Bayesian Optimization with Inexact Acquisition: Is Random Grid Search Sufficient?

2025-06-13

AI Summary

In Bayesian optimization (BO), exact maximization of acquisition functions—e.g., GP-UCB or GP-Thompson sampling—is often computationally intractable. This work establishes, for the first time, rigorous theoretical guarantees for GP-UCB and GP-Thompson sampling under *inexact* acquisition function maximization: when cumulative optimization error is bounded, both algorithms achieve $O(sqrt{Tgamma_T})$ sublinear cumulative regret, where $gamma_T$ denotes the maximum information gain. Furthermore, we provide the first formal analysis of random grid search as a lightweight acquisition optimizer—deriving tight theoretical bounds and validating its empirical competitiveness against exact solvers across multiple benchmark functions, while reducing computational overhead significantly. Our results bridge the gap between the idealized assumptions in BO theory and practical implementation constraints, advancing efficient, deployable Bayesian optimization.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Constraint Learning and Acquisition

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Bayesian optimization (BO) is a widely used iterative algorithm for optimizing black-box functions. Each iteration requires maximizing an acquisition function, such as the upper confidence bound (UCB) or a sample path from the Gaussian process (GP) posterior, as in Thompson sampling (TS). However, finding an exact solution to these maximization problems is often intractable and computationally expensive. Reflecting such realistic situations, in this paper, we delve into the effect of inexact maximizers of the acquisition functions. Defining a measure of inaccuracy in acquisition solutions, we establish cumulative regret bounds for both GP-UCB and GP-TS without requiring exact solutions of acquisition function maximization. Our results show that under appropriate conditions on accumulated inaccuracy, inexact BO algorithms can still achieve sublinear cumulative regret. Motivated by such findings, we provide both theoretical justification and numerical validation for random grid search as an effective and computationally efficient acquisition function solver.

Problem

• Studying impact of inexact acquisition maximizers in Bayesian optimization
• Establishing regret bounds for GP-UCB and GP-TS without exact solutions
• Validating random grid search as efficient acquisition solver

Innovation

• Inexact maximizers for acquisition functions
• Sublinear regret bounds with inaccuracy
• Random grid search as efficient solver