流形上零阶优化的随机李括号逼近
Stochastic Lie-Bracket Approximations for Zeroth-Order Optimization on Manifolds
- University at Buffalo, State University of New York（纽约州立大学布法罗分校）
- University of California San Diego（加州大学圣地亚哥分校）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出一种仅用目标函数测量值驱动的连续时间零阶优化算法，通过随机李括号逼近实现黎曼梯度动力学，并证明其轨迹逼近集中于全局极小值的朗之万扩散，给出有限与无限时域的概率收敛保证。
AI中文摘要：
我们提出了一种用于紧致流形上光滑函数零阶全局优化的连续时间算法，该算法仅利用目标函数的测量值来综合黎曼梯度动力学，而无需计算梯度。该算法被表述为由分段光滑的McShane型维纳过程逼近驱动的常微分方程，并包含一个仅依赖于驱动框架而非目标函数的几何补偿项。我们的主要结果建立了固定逼近参数下的无限时域实用概率保证。为此，我们证明了算法的轨迹在均方意义上逼近一个黎曼朗之万扩散，其唯一的不变吉布斯测度集中于全局极小值集合。我们利用这一逼近来建立有限时域概率收敛保证，并直接分析算法的长时间行为以获得无限时域结果。
英文摘要：
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.
