Bayesian Optimization for Function-Valued Responses under Min-Max Criteria

2025-11-26

AI Summary

Existing Bayesian optimization (BO) methods for expensive black-box functions with functional responses—e.g., time- or wavelength-resolved physical outputs—are primarily designed for scalar objectives and optimize average error (e.g., integrated error), failing to guarantee worst-case performance. Method: We propose min-max Functional Bayesian Optimization (MM-FBO), the first BO framework that directly minimizes the maximum pointwise error of the response function over its entire domain. MM-FBO employs functional principal component analysis (FPCA) for dimensionality reduction, models FPCA scores via Gaussian processes, and introduces an ensemble uncertainty-aware acquisition function balancing worst-case deviation suppression and functional-domain exploration. Contribution/Results: We provide theoretical bounds on discretization error and prove consistency and convergence. Experiments on synthetic benchmarks and real-world physics tasks—including electromagnetic scattering and gas-phase permeation—demonstrate significant improvements over state-of-the-art methods, validating the critical role of explicit functional uncertainty modeling for robust optimization.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Sampling/Simulation-based Search
• Machine Learning: Calibration & Uncertainty Quantification

Application Category

• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

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.

Problem

• Optimizes functional responses under min-max criteria
• Addresses worst-case error in Bayesian optimization
• Models functional uncertainty for scientific applications

Innovation

• Min-max functional Bayesian optimization framework
• Functional principal component analysis for responses
• Integrated uncertainty acquisition function balancing