Bayesian optimization

Bayesian optimization is a sequential design strategy for global optimization of black-box functions that does not assume any functional forms. It is usually employed to optimize expensive-to-evaluate functions. The term is generally attributed to Jonas Mockus and is coined in his work from a series of publications on global optimization in the 1970s and 1980s. Bayesian optimization is typically used on problems of the form , where is a set of points, , which rely upon less than 20 dimensions (), and whose membership can easily be evaluated. Bayesian optimization is particularly advantageous for problems where is difficult to evaluate due to its computational cost. The objective function, , is continuous and takes the form of some unknown structure, referred to as a "black box". Upon its evaluation, only is observed and its derivatives are not evaluated. Since the objective function is unknown, the Bayesian strategy is to treat it as a random function and place a prior over it. The prior captures beliefs about the behavior of the function. After gathering the function evaluations, which are treated as data, the prior is updated to form the posterior distribution over the objective function. The posterior distribution, in turn, is used to construct an acquisition function (often also referred to as infill sampling criteria) that determines the next query point. There are several methods used to define the prior/posterior distribution over the objective function. The most common two methods use Gaussian processes in a method called kriging. Another less expensive method uses the Parzen-Tree Estimator to construct two distributions for 'high' and 'low' points, and then finds the location that maximizes the expected improvement. Standard Bayesian optimization relies upon each being easy to evaluate, and problems that deviate from this assumption are known as exotic Bayesian optimization problems.