On a Gradient Approach to Chebyshev Center Problems with Applications to Function Learning
Abhinav Raghuvanshi, Mayank Baranwal, Debasish Chatterjee
TL;DR
This work addresses the Chebyshev center problem arising in optimal interpolation and function learning, introducing gradOL, a gradient-based solver that reformulates the semi-infinite problem into a differentiable max-min structure amenable to automatic differentiation. By solving an inner convex minimization with a log-barrier and performing outer updates via gradients of the outer objective, gradOL recovers Chebyshev centers and radii under strong convexity and delivers large empirical speedups on CSIP benchmarks. The authors establish a theoretical foundation showing local Lipschitz properties of the reformulated map and provide convergence to a generalized stationary point, while demonstrating up to 4000x speedups over SIPAMPL across 34 Chebyshev center problems and 33 CSIPs. The approach extends naturally to general convex semi-infinite programs, offering a unified, scalable framework for stable optimal interpolants and broader CSIP optimization tasks.
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.