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

Lipschitz Interpolation: Non-parametric Convergence under Bounded Stochastic Noise