# Riemannian Zeroth-Order Gradient Estimation with Structure-Preserving Metrics for Geodesically Incomplete Manifolds

arXiv: https://arxiv.org/abs/2601.08039 · HTML: https://mlsanity.com/paper/2601.08039/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 28 of its 68 references that are in the map.
- First posted on arXiv 12 Jan 2026.

## Builds on (most cited first, 28 of 28)

- [Riemannian Adaptive Optimization Methods](https://mlsanity.com/paper/1810.00760/) (2018)
- [Better Theory for SGD in the Nonconvex World](https://mlsanity.com/paper/2002.03329/) (2020)
- [Combining Differentiable PDE Solvers and Graph Neural Networks for Fluid Flow Prediction](https://mlsanity.com/paper/2007.04439/) (2020)
- [Random Reshuffling: Simple Analysis with Vast Improvements](https://mlsanity.com/paper/2006.05988/) (2020)
- [First-order Methods for Geodesically Convex Optimization](https://mlsanity.com/paper/1602.06053/) (2016)
- [Riemannian SVRG: Fast Stochastic Optimization on Riemannian Manifolds](https://mlsanity.com/paper/1605.07147/) (2016)
- [Optimization Techniques on Riemannian Manifolds](https://mlsanity.com/paper/1407.5965/) (2014)
- [From Nesterov's Estimate Sequence to Riemannian Acceleration](https://mlsanity.com/paper/2001.08876/) (2020)
- [Riemannian Optimization via Frank-Wolfe Methods](https://mlsanity.com/paper/1710.10770/) (2017)
- [Adaptive regularization with cubics on manifolds](https://mlsanity.com/paper/1806.00065/) (2018)
- [Tracking and regret bounds for online zeroth-order Euclidean and Riemannian optimisation](https://mlsanity.com/paper/2010.00211/) (2020)
- [Negative curvature obstructs acceleration for strongly geodesically convex optimization, even with exact first-order oracles](https://mlsanity.com/paper/2111.13263/) (2021)
- [Riemannian conjugate gradient methods: General framework and specific algorithms with convergence analyses](https://mlsanity.com/paper/2112.02572/) (2021)
- [Revisiting Zeroth-Order Optimization: Minimum-Variance Two-Point Estimators and Directionally Aligned Perturbations](https://mlsanity.com/paper/2510.19975/) (2025)
- [Accelerated Gradient Methods for Geodesically Convex Optimization: Tractable Algorithms and Convergence Analysis](https://mlsanity.com/paper/2202.02036/) (2022)
- [Zeroth-order Riemannian Averaging Stochastic Approximation Algorithms](https://mlsanity.com/paper/2309.14506/) (2023)
- [From the Greene--Wu Convolution to Gradient Estimation over Riemannian Manifolds](https://mlsanity.com/paper/2108.07406/) (2021)
- [A new, globally convergent Riemannian conjugate gradient method](https://mlsanity.com/paper/1302.0125/) (2013)
- [Stochastic Zeroth-order Functional Constrained Optimization: Oracle Complexity and Applications](https://mlsanity.com/paper/2210.04273/) (2022)
- [Riemannian Accelerated Zeroth-order Algorithm: Improved Robustness and Lower Query Complexity](https://mlsanity.com/paper/2405.05713/) (2024)
- [On Sharp Stochastic Zeroth Order Hessian Estimators over Riemannian Manifolds](https://mlsanity.com/paper/2201.10780/) (2022)
- [Riemannian Interior Point Methods for Constrained Optimization on Manifolds](https://mlsanity.com/paper/2203.09762/) (2022)
- [Retraction based Direct Search Methods for Derivative Free Riemannian Optimization](https://mlsanity.com/paper/2202.11052/) (2022)
- [Inexact Riemannian Gradient Descent Method for Nonconvex Optimization](https://mlsanity.com/paper/2409.11181/) (2024)
- [On the Optimal Construction of Unbiased Gradient Estimators for Zeroth-Order Optimization](https://mlsanity.com/paper/2510.19953/) (2025)
- [Convergence Rates of Stochastic Zeroth-order Gradient Descent for L ojasiewicz Functions](https://mlsanity.com/paper/2210.16997/) (2022)
- [Registration of algebraic varieties using Riemannian optimization](https://mlsanity.com/paper/2401.08562/) (2024)
- [Riemannian trust-region methods for strict saddle functions with complexity guarantees](https://mlsanity.com/paper/2402.07614/) (2024)

## Abstract (arXiv)

In this paper, we study Riemannian zeroth-order optimization in settings where the underlying Riemannian metric g is geodesically incomplete, and the goal is to approximate stationary points with respect to this incomplete metric. To address this challenge, we construct structure-preserving metrics that are geodesically complete while ensuring that every stationary point under the new metric remains stationary under the original one. Building on this foundation, we revisit the classical symmetric two-point zeroth-order estimator and analyze its mean-squared error from a purely intrinsic perspective, depending only on the manifold's geometry rather than any ambient embedding. Leveraging this intrinsic analysis, we establish convergence guarantees for stochastic gradient descent with this intrinsic estimator. Under additional suitable conditions, an ε-stationary point under the constructed metric g' also corresponds to an ε-stationary point under the original metric g, thereby matching the best-known complexity in the geodesically complete setting. Empirical studies on synthetic problems confirm our theoretical findings, and experiments on a practical mesh optimization task demonstrate that our framework maintains stable convergence even in the absence of geodesic completeness.