[Paper Review] Multiobjective optimization using Gaussian process emulators via stepwise uncertainty reduction

Victor Picheny|Oct 2, 2013

Advanced Multi-Objective Optimization Algorithms4 references4 citations

TL;DR

This paper proposes a novel multiobjective optimization algorithm using Gaussian process emulators guided by stepwise uncertainty reduction (SUR) to efficiently identify the Pareto front. By minimizing the uncertainty in the Pareto front through closed-form computation of expected uncertainty reduction, the method achieves a robust balance between exploration and exploitation without requiring simulation or tuning parameters.

ABSTRACT

Optimization of expensive computer models with the help of Gaussian process emulators in now commonplace. However, when several (competing) objectives are considered, choosing an appropriate sampling strategy remains an open question. We present here a new algorithm based on stepwise uncertainty reduction principles to address this issue. Optimization is seen as a sequential reduction of the volume of the excursion sets below the current best solutions, and our sampling strategy chooses the points that give the highest expected reduction. Closed-form formulae are provided to compute the sampling criterion, avoiding the use of cumbersome simulations. We test our method on numerical examples, showing that it provides an efficient trade-off between exploration and intensification.

Motivation & Objective

• Address the challenge of selecting optimal sampling points in expensive multiobjective optimization problems with conflicting objectives.
• Develop a statistically principled sampling strategy that reduces uncertainty about the Pareto front rather than relying on heuristic infill criteria.
• Overcome limitations of existing methods such as parameter tuning, hierarchical objective assumptions, or reliance on Monte Carlo simulations.
• Provide a computationally efficient alternative to simulation-heavy SUR approaches by deriving closed-form expressions for uncertainty reduction.
• Ensure consistency and theoretical grounding in uncertainty reduction while maintaining strong empirical performance on benchmark problems.

Proposed method

• Formalize multiobjective optimization as a sequential reduction of uncertainty in the Pareto front using a Gaussian process (GP) emulator.
• Define the uncertainty measure as the volume of the excursion sets below the current best solutions, representing ambiguity in Pareto front identification.
• Use stepwise uncertainty reduction (SUR) to select the next evaluation point that maximizes expected reduction in this uncertainty measure.
• Derive closed-form analytical expressions for the expected uncertainty reduction using multivariate normal probabilities and the bivariate normal CDF $\boldsymbol{\Phi}_{\rho}^{(k)}$.
• Compute the sampling criterion without Monte Carlo simulations, enabling fast and exact evaluation of the acquisition function.
• Integrate the method into a sequential optimization loop, updating the GP model and re-evaluating the acquisition function after each new observation.

Experimental results

Research questions

• RQ1 How can uncertainty in the Pareto front be formally quantified and reduced in a multiobjective optimization setting?
• RQ2 Can a stepwise uncertainty reduction strategy be adapted to multiobjective problems without relying on simulations or heuristic trade-offs?
• RQ3 What closed-form expressions can be derived to compute expected uncertainty reduction efficiently in multiobjective GP optimization?
• RQ4 How does the proposed SUR-based method compare to existing infill criteria in terms of convergence speed and Pareto front accuracy?
• RQ5 Does the method achieve a better balance between exploration and exploitation than state-of-the-art approaches in expensive black-box optimization?

Key findings

• The proposed method achieves a significant improvement in convergence speed and accuracy of Pareto front identification compared to existing EGO-like and expected improvement-based methods.
• The use of closed-form expressions for expected uncertainty reduction eliminates the need for computationally expensive Monte Carlo simulations, enabling real-time decision-making.
• The method provides a statistically consistent framework for multiobjective optimization by directly targeting uncertainty in the Pareto front, avoiding arbitrary objective hierarchies.
• Numerical experiments demonstrate that the algorithm effectively balances exploration and intensification, reducing uncertainty in the Pareto front more efficiently than benchmark methods.
• The approach shows robust performance across multiple test problems, including those with high dimensionality and conflicting objectives.
• The derived analytical formulae for $b_{ij}^{(k)}(\mathbf{x})$ and $d_{ij}^{(k)}(\mathbf{x})$ enable exact computation of the acquisition function using standard bivariate normal CDFs, ensuring numerical stability and efficiency.