黎曼度量下不完整流形上的零阶梯度估计
Riemannian Zeroth-Order Gradient Estimation with Structure-Preserving Metrics for Geodesically Incomplete Manifolds
AI总结：
本文研究了在底层黎曼度量不完整的情况下，如何通过保持结构的度量近似 stationary 点。基于对称两点零阶估计器的分析，建立了随机梯度下降的收敛性保证，并在合成数据和实际网格优化任务中验证了方法的稳定性。
摘要：
本文研究了在底层黎曼度量 $g$ 不完整的情况下，如何通过保持结构的度量近似 stationary 点。为此，我们构造了保持结构的度量，使其 geodesically 完整，同时确保新度量下的 stationary 点在原度量下仍为 stationary 点。基于此，我们重新审视经典的对称两点零阶估计器，从纯粹内在视角分析其均方误差，仅依赖流形几何而非任何嵌入环境。利用这一内在分析，我们建立了使用此内在估计器的随机梯度下降的收敛性保证。在附加合适条件下，构造的度量 $g'$ 下的 $ε$-stationary 点也对应于原度量 $g$ 下的 $ε$-stationary 点，从而匹配了在 geodesically 完整设置下的最佳复杂度。在合成问题上的实验证实了我们的理论发现，并在实际网格优化任务中证明了我们的框架在缺乏 geodesic 完整性时仍能保持稳定的收敛性。
英文摘要：
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.
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