Bayesian Inverse Transfer in Evolutionary Multiobjective Optimization

TL;DR

This work introduces inverse transfer in multiobjective optimization and the invTrEMO optimizer, which uses Bayesian inverse Gaussian processes to map objective-space performance to target decision spaces. By unifying disparate tasks through a common objective space, invTrEMO enables knowledge transfer even when source and target decision spaces differ, and simultaneously yields high-precision inverse models for on-demand Pareto-optimal solutions. Empirical evidence across eight benchmark problems and a vehicle crashworthiness case shows faster convergence and robust inverse-model accuracy, with results strengthening as source-target correlations increase and more decision variables overlap. The approach broadens transfer optimization to heterogeneous settings and offers a practical tool for efficient, preference-driven multiobjective decision-making.

Abstract

Transfer optimization enables data-efficient optimization of a target task by leveraging experiential priors from related source tasks. This is especially useful in multiobjective optimization settings where a set of trade-off solutions is sought under tight evaluation budgets. In this paper, we introduce a novel concept of \textit{inverse transfer} in multiobjective optimization. Inverse transfer stands out by employing Bayesian inverse Gaussian process models to map performance vectors in the objective space to population search distributions in task-specific decision space, facilitating knowledge transfer through objective space unification. Building upon this idea, we introduce the first Inverse Transfer Evolutionary Multiobjective Optimizer (invTrEMO). A key highlight of invTrEMO is its ability to harness the common objective functions prevalent in many application areas, even when decision spaces do not precisely align between tasks. This allows invTrEMO to uniquely and effectively utilize information from heterogeneous source tasks as well. Furthermore, invTrEMO yields high-precision inverse models as a significant byproduct, enabling the generation of tailored solutions on-demand based on user preferences. Empirical studies on multi- and many-objective benchmark problems, as well as a practical case study, showcase the faster convergence rate and modelling accuracy of the invTrEMO relative to state-of-the-art evolutionary and Bayesian optimization algorithms. The source code of the invTrEMO is made available at https://github.com/LiuJ-2023/invTrEMO.

Figures (14)

• Figure 1: An illustration of the assumed relationship between the source and the target tasks in our problem setup. The common objective space provides the unification through which knowledge transfer could take place, in the inverse setting, between the subset of variables that overlap. Decision variables in distinct tasks are said to overlap if they bear the same physical interpretation in the application domain of interest.
• Figure 2: Influence of $\delta_1$ and $\delta_2$ on source-target correlations. (a) Pearson correlation coefficients between mDTLZ2$^{-1}$-$(1,0)$ and mDTLZ2$^{-1}$-$(\delta_1,0)$ when $\delta_1$ is set to 0.3, 0.7, or 0.9. (b) Pearson correlation coefficients between mDTLZ2$^{-1}$-$(1,0)$ and mDTLZ2$^{-1}$-$(1,\delta_2)$ when $\delta_2$ is set to 0.05, 0.25, or 0.4.
• Figure 3: Comparison of IGD convergence trends averaged over 20 independent runs of ParEGO-UCB, MOEA/D-EGO, K-RVEA, CSEA, qNEHVI, PSL-MOBO, and the invTrEMO. Shaded areas represent one standard deviation on either side of the mean. (a) mDTLZ1-($1,0$). (b) mDTLZ2-($1,0$). (c) mDTLZ3-($1,0$). (d) mDTLZ4-($1,0$). (e) mDTLZ1$^{-1}$-($1,0$). (f) mDTLZ2$^{-1}$-($1,0$). (g) mDTLZ3$^{-1}$-($1,0$). (h) mDTLZ4$^{-1}$-($1,0$).
• Figure 4: Comparisons of RMSE results provided by the inverse models of invTrEMO-HS, invTrEMO, invTrEMO-LS, and invTrEMO-ZeroT on mDTLZ2-($1,0$) and mDTLZ4-($1,0$). invTrEMO-HS, invTrEMO, and invTrEMO-LS indicate that the HS, MS, or LS source datasets are used, respectively. Additionally, no source MOP data is employed in invTrEMO-ZeroT. (a) mDTLZ2-($1,0$). (b) mDTLZ4-($1,0$).
• Figure 5: RMSE results and IGD convergence trends of invTrEMO with different numbers of overlapping decision variables. invTrEMO-HS-3D, invTrEMO-HS-5D, and invTrEMO-HS-8D denote that the decision variables of the source tasks overlap with the first 3, 5, or 8 decision variables of the target tasks, respectively. The shaded areas represent one standard deviation in performance on either side of the mean. (a) RMSE results on mDTLZ2-($1,0$). (b) RMSE results on mDTLZ4-($1,0$). (c) IGD convergence trends on mDTLZ2-($1,0$). (d) IGD convergence trends on mDTLZ4-($1,0$).

Theorems & Definitions (6)

• definition 1: Pareto Dominance
• definition 2: Pareto Optimal Solution
• definition 3: Pareto Set
• definition 4: Pareto Front
• proposition 1
• proof