SCMO: Stochastic Control for Optimization over Probability Measures on Infinite-Dimensional Spaces
Abstract
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.
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BibTeX
@misc{cheung2026scmo,
title = {{SCMO: Stochastic Control for Optimization over Probability Measures on Infinite-Dimensional Spaces}},
author = {Hang Cheung and Jinniao Qiu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14548},
eprint = {2609.14548},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14548},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - SCMO: Stochastic Control for Optimization over Probability Measures on Infinite-Dimensional Spaces
AU - Hang Cheung
AU - Jinniao Qiu
PY - 2026
DA - 2026/09/13
PB - arXiv
DO - 10.48550/arxiv.2609.14548
UR - https://arxiv.org/abs/2609.14548
AB - 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.
ER -