14 research outputs found
AReS and MaRS - Adversarial and MMD-Minimizing Regression for SDEs
Stochastic differential equations are an important modeling class in many
disciplines. Consequently, there exist many methods relying on various
discretization and numerical integration schemes. In this paper, we propose a
novel, probabilistic model for estimating the drift and diffusion given noisy
observations of the underlying stochastic system. Using state-of-the-art
adversarial and moment matching inference techniques, we avoid the
discretization schemes of classical approaches. This leads to significant
improvements in parameter accuracy and robustness given random initial guesses.
On four established benchmark systems, we compare the performance of our
algorithms to state-of-the-art solutions based on extended Kalman filtering and
Gaussian processes.Comment: Published at the Thirty-sixth International Conference on Machine
Learning (ICML 2019
Variational bridge constructs for grey box modelling with Gaussian processes
Latent force models are systems whereby there is a mechanistic model describing the dynamics of the system state, with some unknown forcing term that is approximated with a Gaussian process. If such dynamics are non-linear, it can be difficult to estimate the posterior state and forcing term jointly, particularly when there are system parameters that also need estimating. This paper uses black-box variational inference to jointly estimate the posterior, designing a multivariate extension to local inverse autoregressive flows as a flexible approximater of the system. We compare estimates on systems where the posterior is known, demonstrating the effectiveness of the approximation, and apply to problems with non-linear dynamics, multi-output systems and models with non-Gaussian likelihoods
Statistical inference for generative models with maximum mean discrepancy
While likelihood-based inference and its variants provide a statistically efficient and widely applicable approach to parametric inference, their application to models involving intractable likelihoods poses challenges. In this work, we study a class of minimum distance estimators for intractable generative models, that is, statistical models for which the likelihood is intractable, but simulation is cheap. The distance considered, maximum mean discrepancy (MMD), is defined through the embedding of probability measures into a reproducing kernel Hilbert space. We study the theoretical properties of these estimators, showing that they are consistent, asymptotically normal and robust to model misspecification. A main advantage of these estimators is the flexibility offered by the choice of kernel, which can be used to trade-off statistical efficiency and robustness. On the algorithmic side, we study the geometry induced by MMD on the parameter space and use this to introduce a novel natural gradient descent-like algorithm for efficient implementation of these estimators. We illustrate the relevance of our theoretical results on several classes of models including a discrete-time latent Markov process and two multivariate stochastic differential equation models
Statistical Inference for Generative Models with Maximum Mean Discrepancy
While likelihood-based inference and its variants provide a statistically
efficient and widely applicable approach to parametric inference, their
application to models involving intractable likelihoods poses challenges. In
this work, we study a class of minimum distance estimators for intractable
generative models, that is, statistical models for which the likelihood is
intractable, but simulation is cheap. The distance considered, maximum mean
discrepancy (MMD), is defined through the embedding of probability measures
into a reproducing kernel Hilbert space. We study the theoretical properties of
these estimators, showing that they are consistent, asymptotically normal and
robust to model misspecification. A main advantage of these estimators is the
flexibility offered by the choice of kernel, which can be used to trade-off
statistical efficiency and robustness. On the algorithmic side, we study the
geometry induced by MMD on the parameter space and use this to introduce a
novel natural gradient descent-like algorithm for efficient implementation of
these estimators. We illustrate the relevance of our theoretical results on
several classes of models including a discrete-time latent Markov process and
two multivariate stochastic differential equation models
Monte Carlo simulation of SDEs using GANs
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 âsupervised GANâ. 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 CoxâIngersollâRoss (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
Distributional Gradient Matching for Learning Uncertain Neural Dynamics Models
Differential equations in general and neural ODEs in particular are an
essential technique in continuous-time system identification. While many
deterministic learning algorithms have been designed based on numerical
integration via the adjoint method, many downstream tasks such as active
learning, exploration in reinforcement learning, robust control, or filtering
require accurate estimates of predictive uncertainties. In this work, we
propose a novel approach towards estimating epistemically uncertain neural
ODEs, avoiding the numerical integration bottleneck. Instead of modeling
uncertainty in the ODE parameters, we directly model uncertainties in the state
space. Our algorithm - distributional gradient matching (DGM) - jointly trains
a smoother and a dynamics model and matches their gradients via minimizing a
Wasserstein loss. Our experiments show that, compared to traditional
approximate inference methods based on numerical integration, our approach is
faster to train, faster at predicting previously unseen trajectories, and in
the context of neural ODEs, significantly more accurate.Comment: Published at NeurIPS 202