A Deep Dive into Effects of Structural Bias on CMA-ES Performance along Affine Trajectories

TL;DR

The paper examines how structural bias affects the performance of modular CMA-ES configurations (modCMA) by exhaustively screening $435{,}456$ module combinations with the Deep-BIAS SB classifier and SHAP-based attribution. It links SB to performance on affine blends of BBOB landscapes, implemented via the Vermetten generator across 816 function variants and four optimum placements, using a fixed CMA-ES budget and 30 repeats per setting. Key findings show that modules governing covariance adaptation, elitism, threshold, bound correction, and step-size adaptation drive center- and bound-focused SB, with optimum location strongly modulating these effects; performance generally improves as the Sphere component increases but varies with landscape geometry. The study provides design guidance for robust modCMA configurations and motivates future SB analyses in higher-dimensional spaces, highlighting the nuanced interplay between algorithm structure and landscape structure in iterative optimization.

Abstract

To guide the design of better iterative optimisation heuristics, it is imperative to understand how inherent structural biases within algorithm components affect the performance on a wide variety of search landscapes. This study explores the impact of structural bias in the modular Covariance Matrix Adaptation Evolution Strategy (modCMA), focusing on the roles of various modulars within the algorithm. Through an extensive investigation involving 435,456 configurations of modCMA, we identified key modules that significantly influence structural bias of various classes. Our analysis utilized the Deep-BIAS toolbox for structural bias detection and classification, complemented by SHAP analysis for quantifying module contributions. The performance of these configurations was tested on a sequence of affine-recombined functions, maintaining fixed optimum locations while gradually varying the landscape features. Our results demonstrate an interplay between module-induced structural bias and algorithm performance across different landscape characteristics.

Figures (5)

• Figure 1: The summary of the overall methodology used. Full details of all steps are provided in the corresponding sections. Green blocks highlight the contributions of this paper.
• Figure 2: SHAP values showing module contributions to (from left to right) no structural bias, centre bias and bounds bias classes, respectively. The baseline prediction of these classes are $0.094$, $0.559$, $0.054$ respectively, meaning that a SHAP value of 0 would result in the given baseline class probability.
• Figure 3: Examples of the final best point distributions from CMA-ES configurations run on the SB test function $f_0$ in $2D$ from $500$ independent runs (using different random seed) of configurations (re)classified as no, centre, bounds and mixed SB.
• Figure 4: Example $2D$ landscapes for affine combinations of functions f3, f15, f16, f21 (top to bottom) with f1 for 5 affine weights $\alpha$ shown as labels below individual plots, where increasing $\alpha$ corresponds to increasing the proportion of f1. On the instances shown here, the location of the global optimum is fixed near the centre of the domain and marked in red.
• Figure 5: Median and variance for AUC over 30 runs for CMA-ES variants split into bias classes across increasing $\alpha$ for optimum placements (left to right) bounds, centre, centre of bounds, and random --- and for base BBOB function pairs (rows, top to bottom): f3-f1; f15-f1; f16-f1; and f21-f1.