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

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.

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