# On Sharp Stochastic Zeroth Order Hessian Estimators over Riemannian Manifolds

arXiv: https://arxiv.org/abs/2201.10780 · HTML: https://mlsanity.com/paper/2201.10780/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 2 of its 23 references that are in the map.
- First posted on arXiv 26 Jan 2022.

## Builds on (most cited first, 2 of 2)

- [Optimal rates for zero-order convex optimization: the power of two function evaluations](https://mlsanity.com/paper/1312.2139/) (2013)
- [From the Greene--Wu Convolution to Gradient Estimation over Riemannian Manifolds](https://mlsanity.com/paper/2108.07406/) (2021)

## Built on by (most cited first, 4 of 4)

- [Stochastic Zeroth Order Gradient and Hessian Estimators: Variance Reduction and Refined Bias Bounds](https://mlsanity.com/paper/2205.14737/) (2022)
- [Zeroth-order Riemannian Averaging Stochastic Approximation Algorithms](https://mlsanity.com/paper/2309.14506/) (2023)
- [Convergence Rates of Stochastic Zeroth-order Gradient Descent for L ojasiewicz Functions](https://mlsanity.com/paper/2210.16997/) (2022)
- [Riemannian Zeroth-Order Gradient Estimation with Structure-Preserving Metrics for Geodesically Incomplete Manifolds](https://mlsanity.com/paper/2601.08039/) (2026)

## Abstract (arXiv)

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( γ δ2 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.