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：预期改进（EI）的顺序最大化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。特别地，在噪声设定下，改进函数常以最优后验均值作为当前最优解。然而，许多解析型EI方法常忽略最优解对应的不确定性：尽管在无噪声设定下推导了闭式采集函数，却直接应用于带噪声观测的场景。为解决此局限，我们提出一种改进的EI方法，通过融合高斯过程（GP）模型提供的协方差信息来校正其闭式表达式。该采集函数可退化至经典无噪声结果，我们主张应以此公式替代贝叶斯优化软件包、教程及教材中的既有公式。这种增强型采集函数对噪声与无噪声场景均具良好普适性。我们证明，在异方差观测噪声下，该方法在累积遗憾界上达到亚线性收敛速率。实验结果表明，所提出的采集函数在含噪黑箱优化基准函数及神经网络模型压缩的参数搜索中，均能优于传统EI方法。