提出新型零阶海森矩阵估计方法，显著提升高维优化精度与效率。
Revisiting Zeroth-Order Hessian Approximation: A Single-Step Policy Optimization Lens
- 从单步策略优化视角重构零阶海森估计，统一经典随机方法
- 设计最优基线与查询复用策略，降低方差并提高样本效率
- 理论证明无偏性与收敛性，适用于高维贝叶斯推断等任务
准确的零阶（ZO）海森矩阵估计是无导数方法的核心，对双层优化、贝叶斯推断和不确定性量化等任务至关重要。然而，在高维场景下获得完整且低方差的海森矩阵及其逆矩阵估计仍具挑战。本文提出统一框架，将ZO海森估计重新诠释为单步策略优化（PO）问题，揭示一般ZO估计器与平滑策略优化目标海森矩阵之间的理论等价性，将不同经典随机估计器统一为基线选择的特例。基于此，我们提出ZoVH，一套完整的方差缩减估计器，涵盖完整海森矩阵、正则化逆矩阵及偏差校正的逆海森-梯度乘积。ZoVH采用两项关键技术：(1) 推导出可严格最小化方差的最优基线；(2) 通过历史函数查询复用提升样本效率，不增加计算成本。严格的理论分析证实了海森估计的无偏性，验证了基线的方差最优性，给出了整个ZoVH套件的误差界，并建立了所得曲率感知零阶算法的收敛性。大量实证结果验证了理论发现，表明ZoVH在真实应用中实现更优的估计精度与收敛性能。代码已开源：https://github.com/Qjbtiger/ZoVH。
原文摘要 · Abstract (English)
Accurate Zeroth-Order (ZO) Hessian estimation is a cornerstone of derivative-free methods, essential for tasks such as bilevel optimization, Bayesian inference, and uncertainty quantification. However, obtaining a complete suite of low-variance estimators for the Hessian and its inverse in high-dimensional settings remains a significant challenge. To address this, we propose a unified framework that reinterprets ZO Hessian approximation through the lens of single-step Policy Optimization (PO). This perspective establishes a theoretical equivalence between general ZO Hessian estimators and the Hessian of a smoothed PO objective, unifying distinct classical randomized estimators as specific instances of baseline selection. Building on this foundation, we introduce ZoVH, a comprehensive suite of variance-reduced estimators for the full Hessian matrix, its regularized inverse, and the bias-corrected inverse Hessian-gradient product. ZoVH leverages two key techniques: (1) a unique optimal baseline derived to provably minimize variance, and (2) a query reuse strategy that incorporates historical function queries to enhance sample efficiency without inflating costs. Our rigorous theoretical analysis confirms the unbiasedness of the Hessian estimator, validates the variance optimality of our baseline, provides error bounds for the entire ZoVH suite, and establishes convergence guarantees for the resulting curvature-aware ZO algorithm. Extensive empirical results validate our theoretical findings, demonstrating that ZoVH achieves superior estimation accuracy and convergence performance in real-world applications. Code is available at https://github.com/Qjbtiger/ZoVH
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。