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

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.

Figures (8)

• Figure 1: Comparison of root mean squared error (RMSE; $y$-axis) versus stochastic truncation dimension ($x$-axis) for standard Monte Carlo (gray horizontal line), finite-dimensional SRD (dashed) and for the proposed method (solid). Results are for Gaussian process example detailed in \ref{['SEC:NumGPR']}, but with 1000 repeats to estimate the RMSE.
• Figure 1: RMSE difference between standard MC and SRD ($y$-axis) versus approximation dimension ($x$-axis). The dashed line indicates the slope theoretically expected from \ref{['thm:estimation-degeneration']}. Results are for Gaussian process example detailed in \ref{['SEC:NumGPR']}, but with 1000 repeats.
• Figure 1: Illustration of standard SRD (left) and hybrid SRD (right) to estimate the probability for the three-dimensional set given by the intersection of half spaces defined by hyperplanes (light blue). The standard SRD integrates the three-dimensional $\chi$-density along 3D rays (black lines in left figure). The hybrid method uses Monte Carlo in the $z$-direction and SRD in the $x,y$-direction, integrating the two-dimensional $\chi$-density along rays (black lines in right figure).
• Figure 1: Left: Root mean squared error (RMSE, $y$-axis) of probability estimation \ref{['eq:chance']} for different number $N$ of MC samples ($x$-axis). Compared are the finite-dimensional SRD for various KL-mode truncations $K$, and the proposed hybrid method. The "exact" probability, computed with $10^8$ standard Monte Carlo samples, is $p\approx 0.648926$. Right: Gradient at the nominal control, which is largest close to $\partial {\mathcal{D}}_2$, where the stochastic Neumann boundary force is applied.
• Figure 1: Gaussian process regression: Shown on the left is a comparison of the root mean squared error (RMSE; $y$-axis) of probability estimation with $N=10^4$ samples versus KL dimension $K$ ($x$-axis) for finite-dimensional SRD (dashed) and for the proposed method (solid), with both MC and QMC sampling. Shown for comparison is the result with standard Monte Carlo (gray horizontal line). Shown on the right are, for $N=500$ samples, the individual components discussed in \ref{['subsec:variance-analysis']} contributing to the variance of hiSRD-MC sampling. The larger variance compared to the left figure is due to the smaller $N$.

Theorems & Definitions (11)

• Example 1.1: PDE optimal control
• Example 1.2: Gaussian process regression
• Lemma 2.1
• Proof 1
• Theorem 2.2
• Lemma 3.1
• Proof 2
• Proposition 3.3
• Proof 3
• Proposition 3.4