将概率函数估计方法拓展至无限维，提升约束优化与高斯过程的精度与效率。
Infinite-dimensional spherical-radial decomposition for probabilistic functions, with application to constrained optimal control and Gaussian process regression
- 结合子空间SRD与蒙特卡洛法，构建无偏低方差的无限维估计器。
- 在随机偏微分方程控制与核参数优化中，实现联合概率约束下的高效求解。
- 适用于需精确计算概率梯度的高维约束优化问题，如风险敏感控制与贝叶斯建模。
球-径向分解（SRD）是一种高效的有限维椭球分布上概率函数及其梯度估计方法。本文将其推广至无限维情形，通过结合子空间SRD与标准蒙特卡洛方法，提出混合无限维球-径向分解（hiSRD）。该方法为凸集（如机会约束优化中的集合）提供无偏、低方差估计器。我们分析了有限维SRD随维度增长的方差变化，并证明所提方法可消除截断偏差、降低方差并支持概率函数导数计算。在无风险偏好随机偏微分方程最优控制问题（含联合机会状态约束）以及满足联合机会约束的高斯过程回归中优化核参数方面进行了全面数值实验。
原文摘要 · Abstract (English)
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.
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