## Abstract

Gaussian process-based Bayesian optimization (GPBO) is used to search parameters in machine learning, material design, etc. It is a method for finding optimal solutions in a search space through the following four procedures. (1) Develop a Gaussian process regression (GPR) model using observed data. (2) The GPR model is used to obtain the estimated mean and estimated variance for the search space. (3) The point where the sum of the estimated mean and the weighted estimated variance (upper confidence bound, UCB) is largest is the next search point (in the case of a maximum search). (4) Repeat the above procedures. Thus, the generalization performance of the GPR is directly related to the search performance of the GPBO. In procedure (1), the kernel parameters (KPs) of the GPR are tuned via gradient descent (GD) using the log-likelihood as the objective function. However, if the number of iterations of the GD is too high, there is a risk that the KPs will overfit the observed data. In this case, because the estimated mean and variance output by the GPR model are inappropriate, the next search point cannot be properly determined. Therefore, overtuned KPs degrade the GPBO search performance. However, this negative effect can be mitigated by changing the parameters of the GPBO. We focus on the weight of the estimated variances (exploration weight) of the UCB as one of these parameters. In a GPBO with a large exploration weight, the observed data appear in various regions in the search space. If the KP is tuned using such data, the GPR model can estimate the diverse regions somewhat correctly, even if the KP overfits the observed data, i.e., the negative effect of overtuned KPs on the GPR is mitigated by setting a larger exploration weight for the UCB. This suggests that the negative effect of overtuned KPs on the GPBO search performance may be related to the UCB exploration weight. In the present study, this hypothesis was tested using simple numerical simulations. Specifically, GPBO was applied to a simple black-box function with two optimal solutions. As parameters of GPBO, we set the number of KP iterations of GD in the range of 0–500 and the exploration weight as (Formula presented.). The number of KP iterations expresses the degree of overtuning, and the exploration weight expresses the strength of the GPBO search. The results indicate that, in the overtuned KP situation, GPBO with a larger exploration weight has better search performance. This suggests that, when searching for solutions with a small GPBO exploration weight, one must be careful about overtuning KPs. The findings of this study are useful for successful exploration with GPBO in all situations where it is used, e.g., machine learning hyperparameter tuning.

Original language | English |
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Article number | 3067 |

Journal | Mathematics |

Volume | 11 |

Issue number | 14 |

DOIs | |

Publication status | Published - Jul 2023 |

## Keywords

- Bayesian optimization
- Gaussian process
- machine learning
- overfitting