Sequential maximization of expected improvement (EI) is one of the most widely used policies in Bayesian optimization because of its simplicity and ability to handle noisy observations. In particular, the improvement function often uses the best posterior mean as the best incumbent in noisy settings. However, the uncertainty associated with the incumbent solution is often neglected in many analytic EI-type methods: a closed-form acquisition function is derived in the noise-free setting, but then applied to the setting with noisy observations. To address this limitation, we propose a modification of EI that corrects its closed-form expression by incorporating the covariance information provided by the Gaussian Process (GP) model. This acquisition function specializes to the classical noise-free result, and we argue should replace that formula in Bayesian optimization software packages, tutorials, and textbooks. This enhanced acquisition provides good generality for noisy and noiseless settings. We show that our method achieves a sublinear convergence rate on the cumulative regret bound under heteroscedastic observation noise. Our empirical results demonstrate that our proposed acquisition function can outperform EI in the presence of noisy observations on benchmark functions for black-box optimization, as well as on parameter search for neural network model compression.

翻译：序贯最大化期望改进（EI）是贝叶斯优化中最广泛使用的策略之一，因其简洁性及处理噪声观测的能力。具体而言，在噪声设置下，改进函数常采用后验均值的最优值作为当前最优解。然而，许多解析型EI方法往往忽略当前最优解的不确定性：其在无噪声场景下推导出闭式采集函数，却直接应用于带噪声观测的场景。为突破这一局限，我们提出一种修正的EI方法，通过融合高斯过程（GP）模型提供的协方差信息，校正其闭式表达式。该采集函数可退化为经典无噪声情形下的结果，我们认为应以此替代贝叶斯优化软件包、教程及教材中的原始公式。这一增强型采集函数在噪声与无噪声场景下均具有良好的通用性。理论分析表明，在异方差观测噪声下，本方法对累积遗憾界可实现亚线性收敛速率。实验结果表明，在黑箱优化基准函数及神经网络模型压缩参数搜索任务中，当存在噪声观测时，本文提出的采集函数性能优于标准EI方法。