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Knowledge Gradient for Multi-Objective Bayesian Optimization with Decoupled Evaluations

Jack M. Buckingham, Sebastian Rojas Gonzalez, Juergen Branke

TL;DR

This work extends multi-objective Bayesian optimization to decoupled objective evaluations with known costs by introducing the Cost-Weighted Multi-Objective Knowledge Gradient (C-MOKG). The acquisition function divides the multi-objective knowledge gradient by the corresponding evaluation cost and averages over linear scalarizations, enabling cost-efficient learning of the Pareto front. The authors prove non-negativity and asymptotic consistency of the method under both random and expectation over scalarizations and demonstrate competitive performance against HVKG and JES-LB on synthetic problems, with clear benefits from exploiting objective decoupling. The approach provides a principled, data-efficient framework for optimizing heterogeneous, expensive objectives in settings where objective costs vary across evaluations.

Abstract

Multi-objective Bayesian optimization aims to find the Pareto front of trade-offs between a set of expensive objectives while collecting as few samples as possible. In some cases, it is possible to evaluate the objectives separately, and a different latency or evaluation cost can be associated with each objective. This decoupling of the objectives presents an opportunity to learn the Pareto front faster by avoiding unnecessary, expensive evaluations. We propose a scalarization based knowledge gradient acquisition function which accounts for the different evaluation costs of the objectives. We prove asymptotic consistency of the estimator of the optimum for an arbitrary, D-dimensional, real compact search space and show empirically that the algorithm performs comparably with the state of the art and significantly outperforms versions which always evaluate both objectives.

Knowledge Gradient for Multi-Objective Bayesian Optimization with Decoupled Evaluations

TL;DR

This work extends multi-objective Bayesian optimization to decoupled objective evaluations with known costs by introducing the Cost-Weighted Multi-Objective Knowledge Gradient (C-MOKG). The acquisition function divides the multi-objective knowledge gradient by the corresponding evaluation cost and averages over linear scalarizations, enabling cost-efficient learning of the Pareto front. The authors prove non-negativity and asymptotic consistency of the method under both random and expectation over scalarizations and demonstrate competitive performance against HVKG and JES-LB on synthetic problems, with clear benefits from exploiting objective decoupling. The approach provides a principled, data-efficient framework for optimizing heterogeneous, expensive objectives in settings where objective costs vary across evaluations.

Abstract

Multi-objective Bayesian optimization aims to find the Pareto front of trade-offs between a set of expensive objectives while collecting as few samples as possible. In some cases, it is possible to evaluate the objectives separately, and a different latency or evaluation cost can be associated with each objective. This decoupling of the objectives presents an opportunity to learn the Pareto front faster by avoiding unnecessary, expensive evaluations. We propose a scalarization based knowledge gradient acquisition function which accounts for the different evaluation costs of the objectives. We prove asymptotic consistency of the estimator of the optimum for an arbitrary, D-dimensional, real compact search space and show empirically that the algorithm performs comparably with the state of the art and significantly outperforms versions which always evaluate both objectives.
Paper Structure (24 sections, 3 theorems, 25 equations, 4 figures)

This paper contains 24 sections, 3 theorems, 25 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Related Work
  3. Problem definition
  4. Background
  5. Bayesian Optimization
  6. Multi-Attribute Knowledge Gradient (maKG)
  7. Cost Weighted Multi-Objective Knowledge Gradient
  8. Efficient Calculation and Optimization
  9. Discrete Approximation
  10. Quasi-Monte-Carlo
  11. Theoretical Results
  12. Experiments
  13. Synthetic Problems
  14. Bayesian Regret Performance Metric
  15. Hypervolume performance metric
  16. ...and 9 more sections

Key Result

lemma thmcounterlemma

Both forms of the cost-aware multi-objective knowledge gradient are non-negative. That is, for all $\bm{x} \in \mathcal{X}$, $m \in \{1, \dots, M\}$ and all $\bm{\lambda} \in \Lambda$, almost surely.

Figures (4)

  • Figure 1: A comparison of the evolution of the expected Bayesian regret between C-MOKG/maKG (blue circles), HVKG (red diamonds) and JES-LB (green triangles). For HVKG and JES-LB, a suffix of '-c' or '-d' is used to distinguish the coupled and decoupled algorithms, respectively. The coupled versions are solid while the decoupled versions are dashed. The shaded areas show a 95% confidence interval in the expected value across the family of test problems (two standard errors around the mean).
  • Figure 2: Results from the experiments in \ref{['fig:results-bayesregret']}, using hypervolume regret.
  • Figure 3: Convergence of the GP surrogate as samples are collected with C-MOKG.
  • Figure 4: A comparison of the expected Bayesian regret when using random scalarizations (orange triangles) and expectation over scalarizations (blue circles), for the first family of test problems. The decoupled C-MOKG (dashed), significantly benefits from using expectation over scalarizations.

Theorems & Definitions (4)

  • lemma thmcounterlemma
  • theorem thmcountertheorem: Consistency of C-MOKG
  • definition thmcounterdefinition
  • lemma thmcounterlemma