Stochastic Lie-Bracket Approximations for Zeroth-Order Optimization on Manifolds
Abstract
We propose a continuous-time algorithm for zeroth-order global optimization of smooth functions on compact manifolds that synthesizes Riemannian gradient dynamics using only objective measurements, without evaluating gradients. The algorithm is formulated as an ordinary differential equation driven by piecewise-smooth McShane-type approximations of the Wiener process, with a geometric compensation term that depends only on the actuation frame and not on the objective function. Our main result establishes infinite-horizon practical probabilistic guarantees for fixed approximation parameters. To this end, we show that the algorithm's trajectories approximate, in the mean-square sense, a Riemannian Langevin diffusion whose unique invariant Gibbs measure concentrates on the set of global minimizers. We exploit this approximation to establish finite-horizon probabilistic convergence guarantees and directly analyze the algorithm's long-time behavior to obtain the infinite-horizon result.
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BibTeX
@misc{abdelgalil2026stochastic,
title = {{Stochastic Lie-Bracket Approximations for Zeroth-Order Optimization on Manifolds}},
author = {Mahmoud Abdelgalil and Miroslav Krstic and Jorge I. Poveda},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.38069},
eprint = {2609.38069},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.38069},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Stochastic Lie-Bracket Approximations for Zeroth-Order Optimization on Manifolds
AU - Mahmoud Abdelgalil
AU - Miroslav Krstic
AU - Jorge I. Poveda
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.38069
UR - https://arxiv.org/abs/2609.38069
AB - We propose a continuous-time algorithm for zeroth-order global optimization of smooth functions on compact manifolds that synthesizes Riemannian gradient dynamics using only objective measurements, without evaluating gradients. The algorithm is formulated as an ordinary differential equation driven by piecewise-smooth McShane-type approximations of the Wiener process, with a geometric compensation term that depends only on the actuation frame and not on the objective function. Our main result establishes infinite-horizon practical probabilistic guarantees for fixed approximation parameters. To this end, we show that the algorithm's trajectories approximate, in the mean-square sense, a Riemannian Langevin diffusion whose unique invariant Gibbs measure concentrates on the set of global minimizers. We exploit this approximation to establish finite-horizon probabilistic convergence guarantees and directly analyze the algorithm's long-time behavior to obtain the infinite-horizon result.
ER -