论文标题
在尖锐的随机零订单上，riemannian歧管上的黑森估计器
On Sharp Stochastic Zeroth Order Hessian Estimators over Riemannian Manifolds
论文作者
论文摘要
我们研究了Hessian估计器的功能，该功能定义了$ n $维的完整分析性Riemannian歧管。我们使用$ o（1）$函数评估介绍了新的随机零阶Hessian估计器。我们表明，对于一个分析实现的函数$ f $，我们的估计器达到了$ o \ left（γδ^2 \ right）$的偏置，其中$γ$依赖于Levi-Civita连接和函数$ f $，而$δ$是有限差步长。据我们所知，我们的结果为Hessian估计量提供了第一个偏见，该估计量明确取决于基础Riemannian歧管的几何形状。我们还根据您的Hessian估计器研究下游计算。经验评估证明了我们方法的至高无上。
We study Hessian estimators for functions defined over an $n$-dimensional complete analytic Riemannian manifold. We introduce new stochastic zeroth-order Hessian estimators using $O (1)$ function evaluations. We show that, for an analytic real-valued function $f$, our estimator achieves a bias bound of order $ O \left( γδ^2 \right) $, where $ γ$ depends on both the Levi-Civita connection and function $f$, and $δ$ is the finite difference step size. To the best of our knowledge, our results provide the first bias bound for Hessian estimators that explicitly depends on the geometry of the underlying Riemannian manifold. We also study downstream computations based on our Hessian estimators. The supremacy of our method is evidenced by empirical evaluations.
