Fitness-based Linkage Learning and Maximum-Clique Conditional Linkage Modelling for Gray-box Optimization with RV-GOMEA

TL;DR

The paper tackles gray-box optimization where partial evaluations are feasible but the Variable Interaction Graph (VIG) is not known in advance. It combines fitness-based Dependency Strength Matrix (DSM) learning with conditional linkage models in RV-GOMEA and introduces clique-seeding to better capture overlapping dependencies. The results show that learned conditional linkage models perform on par with or better than VkD-CMA on most problems, with negligible overhead, and that clique seeding can provide additional gains. This work enables efficient, structure-aware optimization in real-world problems and supports transferring problem-structure insights to related instances.

Abstract

For many real-world optimization problems it is possible to perform partial evaluations, meaning that the impact of changing a few variables on a solution's fitness can be computed very efficiently. It has been shown that such partial evaluations can be excellently leveraged by the Real-Valued GOMEA (RV-GOMEA) that uses a linkage model to capture dependencies between problem variables. Recently, conditional linkage models were introduced for RV-GOMEA, expanding its state-of-the-art performance even to problems with overlapping dependencies. However, that work assumed that the dependency structure is known a priori. Fitness-based linkage learning techniques have previously been used to detect dependencies during optimization, but only for non-conditional linkage models. In this work, we combine fitness-based linkage learning and conditional linkage modelling in RV-GOMEA. In addition, we propose a new way to model overlapping dependencies in conditional linkage models to maximize the joint sampling of fully interdependent groups of variables. We compare the resulting novel variant of RV-GOMEA to other variants of RV-GOMEA and VkD-CMA on 12 problems with varying degree of overlapping dependencies. We find that the new RV-GOMEA not only performs best on most problems, also the overhead of learning the conditional linkage models during optimization is often negligible.

Figures (7)

• Figure 1: A case study of fitness-based linkage learning: the REBGrid problem described in Section \ref{['sec:experiments:problems']}, with $\ell=9$.
• Figure 2: VIG factorizations and derived conditional distributions, on the REBGrid problem ($\ell=9$). Shown are the original MCond partitioning technique (starting from randomly selected vertex 8), in (a) and (b), and the proposed clique seeding technique, in (c) and (d).
• Figure 3: Required corrected number of evaluations for different linkage models on $f^{REB}(\bm{x}, c, \theta, k=2, s=1)$ with $\ell=20$, as determined by bisection. The median of 5 repeats is shown.
• Figure 4: Heatmaps depicting a part ($\mathcal{I}_{[0:19]} \times \mathcal{I}_{[0:19]}$) of the computed DSM for each tested problem at a dimensionality $\ell > 30$. Variable pairs which are deemed independent are depicted in white.
• Figure 5: Results of scalability experiments for the tested benchmark problems and optimization approaches at different dimensionalities, as determined by bisection. The mean over 5 repeated bisections is shown.