Concentration Inequalities for Cross-validation in Scattered Data Approximation

TL;DR

This paper develops a concentration inequality for cross-validation scores in scattered data approximation, providing non-asymptotic probabilistic bounds that enhance understanding of model selection reliability.

Contribution

It introduces a novel concentration inequality for cross-validation in scattered data approximation, offering non-asymptotic bounds with broad applicability and precise constants for Shepard's model.

Findings

Provides a high-probability bound on cross-validation error

Applies the inequality to Shepard's model with explicit constants

Numerical experiments demonstrate practical relevance

Abstract

Choosing models from a hypothesis space is a frequent task in approximation theory and inverse problems. Cross-validation is a classical tool in the learner's repertoire to compare the goodness of fit for different reconstruction models. Much work has been dedicated to computing this quantity in a fast manner but tackling its theoretical properties occurs to be difficult. So far, most optimality results are stated in an asymptotic fashion. In this paper we propose a concentration inequality on the difference of cross-validation score and the risk functional with respect to the squared error. This gives a pre-asymptotic bound which holds with high probability. For the assumptions we rely on bounds on the uniform error of the model which allow for a broadly applicable framework. We support our claims by applying this machinery to Shepard's model, where we are able to determine precise constants of the concentration inequality. Numerical experiments in combination with fast algorithms indicate the applicability of our results.