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

Relative Entropy Minimization over Hilbert Spaces via Robbins-Monro