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

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.

💡 Research Summary

Bayesian optimization (BO) has become the workhorse for optimizing expensive black‑box functions, yet virtually all existing BO frameworks assume a scalar objective. In many scientific and engineering domains the response is a smooth function of an index such as time, wavelength, or spatial coordinate, and the quality of a design is often dictated not by the average error but by the worst deviation over the entire functional domain. The paper “Bayesian Optimization for Function‑Valued Responses under Min‑Max Criteria” introduces a principled solution to this gap by formulating a true min‑max objective for functional responses and by developing a surrogate‑based acquisition strategy that explicitly targets the worst‑case error while still encouraging global exploration.

The methodological core consists of two stages. First, the functional output y(t;x) (t∈𝒯, x∈𝒳) is projected onto a low‑dimensional basis using functional principal component analysis (FPCA): y(t;x)≈∑_{k=1}^K a_k(x) φ_k(t). The basis functions φ_k(t) are computed once from a modest training set, and the scores a_k(x) capture all variability relevant to optimization. Second, each score a_k(x) is modeled independently with a Gaussian process (GP_k). This decomposition avoids the prohibitive cost of building a single GP over the entire high‑dimensional functional space while preserving a full probabilistic description of uncertainty for every point in the index set.

On top of this representation the authors propose the Integrated Uncertainty Acquisition Function (IUAF). IUAF combines (i) a conservative estimate of the worst‑case expected error, given by max_{t∈𝒯} μ_t(x) + β·max_{t∈𝒯} σ_t(x), where μ_t(x) and σ_t(x) are the mean and standard deviation of the GP‑induced prediction for the score at index t, and (ii) a global exploration term, the integral of σ_t(x) over the whole domain, ∫_𝒯 σ_t(x) dt. The final acquisition value is a weighted sum α·