[Paper Review] Practical Batch Bayesian Optimization for Less Expensive Functions

Loc Nguyen, Sunil Gupta+3 more|arXiv (Cornell University)|Nov 5, 2018

Advanced Bandit Algorithms ResearchDecision Sciences22 references3 citations

TL;DR

This paper proposes UCB-DE, a practical batch Bayesian optimization method for less expensive black-box functions, using a distance-based exploration strategy guided by Sobol sequences to avoid costly global optimization for each batch point. It achieves state-of-the-art performance with significantly reduced computation time—up to 6× faster than BUCB—while maintaining competitive convergence, especially in low-to-moderate cost evaluation settings.

ABSTRACT

Bayesian optimization (BO) and its batch extensions are successful for optimizing expensive black-box functions. However, these traditional BO approaches are not yet ideal for optimizing less expensive functions when the computational cost of BO can dominate the cost of evaluating the blackbox function. Examples of these less expensive functions are cheap machine learning models, inexpensive physical experiment through simulators, and acquisition function optimization in Bayesian optimization. In this paper, we consider a batch BO setting for situations where function evaluations are less expensive. Our model is based on a new exploration strategy using geometric distance that provides an alternative way for exploration, selecting a point far from the observed locations. Using that intuition, we propose to use Sobol sequence to guide exploration that will get rid of running multiple global optimization steps as used in previous works. Based on the proposed distance exploration, we present an efficient batch BO approach. We demonstrate that our approach outperforms other baselines and global optimization methods when the function evaluations are less expensive.

Motivation & Objective

• To address the gap in batch Bayesian optimization for functions where evaluations are not highly expensive, but computational cost of BO itself becomes a bottleneck.
• To design a scalable batch BO algorithm that minimizes computational overhead while maintaining effective exploration and exploitation.
• To replace iterative global optimization for batch selection with a single global optimization step followed by space-filling via Sobol sequences.
• To validate the method on real-world applications with moderate evaluation costs, such as simulator-based experiments and acquisition function optimization.
• To demonstrate that computational efficiency can be significantly improved without sacrificing optimization performance in less expensive function settings.

Proposed method

• Proposes a novel distance exploration (DE) strategy that selects batch points based on geometric distance from existing observations to encourage exploration.
• Uses Sobol sequences to efficiently generate space-filling points that are far from observed locations, replacing multiple global optimization steps.
• Introduces UCB-DE, a two-stage batch BO algorithm: first point selected via standard UCB acquisition, remaining points via DE using Sobol sequences.
• Leverages the low-discrepancy property of Sobol sequences to ensure even coverage of the search space without retraining the GP model for each point.
• Avoids repeated GP posterior updates and complex optimization for each batch point, reducing computational cost compared to BUCB, DPP, and LP methods.
• Employs a single global optimization for the first point (via UCB) and deterministic sampling via Sobol sequences for the rest, enabling fast and scalable batch generation.

Experimental results

Research questions

• RQ1 Can a batch Bayesian optimization method be designed to be computationally efficient when function evaluations are not highly expensive?
• RQ2 Does a distance-based exploration strategy using Sobol sequences outperform iterative global optimization in batch selection for less expensive functions?
• RQ3 Can a single global optimization step followed by deterministic space-filling achieve better computational efficiency than existing batch BO methods?
• RQ4 How does the proposed UCB-DE method compare to baselines in terms of convergence speed and optimization performance on real-world, moderately expensive functions?
• RQ5 To what extent does the use of Sobol sequences maintain exploration quality while reducing computational cost in batch Bayesian optimization?

Key findings

• UCB-DE achieves the best optimization performance in the shortest time across all benchmark and real-world applications, outperforming BUCB, L-BFGS-B, and Direct methods.
• For the PES acquisition function optimization, UCB-DE reduces optimization time from 800 seconds to 100 seconds for the same optimal value, achieving a 6× speedup.
• With increasing batch size, UCB-DE’s computation time remains nearly constant, while BUCB’s time increases significantly due to repeated global optimization steps.
• In 6D Hartmann function experiments, UCB-DE is 3–6 times faster than BUCB while achieving comparable or better convergence to the optimal value.
• The method maintains high performance even in high-dimensional settings (d=10), demonstrating robustness and scalability.
• Random search with UCB (UCB-Rand) performs surprisingly well, but UCB-DE still outperforms it in both speed and accuracy, especially in complex or high-dimensional problems.