Improving population size adapting CMA-ES algorithm on step-size blow-up in weakly-structured multimodal functions (2506.00825v1)

Abstract: Multimodal optimization requires both exploration and exploitation. Exploration identifies promising attraction basins, while exploitation finds the best solutions within these basins. The balance between exploration and exploitation can be maintained by adjusting parameter settings. The population size adaptation covariance matrix adaption evolutionary strategy algorithm (PSA-CMA-ES) achieves this balance by dynamically adjusting population size. PSA-CMA-ES performs well on well-structured multimodal benchmark problems. In weakly structured multimodal problems, however, the algorithm struggles to effectively manage step-size increases, resulting in uncontrolled step-size blow-ups that impede convergence near the global optimum. In this study, we reformulated the step-size correction strategy to overcome this limitation. We analytically identified the cause of the step-size blow-up and demonstrate the existence of a significance level for population size change guiding a safe passage to step-size correction. These insights were incorporated to form the reformulation. Through computer experiments on two weakly structured multimodal benchmark problems, we evaluated the performance of the new approach and compared the results with the state-of-the-art algorithm. The improved algorithm successfully mitigates step-size blow-up, enabling a better balance between exploration and exploitation near the global optimum enhancing convergence.

• The paper introduces a reformulated PSA-CMA-ES algorithm that mitigates uncontrolled step-size blow-ups in weakly-structured multimodal functions.
• It employs selective step-size corrections based on significant population size changes to enhance convergence and computational efficiency.
• Experimental results on Rastrigin and Schaffer functions demonstrate reduced CPU time, improved accuracy, and fewer function evaluations.

Improving Population Size Adapting CMA-ES Algorithm on Step-Size Blow-Up

Introduction

The paper "Improving population size adapting CMA-ES algorithm on step-size blow-up in weakly-structured multimodal functions" addresses a critical challenge in evolutionary optimization algorithms, specifically the Covariance Matrix Adaptation Evolutionary Strategy (CMA-ES) and its variant, Population Size Adaptation (PSA-CMA-ES). While CMA-ES is efficient in exploring and exploiting well-structured multimodal benchmark problems, it encounters difficulties with weakly structured multimodal functions. The paper proposes a reformulated strategy to tackle uncontrolled step-size blow-ups that hinder convergence near global optima in PSA-CMA-ES, focusing on Rastrigin and Schaffer benchmark functions.

Problem Analysis

PSA-CMA-ES dynamically adjusts population size to maintain exploration and exploitation balance. However, for weakly structured multimodal functions, PSA-CMA-ES struggles with step-size increases that lead to uncontrolled blow-ups, preventing convergence. The paper's analytical approach identifies the core reason behind these blow-ups, emphasizing the need for a significant level of population size change to enable safe step-size corrections. Empirical evidence with weakly structured problems supports these findings, confirming inefficiencies in the current PSA-CMA-ES mechanism (Figure 1).

Figure 1: Visualization of the benchmark test functions; Rastrigin function is depicted in the upper panel (a), (b) and Schaffer function is depicted in the lower panel (c) and (d). The global optimum of each function (origin) are shown in the contour plots (b) and (d) respectively.

Reformulated Step-Size Correction

The reformulation of the step-size correction mechanism addresses the shortcomings of the existing PSA-CMA-ES algorithm. By selectively applying step-size corrections conditioned on the insignificance of population size changes, this new strategy avoids excessive step-size blow-ups. The reformulation leverages insights on the significance level derived from theoretical analysis, ensuring that corrections scale down when population size changes are below a certain threshold, effectively balancing exploration and convergence towards optima.

Experimental Evidence

Empirical results demonstrate the improved performance of the reformulated PSA-CMA-ES algorithm over the general version across Rastrigin and Schaffer benchmark functions. Through tailored experiments, the paper confirms the effectiveness of the selective correction mechanism in enhancing convergence and computational efficiency. The reformulated algorithm consistently outperforms the general algorithm in terms of CPU time, accuracy of the optimum, and average number of function evaluations, making it superior in practical applications (Figure 2).

(Figure 2)

Figure 2: Performance comparison of the Reformulation against General PSA-CMA-ES on the 2D Rastrigin and 2D Schaffer functions over 20 independent runs.

Implications and Future Directions

The reformulation offers significant implications for practical multimodal optimization, particularly in fields requiring precise convergence around global optima. Future research should focus on refining the population size scaling mechanisms within the PSA-CMA-ES framework to further enhance efficiency during convergence. Notably, such refinements should aim to optimize computational resources while maintaining the algorithm’s exploratory capabilities in complex, higher-dimensional scenarios.

Conclusion

This paper effectively addresses inherent limitations in the PSA-CMA-ES algorithm, proposing a robust reformulated strategy to mitigate step-size blow-ups. By emphasizing a thorough analytical basis for these corrections, it sets the stage for improved optimization techniques applicable to a broader range of multimodal functions. The demonstrated computational efficiency and convergence enhancements indicate promising advancements for future evolutionary algorithm research and applications in multimodal optimization domains.