Function learning with local linear regression models: An analysis based on discrepancy

In this work local linear regression models are introduced and analyzed in the context of empirical risk minimization (ERM) for function learning. This kind of models can be seen as a more sophisticated version of classic kernel smoothing models, based on the principle of local estimation. In particular, we analyze the conditions under which consistency of the ERM procedure is guaranteed, pointing out assumptions on the way the input space is sampled to obtain the observation data. This allows to extend the tractation to the case where the choice of the training set is part of the learning process. To this purpose, a choice of the observation points based on low-discrepancy sequences, a family of sampling methods commonly employed for efficient numerical integration, is analyzed. Simulation results involving two different examples of function learning are provided.