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，因其简洁性和处理含噪观测的能力而备受青睐。特别地，改进函数在含噪场景中常将最优后验均值作为当前最优解。然而，许多解析型EI方法往往忽略与当前最优解相关的不确定性：其闭式获取函数是在无噪场景下推导的，却直接应用于含噪观测场景。为解决这一局限，我们提出了一种修正的EI方法，通过引入高斯过程（GP）模型提供的协方差信息校正其闭式表达式。该获取函数可退化为经典无噪结果，我们认为它应取代贝叶斯优化软件包、教程及教材中的现有公式。这一增强型获取函数在含噪与无噪场景中均具有良好的普适性。我们证明，在异方差观测噪声条件下，该方法在累计遗憾界上可实现次线性收敛速率。实验结果表明，在基准黑箱优化函数的含噪观测场景以及神经网络模型压缩的参数搜索任务中，我们提出的获取函数性能优于传统EI方法。