Title: Lazily Adapted Constant Kinky Inference for Nonparametric Regression and Model-Reference Adaptive Control,
Abstract: Techniques known as Nonlinear Set Membership prediction, Lipschitz
Interpolation or Kinky Inference are approaches to machine learning that
utilise presupposed Lipschitz properties to compute inferences over unobserved
function values. Provided a bound on the true best Lipschitz constant of the
target function is known a priori they offer convergence guarantees as well as
bounds around the predictions. Considering a more general setting that builds
on Hoelder continuity relative to pseudo-metrics, we propose an online method
for estimating the Hoelder constant online from function value observations
that possibly are corrupted by bounded observational errors. Utilising this to
compute adaptive parameters within a kinky inference rule gives rise to a
nonparametric machine learning method, for which we establish strong universal
approximation guarantees. That is, we show that our prediction rule can learn
any continuous function in the limit of increasingly dense data to within a
worst-case error bound that depends on the level of observational uncertainty.
We apply our method in the context of nonparametric model-reference adaptive
control (MRAC). Across a range of simulated aircraft roll-dynamics and
performance metrics our approach outperforms recently proposed alternatives
that were based on Gaussian processes and RBF-neural networks. For
discrete-time systems, we provide guarantees on the tracking success of our
learning-based controllers both for the batch and the online learning setting.