Bayesian Optimization for Function-Valued Responses under Min-Max Criteria
Pouya Ahadi, Reza Marzban, Ali Adibi, Kamran Paynabar
TL;DR
This paper tackles optimizing function-valued responses under a worst-case (min–max) criterion by introducing MM-FBO, which combines FPCA-based dimensionality reduction with Gaussian process surrogates for the principal component scores. The authors derive a tractable acquisition function that blends worst-case exploitation with integrated uncertainty and prove theoretical guarantees: a discretization bound connecting the continuous and discrete objectives and a consistency result ensuring convergence of the acquisition to the true min–max objective. Empirical results on synthetic benchmarks and physics-inspired case studies in metasurface design and vapor-phase infiltration show MM-FBO consistently achieves faster, more stable convergence and significantly stronger worst-case performance than standard BO baselines. The work highlights the value of explicitly modeling functional uncertainty in design optimization and provides a practical framework for robust function-valued optimization in engineering and physics-oriented applications.
Abstract
Bayesian optimization is widely used for optimizing expensive black box functions, but most existing approaches focus on scalar responses. In many scientific and engineering settings the response is functional, varying smoothly over an index such as time or wavelength, which makes classical formulations inadequate. Existing methods often minimize integrated error, which captures average performance but neglects worst case deviations. To address this limitation we propose min-max Functional Bayesian Optimization (MM-FBO), a framework that directly minimizes the maximum error across the functional domain. Functional responses are represented using functional principal component analysis, and Gaussian process surrogates are constructed for the principal component scores. Building on this representation, MM-FBO introduces an integrated uncertainty acquisition function that balances exploitation of worst case expected error with exploration across the functional domain. We provide two theoretical guarantees: a discretization bound for the worst case objective, and a consistency result showing that as the surrogate becomes accurate and uncertainty vanishes, the acquisition converges to the true min-max objective. We validate the method through experiments on synthetic benchmarks and physics inspired case studies involving electromagnetic scattering by metaphotonic devices and vapor phase infiltration. Results show that MM-FBO consistently outperforms existing baselines and highlights the importance of explicitly modeling functional uncertainty in Bayesian optimization.