Saddle points provide a hierarchical view of the energy landscape, revealing transition pathways and interconnected basins of attraction, and offering insight into the global structure, metastability, and possible collective mechanisms of the underlying system. In this work, we propose a stochastic saddle-search algorithm to circumvent exact derivative and Hessian evaluations that have been used in implementing traditional and deterministic saddle dynamics. At each iteration, the algorithm uses a stochastic eigenvector-search method, based on a stochastic Hessian, to approximate the unstable directions, followed by a stochastic gradient update with reflections in the approximate unstable direction to advance toward the saddle point. We carry out rigorous numerical analysis to establish the almost sure convergence for the stochastic eigenvector search and local almost sure convergence with an $O(1/n)$ rate for the saddle search, and present a theoretical guarantee to ensure the high-probability identification of the saddle point when the initial point is sufficiently close. Numerical experiments, including the application to a neural network loss landscape and a Landau-de Gennes type model for nematic liquid crystal, demonstrate the practical applicability and the ability for escaping from "bad" areas of the algorithm.

翻译：鞍点为能量景观提供了层次化视角，揭示了过渡路径和相互连通的吸引盆，有助于深入理解底层系统的全局结构、亚稳态特性及可能的集体机制。本研究提出一种随机鞍点搜索算法，以规避传统确定性鞍点动力学实现中所需的精确导数和Hessian矩阵计算。该算法在每次迭代中，首先基于随机Hessian矩阵采用随机特征向量搜索方法逼近不稳定方向，随后沿近似不稳定方向进行带反射操作的随机梯度更新以逼近鞍点。我们通过严格数值分析证明了随机特征向量搜索的几乎必然收敛性，以及鞍点搜索算法在局部范围内以$O(1/n)$速率达到几乎必然收敛，并提供了当初始点足够接近时高概率识别鞍点的理论保证。数值实验（包括神经网络损失景观和向列相液晶Landau-de Gennes型模型的应用）验证了该算法的实际适用性及其逃离"不良"区域的能力。