We propose an Expected Free Energy-based acquisition function for Bayesian optimization to solve the joint learning and optimization problem, i.e., optimize and learn the underlying function simultaneously. We show that, under specific assumptions, Expected Free Energy reduces to Upper Confidence Bound, Lower Confidence Bound, and Expected Information Gain. We prove that Expected Free Energy has unbiased convergence guarantees for concave functions. Using the results from these derivations, we introduce a curvature-aware update law for Expected Free Energy and show its proof of concept using a system identification problem on a Van der Pol oscillator. Through rigorous simulation experiments, we show that our adaptive Expected Free Energy-based acquisition function outperforms state-of-the-art acquisition functions with the least final simple regret and error in learning the Gaussian process.

翻译：我们提出一种基于期望自由能的采集函数用于贝叶斯优化，以解决联合学习与优化问题，即同时优化和学习潜在函数。我们证明，在特定假设下，期望自由能可退化为上置信界、下置信界和期望信息增益。我们证明期望自由能对凹函数具有无偏收敛保证。基于这些推导结果，我们引入一种感知曲率的期望自由能更新律，并通过范德波尔振荡器上的系统辨识问题展示其概念验证。通过严格的仿真实验，我们表明，基于自适应期望自由能的采集函数在最终简单遗憾值和高斯过程学习误差方面均优于最先进的采集函数。