How does bayesian optimization with gaussian processes work?

Gaussian processes are a way of interpolating between data using a kernel fit, but with a covariance matrix. Think of it as kernel regression, but with confidence intervals that can be drawn at every point.

This diagram from this medium article is explains how a combination of a Gaussian process and an acquisition function works in Bayesian optimization:

An optimization problem is one that has an objective, for example, you might want to find a global minimum. Given the predictions and the confidence interval of a Gaussian process, Bayesian optimization uses an acquisition function to choose observations that would advance that objective. The acquisition function is a combination of that objective, and a balance between exploration and exploitation.

The key concepts you should grasp are:

1. Bayesian optimization balances between exploring new and uninformed areas without data, and exploiting known information from pre-existing data.
2. This continually improves a Gaussian process model, so that it makes better decisions about what to observe next.
3. All of this is to optimize for a particular objective