Abstract
Multi-objective Bayesian optimization (MOBO) is broadly used for applications with high cost observations such as materials discovery. In BO, a derivative-free optimization algorithm is generally employed to maximize the acquisition function. In this study, we present a method for acquisition function maximization based on a \((1 + 1)\)-evolutionary strategy in MOBO for materials discovery, which is a simple and easy-to-use approach with low computational complexity compared to conventional algorithms. In MOBO, weight vectors are used for scalarizing MO functions, typically employed to convert MO optimization into single-objective optimization. The weight vectors at each round of MOBO are generally obtained using either stochastic (random sampling) or deterministic methods based on searched results. To clarify the effect of both the scalarizing methods on MOBO, we examine the effectiveness of random sampling methods versus two deterministic methods: reference-vector-based and self-organizing map-based decomposition methods. Experimental results from four test functions and a hydrogen storage material database as a concrete application show the effectiveness of the proposed method and the random sampling method. These results implied that the proposed method was useful for real-world MOBO experiments in materials discovery.
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Notes
The database was constructed from the datasets obtained from http://hydrogenmaterialssearch.govtools.us (data last accessed on Apr., 2021).
To improve convergence of the algorithms, the initial values were adjusted as follows. For L-BFGS-B, the initial values were 0.1 for TF1 and TF2; for CMA-ES, the initial values were 0.01 for TF1. CMA-ES was implemented by referring to https://github.com/CMA-ES/c-cmaes. L-BFGS-B was used in the SciPy (1.9.3) library in Python.
For Tf2 and Tf3, there were significant differences (\( < 5 \% \) significant level) between ES and L-BFGS-B; for Tf2, there was significant difference between ES and CMA-ES; for Tf4, CMA-ES and L-BFGS-B significantly outperformed ES.
For qParEGO, qEHVI, and qNEHVI, we used BoTorch (0.8.4) in Python (3.10.9), PyTorch (2.0.0) and Cuda (11.7).
In the comparisons, we used the HVR metric since two state-of-the-art algorithms use the hypervolume.
For F3, the state-of-the-art algorithms significantly outperformed ES. However, there were no significant differences (\( > 5 \% \) significant level) between them for F1, F2, and F4.
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The author would like to thank Dr. Kazutoshi Miwa for helpful discussions about hydrogen storage materials.
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Appendices
Loss curves of ES and DIR
In Fig. 16, the shaded areas indicate the standard deviation. As F1 and F2 results, as increasing the noise level, the convergence of ES became slower.
Loss curves of ES and DIR. The blue line denotes ES result, and the red line denotes DIR result. The Tchebychev scalarization was used
Actual iteration number of ES with the Tchebychev scalarization for the test functions. The parentheses denote the noise level
Actual iteration number of ES
For the experiments of the test functions, the actual iteration number of ES (Algorithm 2) at each round is shown in Fig. 17. The solid line denotes the averaged iteration number, and the shaded areas indicate the standard deviation. In the sub-captions, the noise level is shown in the parentheses. As can be seen, the standard deviations were relatively large because \( M = 200 \).
Approximated PFs for F1. The red cure denotes PF (ground truth). DIR, orig-ES, and ES used the Tchebychev scalarization
Wall time comparison between ES and DIR
The wall time results (mean and standard deviation values) for the test functions are shown in Table 9. Here, we implemented the algorithms by the C language on a Linux machine with Intel Core i7-5830K processor (3.3GHz) and memory 64GB. From this table, the average speedup of ES was about \( 15 \% \), while the variation of the wall time of ES became larger than that of DIR.
Approximated PFs for F2. The red cure denotes PF (ground truth). DIR, orig-ES, and ES used the Tchebychev scalarization
Approximated PFs for F3. The red cure denotes PF (ground truth). DIR, orig-ES, and ES used the Tchebychev scalarization
Approximated PFs for F4. The red cure denotes PF (ground truth). DIR, orig-ES, and ES used the Tchebychev scalarization
Approximated PFs
Test functions for function optimization
See Table 10.
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Ohno, H. Empirical study of evolutionary computation-based multi-objective Bayesian optimization for materials discovery. Soft Comput 28, 8807–8834 (2024). https://doi.org/10.1007/s00500-023-09058-z
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DOI: https://doi.org/10.1007/s00500-023-09058-z