Expected Hypervolume Improvement Is a Particular Hypervolume Improvement

Multi-objective Bayesian optimization (MOBO) aims to optimize multiple competing objective functions in the expensive-to-evaluate scenario. The Expected Hypervolume Improvement (EHVI) is a commonly used acquisition function for MOBO and shows a good performance. However, the computation of EHVI becomes challenging as the number of objective functions grows. In this paper, we revisit the formulation of EHVI, as well as its multi-point counterpart qEHVI, and derive much simpler analytic expressions for them. The main contributions of this paper include: (1) first formulating EHVI as a particular hypervolume improvement, and thus immediately obtaining a formal proof of its NP-hardness, faster algorithms in both theory and practice, and more results on its derivatives; (2) first obtaining the analytic expressions of qEHVI for any q > 1 and m ≥ 2 where m is the number of objectives; and (3) demonstrating the advantages of our formulation over existing exact and approximation methods for computing EHVI and qEHVI through a large number of numerical experiments.