Infinite-dimensional spherical-radial decomposition for probabilistic functions, with application to constrained optimal control and Gaussian process regression

TL;DR

This paper introduces a novel infinite-dimensional spherical-radial decomposition method that improves probabilistic function estimation, with applications in constrained optimal control and Gaussian process regression, by reducing bias and variance.

Contribution

It generalizes the spherical-radial decomposition to infinite dimensions, combining subspace methods with Monte Carlo to enhance probabilistic estimations in complex settings.

Findings

The hybrid method is unbiased and reduces variance in infinite-dimensional settings.

It enables derivative computation of probabilistic functions under constraints.

Numerical studies demonstrate effectiveness in stochastic PDE control and Gaussian process optimization.

Abstract

The spherical-radial decomposition (SRD) is an efficient method for estimating probabilistic functions and their gradients defined over finite-dimensional elliptical distributions. In this work, we generalize the SRD to infinite stochastic dimensions by combining subspace SRD with standard Monte Carlo methods. The resulting method, which we call hybrid infinite-dimensional SRD (hiSRD) provides an unbiased, low-variance estimator for convex sets arising, for instance, in chance-constrained optimization. We provide a theoretical analysis of the variance of finite-dimensional SRD as the dimension increases, and show that the proposed hybrid method eliminates truncation-induced bias, reduces variance, and allows the computation of derivatives of probabilistic functions. We present comprehensive numerical studies for a risk-neutral stochastic PDE optimal control problem with joint chance state constraints, and for optimizing kernel parameters in Gaussian process regression under the constraint that the posterior process satisfies joint chance constraints.