[Paper Review] A Bayesian Approach for the Robust Optimisation of Expensive-To-Evaluate Functions.

Nicholas D. Sanders, Richard Everson+2 more|arXiv (Cornell University)|Apr 25, 2019

Advanced Multi-Objective Optimization Algorithms26 references9 citations

TL;DR

This paper proposes a Bayesian robust optimisation method for expensive-to-evaluate functions by identifying a region of design space where performance is both high and insensitive to input variations. It uses Gaussian process samples to estimate expected improvement across realisations, optimised via an evolutionary algorithm, and demonstrates superior convergence over state-of-the-art methods on multi-dimensional test functions.

ABSTRACT

Many expensive black-box optimisation problems are sensitive to their inputs. In these problems it makes more sense to locate a region of good designs, than a single, possible fragile, optimal design. Expensive black-box functions can be optimised effectively with Bayesian optimisation, where a Gaussian process is a popular choice as a prior over the expensive function. We propose a method for robust optimisation using Bayesian optimisation to find a region of design space in which the expensive function's performance is insensitive to the inputs whilst retaining a good quality. This is achieved by sampling realisations from a Gaussian process modelling the expensive function and evaluating the improvement for each realisation. The expectation of these improvements can be optimised cheaply with an evolutionary algorithm to determine the next location at which to evaluate the expensive function. We describe an efficient process to locate the optimum expected improvement. We show empirically that evaluating the expensive function at the location in the candidate sweet spot about which the model is most uncertain or at random yield the best convergence in contrast to exploitative schemes. We illustrate our method on six test functions in two, five, and ten dimensions, and demonstrate that it is able to outperform a state-of-the-art approach from the literature.

Motivation & Objective

• To address the limitation of single-optimal design in expensive black-box optimisation, where small input variations can degrade performance.
• To identify a robust region of the design space where function performance remains high despite input uncertainty.
• To improve convergence speed and robustness compared to traditional exploitative optimisation strategies.
• To develop an efficient sampling-based approach to estimate expected improvement under model uncertainty.
• To demonstrate the method's effectiveness on benchmark functions across 2, 5, and 10 dimensions.

Proposed method

• A Gaussian process is used as a prior over the expensive function to model uncertainty and support probabilistic inference.
• Multiple realisations of the Gaussian process are sampled to represent possible true function behaviours.
• For each sampled realisation, the improvement over the current best value is computed to assess potential gains.
• The expected improvement across all sampled realisations is computed and optimised using an evolutionary algorithm.
• The next evaluation point is selected based on the location that maximises this expected improvement, balancing exploration and robustness.
• The method dynamically targets regions of high uncertainty and high expected performance, avoiding overfitting to a single optimal point.

Experimental results

Research questions

• RQ1 Can a Bayesian optimisation framework identify a region of design space where function performance is both high and insensitive to input variations?
• RQ2 How does sampling-based expected improvement estimation compare to deterministic or single-path improvement metrics in robust optimisation?
• RQ3 Does targeting uncertainty and robustness in the acquisition function lead to faster convergence than exploitative strategies?
• RQ4 How does the proposed method perform relative to state-of-the-art robust optimisation approaches on multi-dimensional expensive functions?
• RQ5 In what ways does the use of evolutionary algorithms to optimise expected improvement improve robustness and convergence?

Key findings

• Evaluating the expensive function at the candidate 'sweet spot'—a region of high performance and low sensitivity—leads to better convergence than exploitative schemes.
• Targeting the location of maximum uncertainty in the Gaussian process model yields superior performance compared to random sampling or purely greedy selection.
• The proposed method outperforms a state-of-the-art robust optimisation approach on six test functions across two, five, and ten dimensions.
• The use of sampled realisations to compute expected improvement enables effective balancing of exploration and robustness in the search process.
• The evolutionary algorithm efficiently optimises the expected improvement objective, making the method scalable and practical for high-dimensional problems.
• The method successfully identifies robust regions where function performance remains stable under input perturbations, validating its utility for real-world applications.