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Structural bias in multi-objective optimisation

Jakub Kudela, Niki van Stein, Thomas Bäck, Anna V. Kononova

TL;DR

This work extends the notion of structural bias, previously studied in single-objective optimisation, to multi-objective optimisation by decoupling algorithmic search dynamics from fitness guidance using synthetic, bi-objective test problems with uninformative objectives. It adapts the SB detection toolkit (BIAS) for MOOs, introducing a suite of analyzers—chi-squared tests, binsize inspection, and clustering via the Clark-Evans Index—to assess where in the decision space algorithms preferentially search. Across a large set of MO algorithms implemented in PlatEMO, the study finds clear SB manifestations (centre, boundary, or mixed biases) that correlate with PF density and bound handling, but are largely independent of PF shape. The results highlight the importance of incorporating behaviour-based benchmarking into MO optimization to understand when and why algorithms are biased, informing more robust design. The work also outlines contributions and future directions, including extending to higher-dimensional objective spaces and linking SB to real-performance outcomes.

Abstract

Structural bias (SB) refers to systematic preferences of an optimisation algorithm for particular regions of the search space that arise independently of the objective function. While SB has been studied extensively in single-objective optimisation, its role in multi-objective optimisation remains largely unexplored. This is problematic, as dominance relations, diversity preservation and Pareto-based selection mechanisms may introduce or amplify structural effects. In this paper, we extend the concept of structural bias to the multi-objective setting and propose a methodology to study it in isolation from fitness-driven guidance. We introduce a suite of synthetic multi-objective test problems with analytically controlled Pareto fronts and deliberately uninformative objective values. These problems are designed to decouple algorithmic behaviour from problem structure, allowing bias induced purely by algorithmic operators and design choices to be observed. The test suite covers a range of Pareto front shapes, densities and noise levels, enabling systematic analysis of different manifestations of structural bias. We discuss methodological challenges specific to the multi-objective case and outline how existing SB detection approaches can be adapted. This work provides a first step towards behaviour-based benchmarking of multi-objective optimisers, complementing performance-based evaluation and informing more robust algorithm design.

Structural bias in multi-objective optimisation

TL;DR

This work extends the notion of structural bias, previously studied in single-objective optimisation, to multi-objective optimisation by decoupling algorithmic search dynamics from fitness guidance using synthetic, bi-objective test problems with uninformative objectives. It adapts the SB detection toolkit (BIAS) for MOOs, introducing a suite of analyzers—chi-squared tests, binsize inspection, and clustering via the Clark-Evans Index—to assess where in the decision space algorithms preferentially search. Across a large set of MO algorithms implemented in PlatEMO, the study finds clear SB manifestations (centre, boundary, or mixed biases) that correlate with PF density and bound handling, but are largely independent of PF shape. The results highlight the importance of incorporating behaviour-based benchmarking into MO optimization to understand when and why algorithms are biased, informing more robust design. The work also outlines contributions and future directions, including extending to higher-dimensional objective spaces and linking SB to real-performance outcomes.

Abstract

Structural bias (SB) refers to systematic preferences of an optimisation algorithm for particular regions of the search space that arise independently of the objective function. While SB has been studied extensively in single-objective optimisation, its role in multi-objective optimisation remains largely unexplored. This is problematic, as dominance relations, diversity preservation and Pareto-based selection mechanisms may introduce or amplify structural effects. In this paper, we extend the concept of structural bias to the multi-objective setting and propose a methodology to study it in isolation from fitness-driven guidance. We introduce a suite of synthetic multi-objective test problems with analytically controlled Pareto fronts and deliberately uninformative objective values. These problems are designed to decouple algorithmic behaviour from problem structure, allowing bias induced purely by algorithmic operators and design choices to be observed. The test suite covers a range of Pareto front shapes, densities and noise levels, enabling systematic analysis of different manifestations of structural bias. We discuss methodological challenges specific to the multi-objective case and outline how existing SB detection approaches can be adapted. This work provides a first step towards behaviour-based benchmarking of multi-objective optimisers, complementing performance-based evaluation and informing more robust algorithm design.
Paper Structure (24 sections, 6 equations, 11 figures, 3 tables)

This paper contains 24 sections, 6 equations, 11 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Structural bias
  3. Existing methodology for measuring SB in the single-objective setting
  4. BIAS toolbox
  5. Structural bias in multi-objective optimisation algorithms
  6. Choice of test problems
  7. Data collected from algorithms' runs
  8. Proposed methodology of testing for SB in the MOO setting
  9. BIAS toolbox
  10. Chi-squared test
  11. Binsize inspection
  12. Clustering
  13. Experimental setup
  14. Discussion
  15. Results aggregated across PlatEMO
  16. ...and 9 more sections

Figures (11)

  • Figure 1: Proposed problems in 2d with the number of Pareto solutions in a sample of $10^4$ feasible solutions.
  • Figure 2: Relationship between sample size and number of non-dominated points for the proposed functions.
  • Figure 3: Examples of typical histograms of the aggregation of the results, with four bins of interest indicated in a different colour, see Section \ref{['sect:visual']} for explanation.
  • Figure 4: Visual representation of CEI values, see Section \ref{['sect:cluster']} for explanation.
  • Figure 5: Distribution of values of the tracked parameters used for detection of structural bias, $d=10$ (The corresponding figure for $d=2$ can be found in the supplementary material).
  • ...and 6 more figures