On a Gradient Approach to Chebyshev Center Problems with Applications to Function Learning

TL;DR

This paper introduces gradOL, a gradient-based optimization framework for Chebyshev center problems, demonstrating improved accuracy, scalability, and theoretical foundations, with broad applications in function learning and convex semi-infinite programming.

Contribution

The paper presents the first gradient-based method for Chebyshev center problems, combining theoretical guarantees with empirical efficiency and extending to general convex semi-infinite programming.

Findings

Achieves up to 4000x speedup over existing solvers.

Successfully recovers optimal Chebyshev centers under strong convexity.

Demonstrates significant accuracy and efficiency improvements on benchmark problems.

Abstract

We introduce $\textsf{gradOL}$, the first gradient-based optimization framework for solving Chebyshev center problems, a fundamental challenge in optimal function learning and geometric optimization. $\textsf{gradOL}$ hinges on reformulating the semi-infinite problem as a finitary max-min optimization, making it amenable to gradient-based techniques. By leveraging automatic differentiation for precise numerical gradient computation, $\textsf{gradOL}$ ensures numerical stability and scalability, making it suitable for large-scale settings. Under strong convexity of the ambient norm, $\textsf{gradOL}$ provably recovers optimal Chebyshev centers while directly computing the associated radius. This addresses a key bottleneck in constructing stable optimal interpolants. Empirically, $\textsf{gradOL}$ achieves significant improvements in accuracy and efficiency on 34 benchmark Chebyshev center problems from a benchmark $\textsf{CSIP}$ library. Moreover, we extend $\textsf{gradOL}$ to general convex semi-infinite programming (CSIP), attaining up to $4000\times$ speedups over the state-of-the-art $\texttt{SIPAMPL}$ solver tested on the indicated $\textsf{CSIP}$ library containing 67 benchmark problems. Furthermore, we provide the first theoretical foundation for applying gradient-based methods to Chebyshev center problems, bridging rigorous analysis with practical algorithms. $\textsf{gradOL}$ thus offers a unified solution framework for Chebyshev centers and broader CSIPs.