SCMO：无穷维空间上概率测度优化的随机控制方法

中文摘要 AI 辅助

我们研究在可分希尔伯特空间上的概率测度上，对可能非凸且非光滑的泛函进行仅目标优化的方法，允许优化器本质上为非狄拉克测度。我们引入了SCMO（随机控制测度优化器），这是一种源自熵正则化随机控制的无梯度粒子方法。经过有限粒子和伽辽金近似后，Cole-Hopf变换将最优反馈表示为吉布斯加权的终端位移。SCMO通过采样上下文云来近似此反馈，每次用一个候选样本替换一个粒子，对所得经验测度进行评分，并应用指数重加权。SCMO对候选提议和粒子更新使用独立的协方差，当相等时称为匹配，否则称为非匹配；我们的分析涵盖这两种设置。在匹配情况下，我们建立了无PDE的定性收敛性以及一个先投影后定量的界，该界将粒子、伽辽金投影和熵正则化误差分开。对于具有非匹配协方差的实际多上下文实现，我们证明了有限候选一致性，并表明非匹配协方差的有符号、曲率相关效应可以降低近似误差的上界。最后，在函数空间、轨迹分布和接触丰富的操作问题上的实验表明，SCMO能够处理非光滑和非凸目标，逃离次优局部盆地，并恢复规定的多模态、非狄拉克分布结构。复现我们实验结果的代码可在以下网址获取：此https URL。

英文摘要

We study objective-only optimization of possibly nonconvex and nonsmooth functionals over probability measures on a separable Hilbert space, allowing the optimizer to be intrinsically non-Dirac. We introduce SCMO (Stochastic Control Measure Optimizer), a gradient-free particle method derived from entropy regularized stochastic control. After finite-particle and Galerkin approximations, a Cole--Hopf transform represents the optimal feedback as a Gibbs-weighted terminal displacement. SCMO approximates this feedback by sampling context clouds, replacing one particle at a time with candidate draws, scoring the resulting empirical measures, and applying exponential reweighting. SCMO uses separate covariances for candidate proposals and particle updates, termed matched when equal and nonmatched otherwise; our analysis covers both settings. In the matched case, we establish PDE-free qualitative convergence and a projection-first quantitative bound separating particle, Galerkin-projection, and entropy-regularization errors. For the practical multi-context implementation with nonmatched covariance, we prove finite-candidate consistency and show that the signed, curvature-dependent effect of nonmatched covariance can reduce the resulting upper bound on the approximation error. Finally, experiments on function-space, trajectory-law, and contact-rich manipulation problems show that SCMO handles nonsmooth and nonconvex objectives, escapes suboptimal local basins, and recovers prescribed multimodal, non-Dirac law structure. Code for reproducing our experimental results is available at https://github.com/HenryCHEUNG7373/SCMO.

发表机构

• University of Calgary（卡尔加里大学）

机构由 AI 辅助整理，请以论文原文为准。