Infinite-dimensional spherical-radial decomposition for probabilistic functions, with application to constrained optimal control and Gaussian process regression
This work addresses the challenge of handling infinite-dimensional probabilistic functions for researchers and practitioners in constrained optimal control and Gaussian process regression, though it is incremental as it builds on existing finite-dimensional methods.
The authors tackled the problem of estimating probabilistic functions and their gradients in infinite-dimensional settings by generalizing the spherical-radial decomposition to infinite stochastic dimensions, resulting in an unbiased, low-variance estimator called hiSRD. They demonstrated its application in chance-constrained optimization and Gaussian process regression, showing reduced variance and elimination of truncation-induced bias.
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.