This paper examines the asymptotic convergence properties of Lipschitz interpolation methods within the context of bounded stochastic noise. In the first part of the paper, we establish probabilistic consistency guarantees of the classical approach in a general setting and derive upper bounds on the uniform convergence rates. These bounds align with well-established optimal rates of non-parametric regression obtained in related settings and provide new precise upper bounds on the non-parametric regression problem under bounded noise assumptions. Practically, they can serve as a theoretical tool for comparing Lipschitz interpolation to alternative non-parametric regression methods, providing a condition on the behaviour of the noise at the boundary of its support which indicates when Lipschitz interpolation should be expected to asymptotically outperform or underperform other approaches. In the second part, we expand upon these results to include asymptotic guarantees for online learning of dynamics in discrete-time stochastic systems and illustrate their utility in deriving closed-loop stability guarantees of a simple controller. We also explore applications where the main assumption of prior knowledge of the Lipschitz constant is removed by adopting the LACKI framework (Calliess et al. (2020)) and deriving general asymptotic consistency.

翻译：本文研究有界随机噪声背景下Lipschitz插值方法的渐近收敛性质。在第一部分中，我们在一般框架下建立了经典方法的概率一致性保证，并推导了均匀收敛率的上界。这些上界与相关研究中非参数回归的公认最优率相一致，并在有界噪声假设下为非参数回归问题提供了新的精确上界。在实际应用中，该上界可作为理论工具，将Lipschitz插值与其他非参数回归方法进行对比，通过分析噪声在其支撑边界处的行为条件，揭示Lipschitz插值在渐近性能上相对其他方法的优劣预期。第二部分中，我们扩展上述结果，建立了离散时间随机系统动力学在线学习的渐近保证，并展示了其在推导简单控制器闭环稳定性保证中的应用。此外，我们通过采用LACKI框架（Calliess等人，2020）并推导一般渐近一致性，探索了移除Lipschitz常数先验知识这一主要假设的应用场景。