Lipschitz Optimisation for Lipschitz Interpolation

Supervised machine learning methods are algorithms for inductive inference. On the basis of a sample, they construct (learn) a computable model of a data generating process that facilitates inference over the underlying ground truth function and aims to predict its function values at unobserved inputs. Among supervised learning methods, nonparametric algorithms tend to offer greater flexibility to learn rich function classes. Unfortunately, many classical techniques for nonparametric regression, such as the Nadaraya-Watson estimator [21], [14] or the LOESS method, [6] suffer from a practical limitation: their regression performance depends on the choice of hyperparameters. While in principle, it would be possible to tune these to the data (in manner similar in spirit to the one we propose in this work), to the best of our knowledge, currently there is little understanding on how to do so with a global optimiser that offers theoretical performance guarantees on the optimisation solution. This means that in practice, one is left to engineer these hyperparameters (or the settings of an optimiser) by manual tuning in order to ensure good performance on a particular learning problem. Of course, this stands in opposition to the motivation for utilising nonparametric learning, especially in system identification: which is to facilitate flexible and fully automated black-box learning that does not require manual intervention.