Expected Improvement

Expected Improvement is the most widely used acquisition function in Bayesian optimization, quantifying the expected reduction in the best observed objective value if the next evaluation is performed at a given point. Given a surrogate model — typically a Gaussian Process — with predictive mean μ(x) and standard deviation σ(x), and given the current best observed value f*, the Expected Improvement at point x is the expectation of max(0, f* − f(x)) under the posterior distribution. The result has a convenient closed form for Gaussian surrogates, making it computationally tractable and analytically elegant.

The function has an implicit personality: it is optimistically greedy. It samples where improvement is most probable, and it naturally vanishes as uncertainty collapses — once a region is well-understood, Expected Improvement directs search elsewhere. This automatic transition from exploration to exploitation is its primary appeal. But it is also its limitation: Expected Improvement is impatient, prioritizing probable modest gains over uncertain large ones. Alternative acquisition functions like Knowledge Gradient or Information-based methods are more patient, sacrificing immediate improvement for global information gain. The choice between them is not merely technical; it is a decision about what kind of optimizer you want to be.