A numerical algorithm for attaining the Chebyshev bound in optimal learning

TL;DR

This paper introduces a numerically feasible algorithm for finding the Chebyshev center in optimal learning, enabling minimal-radius data fitting in Banach spaces, with practical examples demonstrating its effectiveness.

Contribution

It develops an extended algorithm for computing Chebyshev centers in non-convex hypothesis spaces within Banach spaces, advancing optimal recovery methods.

Findings

Algorithm effectively computes Chebyshev centers in finite-dimensional Banach spaces.

Numerical examples demonstrate the algorithm's practical utility.

The method extends recent convex semi-infinite problem solutions.

Abstract

Given a compact subset of a Banach space, the Chebyshev center problem consists of finding a minimal circumscribing ball containing the set. In this article we establish a numerically tractable algorithm for solving the Chebyshev center problem in the context of optimal learning from a finite set of data points. For a hypothesis space realized as a compact but not necessarily convex subset of a finite-dimensional subspace of some underlying Banach space, this algorithm computes the Chebyshev radius and the Chebyshev center of the hypothesis space, thereby solving the problem of optimal recovery of functions from data. The algorithm itself is based on, and significantly extends, recent results for near-optimal solutions of convex semi-infinite problems by means of targeted sampling, and it is of independent interest. Several examples of numerical computations of Chebyshev centers are included in order to illustrate the effectiveness of the algorithm.