Lazily Adapted Constant Kinky Inference for Nonparametric Regression and Model-Reference Adaptive Control

Techniques known as Nonlinear Set Membership prediction, Lipschitz Interpolation or KinkyInference are approaches to machine learning that utilise presupposed Lipschitz properties tocompute inferences over unobserved function values. Provided a bound on the true best Lipschitzconstant of the target function is known a priori they offer convergence guarantees as well asbounds around the predictions. Considering a more general setting that builds on Hölder conti-nuity relative to pseudo-metrics, we propose an online method for estimating the Hölder constantonline from function value observations that possibly are corrupted by bounded observationalerrors. Utilising this to compute adaptive parameters within a kinky inference rule gives rise to anonparametric machine learning method, for which we establish strong universal approximationguarantees. That is, we show that our prediction rule can learn any continuous function in thelimit of increasingly dense data to within a worst-case error bound that depends on the level ofobservational uncertainty. We apply our method in the context of nonparametric model-referenceadaptive control (MRAC). Across a range of simulated aircraft roll-dynamics and performancemetrics our approach outperforms recently proposed alternatives that were based on Gaussianprocesses and RBF-neural networks. For discrete-time systems, we provide stability guaranteesfor our learning-based controllers both for the batch and the online learning setting.