On the convergence of noisy Bayesian Optimization with Expected Improvement (2501.09262v1)

Abstract: Expected improvement (EI) is one of the most widely-used acquisition functions in Bayesian optimization (BO). Despite its proven success in applications for decades, important open questions remain on the theoretical convergence behaviors and rates for EI. In this paper, we contribute to the convergence theories of EI in three novel and critical area. First, we consider objective functions that are under the Gaussian process (GP) prior assumption, whereas existing works mostly focus on functions in the reproducing kernel Hilbert space (RKHS). Second, we establish the first asymptotic error bound and its corresponding rate for GP-EI with noisy observations under the GP prior assumption. Third, by investigating the exploration and exploitation of the non-convex EI function, we prove improved error bounds for both the noise-free and noisy cases. The improved noiseless bound is extended to the RKHS assumption as well.

On the Convergence of Noisy Bayesian Optimization with Expected Improvement

The paper, "On the convergence of noisy Bayesian Optimization with Expected Improvement," presents significant advancements in the theoretical understanding of Bayesian Optimization (BO) using the Expected Improvement (EI) acquisition function. The research addresses substantial gaps in the asymptotic convergence theory of the EI method when dealing with noisy observations, developing analytical frameworks applicable under Gaussian process (GP) prior assumptions for the objective functions. This contribution is pivotal for extending Bayesian Optimization's applicability across various domains such as machine learning, robotics, and structural design, especially in scenarios where noise is inevitable.

Key Contributions

The authors make three critical contributions to the literature on Bayesian Optimization:

1. GP Prior Assumptions for Objective Functions: Unlike previous studies that often assume the objective functions lie within a reproducing kernel Hilbert space (RKHS), this paper considers functions under the Gaussian process prior assumption. This approach reflects a more practical scenario where objective functions are expected to exhibit properties of a Gaussian distribution.
2. Asymptotic Error Bounds with Noisy Observations: The paper establishes the first asymptotic error bounds for GP-EI with noisy observations. Under the GP prior assumption, it shows that the expected improvement method can achieve upper bounds on errors, thus assuring convergence.
3. Improved Error Bounds for Exploration and Exploitation: The paper achieves improved error bounds by examining the EI function's non-convex properties, specifically focusing on the balance between exploration (via the posterior standard deviation) and exploitation (considering the noise-free best observed sample value). These findings further demonstrate convergence both with and without noise, extending to the scenario where the objective function lies in RKHS.

Numerical Results and Verification

The paper incorporates advanced mathematical techniques and numerical experiments to validate the theoretical claims. By developing new lemmas and leveraging existing theories from information gain metrics and GP properties, the authors provide a robust analytical surface for assessing convergence rates. With comprehensive proofs, they derive improved bounds that apply to both squared exponential (SE) and Matérn kernels.

Implications for Future Research and Applications

The implications of this research extend beyond theoretical contributions to practical applications. The paper's findings can improve BO applications by offering tighter convergence guarantees in noisy environments. These error bounds could influence how practitioners set parameters in BO frameworks and improve convergence rates in real-world applications such as hyperparameter tuning and optimization under uncertainty.

Additionally, the methodology can be expanded to include other acquisition functions or different kernel choices in Gaussian processes, further broadening the applicability of BO with EI. Future work could involve the exploration of different noise models or extending this analysis to multi-fidelity optimization problems.

In summary, this paper provides a rigorous exploration into Bayesian Optimization, moving closer to understanding and applying the Expected Improvement method in noisy settings. It lays a mathematical foundation that supports further exploration into noise-robust optimization methods, promising enhanced outcomes in numerous scientific and engineering applications where uncertainty and noise are prevalent.