  Expected Improvement (EI) is arguably the most popular acquisition function
in Bayesian optimization and has found countless successful applications, but
its performance is often exceeded by that of more recent methods. Notably, EI
and its variants, including for the parallel and multi-objective settings, are
challenging to optimize because their acquisition values vanish numerically in
many regions. This difficulty generally increases as the number of
observations, dimensionality of the search space, or the number of constraints
grow, resulting in performance that is inconsistent across the literature and
most often sub-optimal. Herein, we propose LogEI, a new family of acquisition
functions whose members either have identical or approximately equal optima as
their canonical counterparts, but are substantially easier to optimize
numerically. We demonstrate that numerical pathologies manifest themselves in
"classic" analytic EI, Expected Hypervolume Improvement (EHVI), as well as
their constrained, noisy, and parallel variants, and propose corresponding
reformulations that remedy these pathologies. Our empirical results show that
members of the LogEI family of acquisition functions substantially improve on
the optimization performance of their canonical counterparts and surprisingly,
are on par with or exceed the performance of recent state-of-the-art
acquisition functions, highlighting the understated role of numerical
optimization in the literature.
