Relative Entropy Minimization over Hilbert Spaces via Robbins-Monro

TL;DR

This paper develops a Robbins-Monro based method for minimizing relative entropy to find optimal Gaussian approximations of non-Gaussian measures in infinite-dimensional Hilbert spaces, with demonstrated robustness in numerical examples.

Contribution

It introduces a Robbins-Monro algorithm for relative entropy minimization in infinite dimensions and provides convergence analysis and numerical validation.

Findings

The method converges under specific assumptions.

Numerical examples show robustness across dimensions.

The approach applies to infinite-dimensional path space problems.

Abstract

One way of getting insight into non-Gaussian measures, posed on infinite dimensional Hilbert spaces, is to first obtain best fit Gaussian approximations, which are more amenable to numerical approximation. These Gaussians can then be used to accelerate sampling algorithms. This begs the questions of how one should measure optimality and how the optimizers can be obtained. Here, we consider the problem of minimizing the distance with respect to relative entropy. We examine this minimization problem by seeking roots of the first variation of relative entropy, taken with respect to the mean of the Gaussian, leaving the covariance fixed. Adapting a convergence analysis of Robbins-Monro to the infinite dimensional setting, we can justify the application of this algorithm and highlight necessary assumptions to ensure convergence, not only in the context of relative entropy minimization, but other infinite dimensional problems as well. Numerical examples in path space, showing the robustness of this method with respect to dimension, are provided.