Subsampling in ensemble Kalman inversion

We consider the Ensemble Kalman Inversion which has been recently introduced as an efficient, gradient-free optimisation method to estimate unknown parameters in an inverse setting. In the case of large data sets, the Ensemble Kalman Inversion becomes computationally infeasible as the data misfit needs to be evaluated for each particle in each iteration. Here, randomised algorithms like stochastic gradient descent have been demonstrated to successfully overcome this issue by using only a random subset of the data in each iteration, so-called subsampling techniques. Based on a recent analysis of a continuous-time representation of stochastic gradient methods, we propose, analyse, and apply subsampling-techniques within Ensemble Kalman Inversion. Indeed, we propose two different subsampling techniques: either every particle observes the same data subset (single subsampling) or every particle observes a different data subset (batch subsampling)

Subsampling in ensemble Kalman inversion

We consider the ensemble Kalman inversion (EKI) which has been recently introduced as an efficient, gradient-free optimisation method to estimate unknown parameters in an inverse setting. In the case of large data sets, the EKI becomes computationally infeasible as the data misfit needs to be evaluated for each particle in each iteration. Here, randomised algorithms like stochastic gradient descent have been demonstrated to successfully overcome this issue by using only a random subset of the data in each iteration, so-called subsampling techniques. Based on a recent analysis of a continuous-time representation of stochastic gradient methods, we propose, analyse, and apply subsampling-techniques within EKI. Indeed, we propose two different subsampling techniques: either every particle observes the same data subset (single subsampling) or every particle observes a different data subset (batch subsampling)

Subsampling Error in Stochastic Gradient Langevin Diffusions

The Stochastic Gradient Langevin Dynamics (SGLD) are popularly used to approximate Bayesian posterior distributions in statistical learning procedures with large-scale data. As opposed to many usual Markov chain Monte Carlo (MCMC) algorithms, SGLD is not stationary with respect to the posterior distribution; two sources of error appear: The first error is introduced by an Euler--Maruyama discretisation of a Langevin diffusion process, the second error comes from the data subsampling that enables its use in large-scale data settings. In this work, we consider an idealised version of SGLD to analyse the method's pure subsampling error that we then see as a best-case error for diffusion-based subsampling MCMC methods. Indeed, we introduce and study the Stochastic Gradient Langevin Diffusion (SGLDiff), a continuous-time Markov process that follows the Langevin diffusion corresponding to a data subset and switches this data subset after exponential waiting times. There, we show that the Wasserstein distance between the posterior and the limiting distribution of SGLDiff is bounded above by a fractional power of the mean waiting time. Importantly, this fractional power does not depend on the dimension of the state space. We bring our results into context with other analyses of SGLD

Nested Sampling for Uncertainty Quantification and Rare Event Estimation

Nested Sampling is a method for computing the Bayesian evidence, also called the marginal likelihood, which is the integral of the likelihood with respect to the prior. More generally, it is a numerical probabilistic quadrature rule. The main idea of Nested Sampling is to replace a high-dimensional likelihood integral over parameter space with an integral over the unit line by employing a push-forward with respect to a suitable transformation. Practically, a set of active samples ascends the level sets of the integrand function, with the measure contraction of the super-level sets being statistically estimated. We justify the validity of this approach for integrands with non-negligible plateaus, and demonstrate Nested Sampling's practical effectiveness in estimating the (log-)probability of rare events.Comment: 24 page

Gaussian random fields on non-separable Banach spaces

We study Gaussian random fields on certain Banach spaces and investigate conditions for their existence. Our results apply inter alia to spaces of Radon measures and H\"older functions. In the former case, we are able to define Gaussian white noise on the space of measures directly, avoiding, e.g., an embedding into a negative-order Sobolev space. In the latter case, we demonstrate how H\"older regularity of the samples is controlled by that of the covariance kernel and, thus, show a connection to the Theorem of Kolmogorov-Chentsov

Certified and fast computations with shallow covariance kernels

Many techniques for data science and uncertainty quantification demand efficient tools to handle Gaussian random fields, which are defined in terms of their mean functions and covariance operators. Recently, parameterized Gaussian random fields have gained increased attention, due to their higher degree of flexibility. However, especially if the random field is parameterized through its covariance operator, classical random field discretization techniques fail or become inefficient. In this work we introduce and analyze a new and certified algorithm for the low-rank approximation of a parameterized family of covariance operators which represents an extension of the adaptive cross approximation method for symmetric positive definite matrices. The algorithm relies on an affine linear expansion of the covariance operator with respect to the parameters, which needs to be computed in a preprocessing step using, e.g., the empirical interpolation method. We discuss and test our new approach for isotropic covariance kernels, such as Mat\'ern kernels. The numerical results demonstrate the advantages of our approach in terms of computational time and confirm that the proposed algorithm provides the basis of a fast sampling procedure for parameter dependent Gaussian random fields