Part 1 · Background
A brief history: 1964 → 1998 → 2012
The idea is older than its fame. Kushner (1964) proposed maximizing the probability of improvement over a random-process model of an unknown curve. Mockus (1974–78) introduced expected improvement and gave the field its name. The modern template arrived with Jones, Schonlau & Welch (1998): EGO — “Efficient Global Optimization” — married a kriging (Gaussian process) surrogate to the EI criterion and showed expensive engineering objectives falling in tens of evaluations (~8k citations since). And the field went mainstream when Snoek, Larochelle & Adams (2012) pointed the machinery at machine-learning hyperparameters, beat human experts at tuning convolutional networks, and released Spearmint (~10k citations). Every modern AutoML system descends from that moment.
The tension the whole field resolves: with a handful of evaluations left, should the next one go where the model predicts the best value (exploit), or where the model knows least (explore)? Answering that question precisely requires two things: a model that knows what it doesn’t know, and a rule for converting belief plus ignorance into a decision. Those two things — the Gaussian process posterior and the acquisition function — are the next three sections.