Calibrated and recalibrated expected improvements for Bayesian optimization

Expected improvement (EI), a function of prediction uncertainty σ(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma (\mathbf{x})$$\end{document}and improvement quantity (ξ-y^(x))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {(\xi - {{\hat{y}}}({\mathbf{x}}))}$$\end{document}, has been widely used to guide the Bayesian optimization (BO). However, the EI-based BO can get stuck in sub-optimal solutions even with a large number of samples. The previous studies attribute such sub-optimal convergence problem to the “over-exploitation” of EI. Differently, we argue that, in addition to the “over-exploitation”, EI can also get trapped in querying samples with maximum σ(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \sigma ({\mathbf{x}})$$\end{document} but poor objective function value y(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y(\mathbf{x})$$\end{document}. We call such issue as “over-exploration”, which can be a more challenging problem that leads to the sub-optimal convergence rate of BO. To address the issues of “over-exploration” and “over-exploitation” simultaneously, we propose to calibrate the incumbent ξ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\xi $$\end{document} adaptively instead of fixing it as the present best solution in the EI formulation. Furthermore, we propose two calibrated versions of EI, namely calibrated EI (CEI) and recalibrated EI (REI), which combine the calibrated incumbent ξCalibrated\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\xi ^\text{Calibrated}$$\end{document} with distance constraint to enhance the local exploitation and global exploration of promising areas, respectively. After that, we integrate EI with CEI & REI to devise a novel BO algorithm named as CR-EI. Through tests on seven benchmark functions and an engineering problem of airfoil optimization, the effectiveness of CR-EI has been well demonstrated.