt{d}}{epsilon} right)$ mixing time, without warm start, under the common log-smooth and log-strongly-convex conditions, plus a growth condition on the 3rd-order derivative of the potential of target measures at infinity. This bound improves the best previously known $tilde{mathcal{O}}left( frac{d}{epsilon} right)$ result and is optimal (in terms of order) in both dimension $d$ and accuracy tolerance $epsilon$ for target measures satisfying the aforementioned assumptions. Our theoretical analysis is further validated by numerical experiments.
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.
Stochastic differential equations (SDEs) are a staple of mathematical modelling of temporal dynamics. However, a fundamental limitation has been that such models have typically been relatively inflexible, which recent work introducing Neural SDEs has sought to solve. Here, we show that the current classical approach to fitting SDEs may be approached as a special case of (Wasserstein) GANs, and in doing so the neural and classical regimes may be brought together. The input noise is Brownian motion, the output samples are time-evolving paths produced by a numerical solver, and by parameterising a discriminator as a Neural Controlled Differential Equation (CDE), we obtain Neural SDEs as (in modern machine learning parlance) continuous-time generative time series models. Unlike previous work on this problem, this is a direct extension of the classical approach without reference to either prespecified statistics or density functions. Arbitrary drift and diffusions are admissible, so as the Wasserstein loss has a unique global minima, in the infinite data limit any SDE may be learnt. Example code has been made available as part of the texttt{torchsde} repository.
In this work, we provide an efficient and realistic data-driven approach to simulate astronomical images using deep generative models from machine learning. Our solution is based on a variant of the generative adversarial network (GAN) with progressive training methodology and Wasserstein cost function. The proposed solution generates naturalistic images of galaxies that show complex structures and high diversity, which suggests that data-driven simulations using machine learning can replace many of the expensive model-driven methods used in astronomical data processing.