Temporal True and Surrogate Fitness Landscape Analysis for Expensive Bi-Objective Optimisation

TL;DR

This study investigates how surrogate models used in expensive multi-objective optimisation shape the fitness landscape over time. By jointly analysing true and surrogate landscape features on the bi-objective BBOB suite and tracking these features at multiple points during optimisation, it reveals that surrogate landscapes differ from the true landscape yet exhibit strong correlations, and that both can inform predictive models of algorithm performance. The work demonstrates the potential of temporal landscape analysis to guide online surrogate switching, offering a framework to improve surrogate-assisted MO-EA efficiency. It combines static and dynamic FLA, multiple surrogate models, dimensionality reduction, and random-forest performance modelling to derive actionable insights for designing better surrogate-guided search strategies.

Abstract

Many real-world problems have expensive-to-compute fitness functions and are multi-objective in nature. Surrogate-assisted evolutionary algorithms are often used to tackle such problems. Despite this, literature about analysing the fitness landscapes induced by surrogate models is limited, and even non-existent for multi-objective problems. This study addresses this critical gap by comparing landscapes of the true fitness function with those of surrogate models for multi-objective functions. Moreover, it does so temporally by examining landscape features at different points in time during optimisation, in the vicinity of the population at that point in time. We consider the BBOB bi-objective benchmark functions in our experiments. The results of the fitness landscape analysis reveals significant differences between true and surrogate features at different time points during optimisation. Despite these differences, the true and surrogate landscape features still show high correlations between each other. Furthermore, this study identifies which landscape features are related to search and demonstrates that both surrogate and true landscape features are capable of predicting algorithm performance. These findings indicate that temporal analysis of the landscape features may help to facilitate the design of surrogate switching approaches to improve performance in multi-objective optimisation.

Figures (5)

• Figure 1: A) Visualisation of the main process of the type of SA-EA that we consider in this work where the (orange) inner cycle represents a generational cycle of an EA with a selection and variational operator. Within this cycle, the solutions are evaluated using a surrogate model. After termination of the generational cycles, the (blue) outer cycle starts, where potentially good surrogate evaluated solutions are selected and evaluated with the true fitness function. B) The general methodology is composed of a static FLA and a temporal FLA where the static FLA utilizes a static sampling strategy while the temporal FLA logs the solutions during the optimisation. A feature extraction process then extracts the true and surrogate landscape features. Furthermore, the performance metric is calculated, which is used for algorithm performance prediction.
• Figure 2: t-SNE plot comparing the true features resulting from the temporal FLA to the static FLA. Each marker represents the median fitness landscape feature of a particular bbob-biobj problem. The colour of the point indicates in what phase of the evolution the landscape feature is calculated, as indicated with the colour bar.
• Figure 3: Dynamic t-SNE plot comparing fitness landscape features for the true to the surrogate fitness evaluations acquired in the temporal FLA. Each marker in the scatter plot represents the median fitness landscape feature of a particular bbob-biobj problem
• Figure 4: A) Comparing the true and surrogate landscape feature distributions yielded during different phases of evolution, where black refers to statistically significant ($p$$\leq$ 0.05) differences according to a Wilcoxon test. B) Correlations between the true and surrogate landscape feature distributions yielded during different phases of evolution. C) Difference in median between the true and surrogate landscape feature distributions yielded during different phases of evolution.
• Figure 5: Comparing the individual landscape feature distributions yielded from the static FLA to the feature distributions yielded from temporal FLA during different phases of evolution; where black refers to statistically significant according to a Mann-Whitney U test and and $p$-value $\leq$ 0.05.