推导出对数变换后高斯过程的期望改进闭式解，提升贝叶斯优化稳定性。
Derivation of Closed Form of Expected Improvement for Gaussian Process Trained on Log-Transformed Objective
- 基于对数变换目标函数构建高斯过程，缓解数值精度敏感问题。
- 首次完整给出Hutter等人2009年提出闭式解的推导过程。
- 适合从事贝叶斯优化与高斯过程研究的读者参考。
期望改进（Expected Improvement, EI）是贝叶斯优化中最广泛使用的采集函数。然而，由于对数值精度敏感，其性能常难提升。此前，Hutter等人（2009）通过在对数变换后的目标函数上训练高斯过程，显著提高了预测准确性，从而带来性能改善。尽管该方法给出了闭式表达，但其推导过程未被公开。本文提供了该命题的友好推导，完整还原了中间步骤，为理解并复现该方法提供理论支持。
原文摘要 · Abstract (English)
Expected Improvement (EI) is arguably the most widely used acquisition function in Bayesian optimization. However, it is often challenging to enhance the performance with EI due to its sensitivity to numerical precision. Previously, Hutter et al. (2009) tackled this problem by using Gaussian process trained on the log-transformed objective function and it was reported that this trick improves the predictive accuracy of GP, leading to substantially better performance. Although Hutter et al. (2009) offered the closed form of their EI, its intermediate derivation has not been provided so far. In this paper, we give a friendly derivation of their proposition.
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