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

2026-01-10

Trans. Mach. Learn. Res.

AI Summary

This work addresses the challenge of efficiently solving semi-infinite Chebyshev center problems by proposing gradOL, the first gradient-based optimization framework for this class of problems. The approach reformulates the original semi-infinite problem into a finite-dimensional min-max optimization form, thereby enabling the use of efficient gradient-based methods. gradOL establishes the first theoretical foundation for applying gradient optimization to Chebyshev center problems and, under the assumption of strongly convex norms, leverages automatic differentiation to achieve numerically stable and scalable solutions. The framework naturally extends to general convex semi-infinite programming. Experimental results demonstrate that gradOL significantly improves both accuracy and efficiency on 34 Chebyshev center benchmarks from the CSIP library and achieves speedups of up to 4,000× over the SIPAMPL solver across 67 convex semi-infinite programming instances.

Technology Category

• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Constraint Optimization
• Machine Learning: Optimization

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

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.

Problem

• Chebyshev center
• function learning
• geometric optimization
• convex semi-infinite programming
• optimal interpolants

Innovation

• gradient-based optimization
• Chebyshev center
• semi-infinite programming
• automatic differentiation
• max-min optimization