MO-ELA: Rigorously Expanding Exploratory Landscape Features for Automated Algorithm Selection in Continuous Multi-Objective Optimisation

TL;DR

This work addresses the scarcity of expressive exploratory landscape analysis features for box-constrained continuous multi-objective optimisation by introducing MO-ELA, a 226/233-feature suite spanning five groups (non-dominated sorting, descriptive statistics, PCA, graph-based, and gradient-based) derived from random samples in both decision and objective spaces. The features are integrated into an automated algorithm selection framework (AAS) and evaluated on a large, diverse MO benchmark set, demonstrating that the MO-ELA features substantially improve selector performance and frequently rank among the top contributors after feature selection, bringing the method close to the virtual best solver for bi-objective problems and offering significant gains for tri-objective cases. Key findings include a high stability and low redundancy of the selected MO-ELA features, with feature groups like NDS and graph-based constructs driving performance, and an analysis showing that larger sample sizes generally enhance selector RI while maintaining reasonable runtimes. Overall, the MO-ELA approach advances practical MO landscape characterisation, enabling more effective per-instance algorithm configuration and suggesting promising avenues for extension to other domains and feature-free baselines.

Abstract

Automated Algorithm Selection (AAS) is a popular meta-algorithmic approach and has demonstrated to work well for single-objective optimisation in combination with exploratory landscape features (ELA), i.e., (numerical) descriptive features derived from sampling the black-box (continuous) optimisation problem. In contrast to the abundance of features that describe single-objective optimisation problems, only a few features have been proposed for multi-objective optimisation so far. Building upon recent work on exploratory landscape features for box-constrained continuous multi-objective optimization problems, we propose a novel and complementary set of additional features (MO-ELA). These features are based on a random sample of points considering both the decision and objective space. The features are divided into 5 feature groups depending on how they are being calculated: non-dominated-sorting, descriptive statistics, principal component analysis, graph structures and gradient information. An AAS study conducted on well-established multi-objective benchmarks demonstrates that the proposed features contribute to successfully distinguishing between algorithm performance and thus adequately capture problem hardness resulting in models that come very close to the virtual best solver. After feature selection, the newly proposed features are frequently among the top contributors, underscoring their value in algorithm selection and problem characterisation.

Figures (10)

• Figure 1: Visualisation of the MO-ELA approach for a function $f : \mathcal{X} \subseteq \mathbb{R}^2 \to \mathbb{R}^2$. The decision space with six sampled points $x_1, \ldots, x_6$ is shown to the left while the respective objective space is depicted to the right. Points are coloured by non-domination layers. Here, $L_1 = \{(x_1, y_1), (x_4, y_4), (x_5, y_5)\}$, $L_2 = \{(x_2, y_2), (x_3, y_3)\}$ and $L_3 = \{(x_6, y_6)\}$. Edges in the left plot correspond to the edges of the MST $G^{\text{ST}}_D$ while the edges in the right plot illustrate the respective spanning tree $G^{\text{ST}}_{D \to O}$.
• Figure 2: Domination layers for BiObjBBOB functions $1$ and $86$ and polynomial regression of the corresponding HV values with degrees $1$ and $4$.
• Figure 3: Overview of the algorithm selection training framework for multi-objective problems. The green indicated box highlights our contribution.
• Figure 4: Contingency table for the different combinations of benchmark, dimension and number of objectives.
• Figure 5: Comparison of the predictions of the best selector against the SBS on the test set containing problems of all dimensions and sample sizes.