发表机构
Johns Hopkins University; Washington University in St. Louis（约翰斯·霍普金斯大学; 圣路易斯华盛顿大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究基于采样的可达性极限，通过将其视为几何支持估计，确定正则属性使恢复适定，界定样本复杂度，证明相关属性不可去除，实验表明对抗采样不改变缩放比例，揭示了初始集几何、动力学和采样定律对可达性估计的影响。
AI 中文摘要
可达性分析在安全关键控制、机器人技术和神经网络验证中至关重要，但经典计算方法在状态维度上扩展性差。基于采样的方法成为有前景的替代方案，能提供有限样本保证。本文通过将基于采样的可达集恢复视为几何支持估计来研究初始集几何、动力学和采样定律对估计器准确性的影响。首先确定两个正则属性使恢复适定，概率质量覆盖保证可提升为豪斯多夫距离下的精度\(r\)。其次界定样本复杂度，恢复所需样本数为\(\tilde{\mathcal{O}}\big((e^{3LT}/r)^n\big)\)，与状态维度和时间范围均呈指数关系。最后证明这些属性不可去除，非线性系统实验表明对抗采样可改善常数但不改变缩放比例。
英文摘要
Reachability analysis is central to safety-critical control, robotics, and neural network verification, but classical computational methods, such as Hamilton--Jacobi reachability and set propagation, scale poorly with state dimension. Sampling-based methods have emerged as a promising alternative, often providing finite-sample guarantees that bound the probability-mass left uncovered. However, an explicit account of how the geometry of the initial set, the dynamics, and the sampling law affect the accuracy of the estimator is not fully available in the literature. We study this by casting sampling-based reachable-set recovery as geometric support estimation over a family of problems specified by an initial set, its dynamics, and a sampling law. First, we identify two regularity properties, positive reach of the initial set's complement and Lipschitz continuity of the dynamics, that together make recovery well-posed: a probability-mass coverage guarantee can be upgraded to accuracy $r$ in Hausdorff distance. Second, we bound the resulting sample complexity: recovery is achievable with $\tilde{\mathcal{O}}\big((e^{3LT}/r)^n\big)$ samples, exponential in both the state dimension and the time horizon. Third, we show that neither can be removed: an minimax lower bound of $Ω\big((e^{LT}/r)^n\big)$ holds for every estimator, so the exponential dependence on dimension and the degradation over the horizon are both intrinsic, not artifacts of a particular method. Experiments on nonlinear systems confirm that adversarial sampling improves constants but not the scaling.
