An Analysis of the Preferences of Distribution Indicators in Evolutionary Multi-Objective Optimization

TL;DR

The paper tackles the challenge of comparing Pareto Front Approximations (PFAs) produced by Multi-Objective Evolutionary Algorithms by systematizing Distribution Indicators (DIs) and evaluating nine representative DIs across diverse assessment scenarios. It first proposes a taxonomy that classifies DIs into four classes and highlights nine indicators representing different design principles. Through extensive experiments on standard benchmark suites (DTLZ, WFG, ZCAT) and scenarios such as loss of coverage, loss of uniformity, and pathological distributions, the study reveals that some DIs can be misleading and emphasizes the robustness of SPD and RSE. The findings provide actionable guidance for practitioners in selecting reliable DIs and motivate future work to develop DIs based on Biodiversity and Potential Energy concepts with stronger theoretical grounding.

Abstract

The distribution of objective vectors in a Pareto Front Approximation (PFA) is crucial for representing the associated manifold accurately. Distribution Indicators (DIs) assess the distribution of a PFA numerically, utilizing concepts like distance calculation, Biodiversity, Entropy, Potential Energy, or Clustering. Despite the diversity of DIs, their strengths and weaknesses across assessment scenarios are not well-understood. This paper introduces a taxonomy for classifying DIs, followed by a preference analysis of nine DIs, each representing a category in the taxonomy. Experimental results, considering various PFAs under controlled scenarios (loss of coverage, loss of uniformity, pathological distributions), reveal that some DIs can be misleading and need cautious use. Additionally, DIs based on Biodiversity and Potential Energy show promise for PFA evaluation and comparison of Multi-Objective Evolutionary Algorithms.

Figures (7)

• Figure 1: Our proposed taxonomy for DIs.
• Figure 2: Three-objective PFAs $\mathcal{L}_\gamma$ with $\gamma = 0.1, 0.5, 1.0$, simulating a loss of coverage. All the PFAs have 105 points.
• Figure 3: Preferences of the nine DIs for PFAs with different degrees of Coverage.
• Figure 4: Three-objective PFAs of DTLZ1 for different degrees of Uniformity. All the PFAs have 105 points.
• Figure 5: Preferences of the nine DIs for PFAs with different degrees of Uniformity.