Curvature-Aware Expected Free Energy as an Acquisition Function for Bayesian Optimization

Abstract

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 demonstrate it on a system identification problem in a Van der Pol oscillator. On a two-dimensional benchmark with an oscillatory landscape, our adaptive Expected Free Energy acquisition achieves competitive performance in both regret and mean squared error, unlike standard acquisition functions that typically only perform well in one metric.